Linear FPT Reductions and Computational Lower Bounds - CiteSeerX

Linear FPT Reductions and Computational Lower Bounds Jianer Chen∗ , Xiuzhen Huang∗ , Iyad A. Kanj† , and Ge Xia∗

∗ Department

of Computer Science, Texas A&M University, College Station, TX 77843 email: {chen,xzhuang,gexia}@cs.tamu.edu † School of CTI, DePaul University, 243 S. Wabash Avenue, Chicago, IL 60604 email: [email protected]

Abstract— We introduce the concept of linear fpt-reductions and develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes WEIGHTED SATISFIABILITY, DOMINATING SET, HITTING SET , SET COVER , CLIQUE , and INDEPENDENT SET . For example, although a trivial enumeration can easily test in time O(nk ) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f (k)no(k) for any function f . Under the same assumption, for any function h(n) = O(n² ), where 0 < ² < 1, we prove that no algorithm of running time no(h(n)) can test if a graph of n vertices has a clique of size h(n). Similar strong lower bounds on the computational complexity are also derived for other NP-hard problems in the above class. Our techniques can be further extended to derive computational lower bounds on polynomial time approximation schemes for NP-hard problems. For example, we prove that the NP-hard DISTINGUISHING SUBSTRING SELECTION problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f (1/²)no(1/²) for any function f unless an unlikely collapse occurs in parameterized complexity theory.

I. I NTRODUCTION Parameterized computation is a recently proposed approach dealing with intractable computational problems. By taking the advantages of the small or moderate values of a parameter k, fixed-parameter tractable algorithms, whose running time takes the form f (k)nc where f is a recursive function2 and c is a constant, have been used to solve a variety of difficult optimization problems in practice. For example, the parameterized algorithm of running time O(1.286k + kn) [7] for VERTEX COVER has been quite practical in its applications in the research of multiple sequence alignments [6]. The rich positive toolkit of novel techniques for designing efficient and practical parameterized algorithms is accompanied in the theory by a corresponding negative toolkit that supports a theory of parameter intractability. The concept of W [1]-hardness has been introduced, and a large number of W [1]-hard parameterized problems have been identified [13]. Now it has become commonly accepted that no W [1]-hard problem can be solved in time f (k)nc for any recursive function f and any constant c (i.e., W [1] 6= FPT). Examples include a recent result by Papadimitriou and Yannakakis [22], 1 J. Chen, X. Huang, and G. Xia were supported in part by the NSF grants CCR-0000206 and CCR-0311590, and I. A. Kanj was supported in part by DePaul University Competitive Research Grant. 2 In this paper, we always assume that complexity functions are “nice” with both domain and range being non-negative integers and the values of the functions and their inverses can be easily computed.

showing that the DATABASE QUERY EVALUATION problem is W [1]-hard. This provides strong evidence that the problem cannot be solved by an algorithm whose running time is of the form f (k)nc , thus excluding the possibility of a practical algorithm for the problem even if the parameter k (the size of the query) is small as in most practical cases. Thus, the W [1]-hardness of a parameterized problem implies that any algorithm of running time O(nh ) solving the problem must have h a function of the parameter k. However, this does not completely exclude the possibility that the problem may become feasible for small values of the parameter k. For instance, if the problem is solvable by an algorithm running in time O(nlog log k ), then such an algorithm is still feasible for moderately small values of k.3 The above problem was recently tackled in [8], where it was p shown that by setting k = n/ log n, any no(k) time algorithms for a class of W [1]-hard parameterized problems, such as CLIQUE, would induce unlikely collapses in parameterized complexity theory. Thus, algorithms of uniform running time no(k) for these p problems are unlikely because of the parameter value k = n/ log n. This result, however, does not exclude the possibility of algorithms of running time no(k) for this class of problems for other values of the parameter k. Note that the complexity of p CLIQUE and other problems with parameter values other than n/ log n has been an interesting topic in research. We mention Papadimitriou and Yannakakis’s paper [21] that introduces the classes LOGNP and LOGSNP to study the complexity of a class of problems in which the parameter values are, either implicitly or explicitly, bounded by O(log n). Constructing a clique of size log n in a graph of n vertices is one of the main problems studied in [21]. Feige and Kilian [14] studied the complexity of finding a clique of size log n, and showed that if this problem can be solved in polynomial time then nondeterministic computation can be simulated by deterministic computation in subexponential time. They also showed that if a clique of size logc n can be constructed in time O(nh ), where c is a constant and h = logc−² n for some ² > 0, then nondeterministic circuits can be simulated by randomized or non-uniform deterministic circuits of subexponential size. In this paper, based on the framework of parameterized complexity theory, we develop new techniques and derive much stronger computational lower bounds for a class of well3 An immediate question that might come to mind is whether such a W [1]hard problem exists. The answer is affirmative: by re-defining the parameter, it is not difficult to construct W [1]-hard problems that are solvable in time O(nlog log k ).

known NP-hard problems. We start by proving computational lower bounds for a class of SATISFIABILITY problems, and then extend the lower bound results to other well-known NP-hard problems by introducing the concept of linear fptreductions. In particular, we prove that (A) unless W [1] = FPT, the NP-hard parameterized problems WEIGHTED CNF SAT, DOMINATING SET, HITTING SET, and SET COVER cannot be solved in time f (k)no(k) p(m) for any function f and any polynomial p, where n is the size of the search space from which the k elements are selected and m is the input instance length; and (B) unless all search problems in the syntactic class SNP introduced by Papadimitriou and Yannakakis [20] are solvable in subexponential time, the NP-hard parameterized problems WEIGHTED CNF q- SAT for any constant q ≥ 3, CLIQUE, INDEPENDENT SET, and many others cannot be solved in time f (k)mo(k) for any function f , where m is the input instance length. Note that each of the problems in class (A) (resp. class (B)) can be solved by a trivial algorithm of running time cnk m (resp. cmk+1 ), where c is an absolute constant, which simply enumerates all possible subsets of k elements in the search space. Much research has tended to seek new approaches to improve this trivial upper bound. One of the common approaches is to apply a more careful branch-and-bound search process trying to optimize the manipulation of local structures before each branch [1], [2], [7], [9], [18]. Continuously improved algorithms for these problems have been developed based on improved local structure manipulations (for example, see [25], [17], [23], [4] on the progress for the INDEPENDENT SET problem). It has even been proposed to automate the manipulation of local structures [19], [24] in order to further improve the computational time. Our results, however, provide strong evidence that the power of this approach is quite limited in principle. The lower bounds f (k)nΩ(k) p(m) and f (k)mΩ(k) for any function f and any polynomial p mentioned above indicate that no local structure manipulation running in polynomial time or in time depending only on the value k will obviate the need for exhaustive enumerations. Moreover, we prove that unless unlikely collapses occur in parameterized complexity theory, the problems in group (A) cannot be solved in time no(k) p(m) for any polynomial p, and the problems in group (B) cannot be solved in time mo(k) , even when the parameter k is restricted to be of the order of any particular function k = Θ(µ(n)), where µ(n) = O(n² ) and 0 < ² < 1. This is a very significant improvement over the results given in [8]: the results in [8] establish lower bounds because of a particular value of the parameter k, while the results in the current paper under the same assumption claim the lower bounds for every value of the parameter k. The linear fpt-reduction has also enabled us to derive lower bounds on the computational complexity of polynomial time approximation schemes (PTAS) for certain NPhard problems. We pick the DISTINGUISHING SUBSTRING SELECTION problem ( DSSP ) as an example, for which a PTAS has been developed recently [10], [11]. Gramm, Guo, and

Niedermeier [15] showed that the parameterized version of the DSSP problem is W [1]-hard, thus excluding the possibility that the DSSP problem has a PTAS whose running time takes the form f (1/²)nc , where f is an arbitrary function and c is a constant. Using the technique of linear fpt-reductions, we are able to prove a much stronger result. We first show that the DOMINATING SET problem can be linearly fpt-reduced to the DSSP problem, which thus proves that the parameterized DSSP problem is W [2]-hard (improving the result in [15]). We then show how this lower bound on parameterized complexity can be transformed into a lower bound on the computational complexity for any PTAS for the problem. More specifically, we prove that unless all search problems in SNP are solvable in subexponential time, the DSSP problem has no PTAS of running time f (1/²)no(1/²) for any recursive function f . This essentially excludes the possibility that the DSSP problem has a practically efficient PTAS even for moderate values of the error bound ². To the authors’ knowledge, this is the first time a specific lower bound has been derived on the running time of a PTAS for an NP-hard problem. We give a brief review on parameterized complexity theory. A parameterized problem Q is a subset of Ω∗ ×N , where Ω is a fixed alphabet and N is the set of all non-negative integers. Therefore, each instance of the parameterized problem Q is a pair (x, k), where the non-negative integer k is called the parameter. We say that the parameterized problem Q is fixedparameter tractable [13] if there is an algorithm that decides if an input (x, k) is a yes-instance of Q in time f (k)|x|c , where c is a fixed constant and f (k) is a recursive function. Denote by FPT the class of all fixed-parameter tractable problems. The inherent computational difficulty for solving some problems practically has led to the common belief that many parameterized problems are not fixed-parameter tractable. A hierarchy of fixed-parameter intractability, the W -hierarchy, has been introduced. At the 0-th level of the hierarchy lies the class FPT. The hardness and completeness have been defined for each level W [i] of the W -hierarchy for i > 0 [13]. It is commonly believed that W [1] 6= F P T . Thus, W [1]-hardness has served as the hypothesis for fixed-parameter intractability. For two functions f (n) and g(n), by f (n) = o(g(n)), we mean that there is a non-decreasing and unbounded function λ(n) such that f (n) ≤ g(n)/λ(n). We say that a function f (n) is “subexponential” if f (n) = 2o(n) . II. L OWER BOUNDS FOR WEIGHTED SATISFIABILITY A Πt -circuit C with n input variables is a (t + 1)-levelled circuit in which the 0-th level consists of the input gates (each is labelled either by a positive literal xi or by a negative literal xi , 1 ≤ i ≤ n), the t-th level consists of a unique output gate which is an AND gate, and all AND and OR gates in C are organized into alternating levels with edges only going from a level to the next level. The size of a Πt -circuit C is the number of gates in C. A circuit is monotone (resp. antimonotone) if all its input gates are labelled by positive literals (resp. negative literals). We say that a truth assignment τ to the input variables of a circuit C satisfies a gate g in C if τ makes the gate g 2

Finally, since f −1 is non-decreasing and unbounded,

have value 1, and that τ satisfies the circuit C if τ satisfies the output gate of C. The weight of an assignment τ is the number of variables assigned value 1 by τ . The SATISFIABILITY problem on Πt -circuits, abbreviated SAT[t], is to determine whether a given Πt -circuit C has a satisfying assignment. The parameterized problem WEIGHTED SATISFIABILITY on Πt -circuits, abbreviated WCS [t], is to determine for a given pair (C, k), where C is a Πt -circuit and k is an integer, whether C has a satisfying assignment of weight k. The WEIGHTED MONOTONE SATISFIABILITY (resp. WEIGHTED ANTIMONOTONE SATISFIABILITY) problem on Πt -circuits, abbreviated MONOTONE WCS [t] (resp. ANTI MONOTONE WCS [t]) is defined similarly to WCS[t] except that the circuit C is required to be monotone (resp. antimonotone). It is known that for each even integer t > 1, MONOTONE WCS [t] is W [t]-complete, and that for each odd integer t > 1, ANTIMONOTONE WCS [t] is W [t]-complete. To make our discussion simpler, we will denote by M - WCS[t] the problem MONOTONE WCS [t] if t is even and the problem ANTIMONOTONE WCS [t] if t is odd. The proof of the following lemma is given in the Appendix, here n and m denote the number of variables and the circuit size of a Πt -circuit, respectively. Lemma 2.1: For any integer t > 1 and any three-variable non-decreasing function φ(k, n, m), if the problem M - WCS[t] is solvable in time φ(k, n, m), then for any integer function r(n) ≤ n, the problem SAT [t] can be solved in time φ(k, n0 , m0 ) + O(m0 ), where k = dn/r(n)e, n0 ≤ 2r(n) k, and m0 ≤ 2m + 22r(n)+1 k. From Lemma 2.1, we get the following important result. Theorem 2.2: Let t > 1 be an integer. For any function f and any polynomial p, if M - WCS[t] is solvable in time f (k)no(k) p(m), then SAT [t] can be solved in time 2o(n) p0 (m) for a polynomial p0 . Proof: By Lemma 2.1, there is a non-decreasing and unbounded function λ such that the problem SAT[t] can be solved in time f (k)(n0 )k/λ(k) p(m0 ), where k = dn/r(n)e, n0 ≤ 2r(n) k, m0 ≤ 2m + 22r(n)+1 k, and r(n) ≤ n is an arbitrary integer function. Without loss of generality, we can assume that the function f is non-decreasing, unbounded, and that f (k) ≥ 2k . Define f −1 by f −1 (h) = max{q | f (q) ≤ h}. Since the function f is non-decreasing and unbounded, the function f −1 is also non-decreasing and unbounded, and satisfies f (f −1 (h)) ≤ h. From f (k) ≥ 2k , we have f −1 (h) ≤ log h. Now we let the function r(n) be r(n) = b3n/f −1 (n)c. Then k = dn/r(n)e ≤ f −1 (n) ≤ log n, and f (k) ≤ f (f −1 (n)) ≤ n. Moreover,

p(m0 )

≤ p(4m) + p(4 log n · 26n/f

therefore λ(k) ≥ λ(f −1 (n)/3) is a non-decreasing and unbounded function of n. Thus, ≤

(k2r(n) )k/λ(k) ≤ k k 2kr(n)/λ(k)

≤

k k 23kn/(λ(k)f

−1

(n))

−1

(n)

) = 2o(n) p(4m)

Therefore, the f (k)(n0 )k/λ(k) p(m0 )-time algorithm A for the problem SAT[t] is actually an algorithm of running time 2o(n) p0 (m), where p0 (m) = p(4m) is a polynomial of m, i.e., the problem SAT[t] can be solved in time 2o(n) p0 (m). This completes the proof of the theorem. The proof of Lemma 2.1 has actually shown the difficulty of constructing a satisfying assignment of a particular weight for the M - WCS[t] problem. Corollary 2.3: Let t > 1 be an integer. For any nondecreasing and unbounded function µ(n) = O(n² ), where 0 < ² < 1, if M - WCS[t] is solvable in time no(k) p(m) for a polynomial p and for the parameter values k = Θ(µ(n)), then SAT[t] is solvable in time 2o(n) p0 (m), where p0 (m) is a polynomial of m. Proof: (Sketch) In the proof of Lemma 2.1, from an instance C of SAT[t] with n input variables and size m, we constructed an instance (C 0 , k) for M - WCS[t], where k = dn/r(n)e and C 0 has n0 ≤ k2n/k input variables and size m0 ≤ 2m+22r(n)+1 k, such that C is a yes-instance of SAT[t] if and only if (C 0 , k) is a yes-instance of M - WCS[t]. By carefully selecting the function r(n), we can make k = Θ(µ(n0 )). Thus, the assumed algorithm on (C 0 , k) for this particular value of k runs in time (n0 )o(k) p(m0 ), and can be used to test if C is a yes-instance of SAT[t]. Since µ(n) = O(n² ), it can be verified that for this particular function r(n), such that k = dn/r(n)e, n0 ≤ k2n/k , and m0 ≤ 2m + 22r(n)+1 k, the function (n0 )o(k) p(m0 ) is of order 2o(n) p0 (m) for some polynomial p0 . Now we show that a subexponential time algorithm for SAT[t] would collapse the W -hierarchy. Theorem 2.4: For any t > 1, if SAT [t] is solvable in time 2o(n) p(m) for a polynomial p, then W [t − 1] = FPT. Proof: The theorem for the case t = 2 was proved in [5]. Here we present a proof for the general case t ≥ 3 using different techniques. In particular, our techniques do not apply to the case t = 2. Let C be a Πt−1 -circuit of n input variables x0 , . . ., xn−1 and size m such that C is monotone if t is odd and C is antimonotone if t is even. Without loss of generality, we assume that log n is an integer (otherwise, we add dummy input variables to the input variables of C). Let k ≤ n be a non-negative integer. We first show how to construct a Πt circuit C 0 of k log n input variables from the circuit C and the integer k such that C has a satisfying assignment of weight k if and only if C 0 is satisfiable. The input variables in C 0 are divided into k blocks B10 , . . . , Bk0 , where each block Bi0 consists of r = log n input variables zi,1 , . . . , zi,r . For a nonnegative integer j ≤ n − 1, we denote by binr (j) the lengthr binary representation of the integer j, which can also be interpreted as an assignment to a block Bi0 in the circuit C 0 . We distinguish two cases based on the parity of t.

k = dn/r(n)e ≥ n/r(n) ≥ n/(3n/f −1 (n)) = f −1 (n)/3

(n0 )k/λ(k)

≤ p(2m + 2k22r(n) ) ≤ p(4m) + p(4k22r(n) )

≤ k k 23n/λ(k) = 2o(n) 3

Case 1. t is odd. Then C is a monotone Πt−1 -circuit and all level-1 gates in C are OR gates. For each positive literal xj in C and for each block Bi0 , we make an AND gate gi,j in C 0 such that if the h-th bit in binr (j) is 1 (resp. 0) then zi,h (resp. z i,h ) is an input to gi,j . The outputs of gi,j in C 0 are identical to the outputs of xj in C. Note that for each assignment binr (j) to block Bi0 , exactly one of these new AND gates, i.e., the gate gi,j , is satisfied and outputs 1. Thus, the assignment binr (j) of block Bi0 in C 0 simulates the assignment xj = 1 in C. The circuit C 0 is obtained from the circuit C by removing all input gates in C and adding the kn new AND gates gi,j , 1 ≤ i ≤ k, 0 ≤ j ≤ n − 1, and the literals in blocks B10 , . . ., Bk0 . Moreover, we add an enforcement circuitry to C 0 to make sure that the assignments to different blocks in C 0 simulate assignments to different variables in C. To achieve this, we construct a depth-2 subcircuit Ci,i0 for each pair of blocks Bi0 and Bi00 such that Ci,i0 outputs 0 if and only if blocks Bi0 and Bi00 are assigned the same value. The output of Ci,i0 is an input to the output AND gate of the circuit C 0 . Since t ≥ 3, the enforcement circuitry does not increase the depth of the circuit C 0 . Thus, the circuit C 0 is a Πt -circuit with kr input variables. It is easy to verify that the circuit C has a satisfying assignment of weight k if and only if the circuit C 0 is satisfiable: suppose C is satisfied by a weight-k assignment τ , which assigns value 1 to k variables xj1 , . . ., xjk , and value 0 to all other variables. Then by assigning the value binr (ji ) to block Bi0 for 1 ≤ i ≤ k, we get an assignment τ 0 for the circuit C 0 such that all AND gates gi,ji in C 0 are satisfied. Since the outputs of the AND gates gi,ji are identical to the outputs of the positive literals xji , we conclude that all level-2 OR gates in C 0 corresponding to those level-1 OR gates in C satisfied by the assignment τ are satisfied by the assignment τ 0 . Since the assignment τ satisfies the circuit C and all blocks Bi0 are assigned different values, the assignment τ 0 satisfies the circuit C 0 and the circuit C 0 is satisfiable. Conversely, suppose the circuit C 0 is satisfied by an assignment τ 0 , then the restriction τi0 of τ 0 to block Bi0 satisfies exactly one AND gate gi,ji , where binr (ji ) = τi0 . Because of the enforcement circuitry, these k gates gi,ji correspond to k different positive literals xji . Thus, if we set xji = 1 for all 1 ≤ i ≤ k, and set value 0 to all other variables, we get an assignment τ of weight exactly k that satisfies the circuit C. Case 2. t is even. Then C is an antimonotone Πt−1 -circuit and all level-1 gates in C are AND gates. For each input variable xj , 0 ≤ j ≤ n − 1, and for each block Bi0 , we make an OR gate gi,j such that if the h-th bit in binr (j) is 0 (resp. 1) then zi,h (resp. z i,h ) is an input to gi,j . The outputs of gi,j in C 0 are identical to the outputs of xj in C. Note that for each assignment binr (j) of block Bi0 , exactly one of these new OR gates, i.e., the gate gi,j , is not satisfied and outputs 0. Thus, the assignment binr (j) of block Bi0 in C 0 simulates the assignment xj = 0 (or equivalently xj = 1) in C. As in Case 1, we also add an enforcement circuitry to C 0 to make sure that no two blocks in C 0 are assigned the same value. The circuit C 0 is a Πt -circuit with kr input variables.

To verify that the circuit C has a satisfying assignment of weight k if and only if the circuit C 0 is satisfiable, suppose C is satisfied by a weight-k assignment τ , which assigns value 1 to k variables xj1 , . . ., xjk , and value 0 to all other variables. Then by assigning the value binr (ji ) to block Bi0 for 1 ≤ i ≤ k, we get an assignment τ 0 to the circuit C 0 such that for each i, only the OR gate gi,ji is not satisfied and outputs 0. Thus, for each level-1 AND gate g1 satisfied by the assignment τ in C, since no negative literals xj1 , . . ., xjk are inputs to g1 in C, no gates g1,j1 , . . ., gk,jk are inputs to g1 in C 0 . Thus, the assignment τ 0 satisfies the gate g1 . Since g1 is an arbitrary level-1 AND gate satisfied by τ in C, we conclude that the assignment τ 0 satisfies all level-2 AND gates that correspond to the level-1 AND gates satisfied by the assignment τ in C. Since τ satisfies the circuit C and all blocks Bi0 are assigned different values, τ 0 satisfies the circuit C 0 and C 0 is satisfiable. Conversely, suppose the circuit C 0 is satisfied by an assignment τ 0 , then the restriction τi0 of τ 0 to block Bi0 satisfies all OR gates gi,j except the gate gi,ji , where binr (ji ) = τi0 . Because of the enforcement circuitry in C 0 , assignments τi0 and τi00 to two different blocks in C 0 are different. Thus, the assignments to the k blocks induce k different input variables xji . If we set xji = 1 for all 1 ≤ i ≤ k and set value 0 for all other input variables in C, we get an assignment τ of weight exactly k that satisfies the circuit C. In summary, we have verified that for any t ≥ 3, for a given Πt−1 -circuit C of n input variables and size m, and for a given k ≤ n, where C is monotone if t is odd and antimonotone if t is even, we can construct a Πt -circuit C 0 such that C has a satisfying assignment of weight k if and only if C 0 is satisfiable. The circuit C 0 has n0 = kr = k log n input variables and size m0 bounded by m+kn+3k 2 log2 n ≤ 3m3 , where the term kn is the number of the gates gi,j , 1 ≤ i ≤ k, 0 ≤ j ≤ n − 1 in the construction of the circuit C 0 , and 3k 2 log2 n is an upper bound on the size of the enforcement circuitry. The circuit C 0 can be constructed from (C, k) in time O(n0 + m0 ). By the hypothesis of the theorem, there is an algorithm A0 that determines whether the circuit C 0 is satisfiable in 0 time 2o(n ) p(m0 ) for a polynomial p. Thus, there is a nondecreasing and unbounded function λ such that the running 0 0 time of the algorithm A0 is bounded by 2n /λ(n ) p(m0 ). 0 This, plus the construction of the circuit C from (C, k), 0 0 gives an algorithm A00 of running time 2n /λ(n ) p(m0 ) that determines whether the Πt−1 -circuit C has a satisfying assign0 0 ment of weight k. Note that 2n /λ(n ) = 2k log n/λ(k log n) ≤ k log n/λ(log n) 2 . This gives the following algorithm A to determine if the circuit C has a weight-k satisfying assignment: For the given pair (C, k), where C is a Πt−1 -circuit of n variables and size m, if k > λ(log n), then enumerate all assignments to C and check if there is a satisfying assignment of weight k to C; if k ≤ λ(log n), then call the algorithm A00 to decide if there is a satisfying assignment of weight k to C. We analyze the algorithm A. First note that m0 ≤ 3m3 , thus, p(m0 ) is bounded by a polynomial p0 (m) of m. Define 4

λ−1 (r) = min{q | λ(q) ≥ r}. Since λ is non-decreasing and unbounded, λ−1 is also a non-decreasing and unbounded λ−1 (k) function. Let f (k) = 22 . We claim that the running time of the algorithm A is bounded by f (k)np0 (m). In effect, if k > λ(log n), we have λ−1 (k) ≥ log n, and f (k) ≥ 2n . Therefore, in this case, the running time of the algorithm A is bounded by 2n p0 (m) ≤ f (k)p0 (m). On the other hand, if k ≤ λ(log n), then the algorithm A calls the algorithm A00 to solve the problem, which runs in time 2k log n/λ(log n) ≤ 2log n = n. This concludes our theorem: in case t ≥ 3 is odd, the algorithm A can determine whether a monotone Πt−1 -circuit C has a satisfying assignment of weight k in time f (k)np0 (m) – thus the W [t−1]-complete problem MONOTONE WCS[t−1] is fixedparameter tractable and W [t−1] = FPT; while in case t ≥ 3 is even, the algorithm A can determine whether an antimonotone Πt−1 -circuit C has a satisfying assignment of weight k in time f (k)np0 (m) – thus the W [t − 1]-complete problem ANTIMONOTONE WCS[t − 1] is fixed-parameter tractable and W [t − 1] = FPT. In both cases, we get W [t − 1] = FPT. This completes the proof of the theorem. From Theorem 2.2 and Theorem 2.4, we get Theorem 2.5: For any integer t > 1, if M - WCS[t] is solvable in time f (k)no(k) p(m) for a recursive function f and a polynomial p, then W [t − 1] = FPT. From Corollary 2.3 and Theorem 2.4, we get Theorem 2.6: Let t > 1 be an integer. For any nondecreasing and unbounded function µ(n) = O(n² ), where 0 < ² < 1, if M - WCS[t] is solvable in time no(k) p(m) for a polynomial p and for the parameter values k = Θ(µ(n)), then W [t − 1] = FPT.

there is a subcollection of k subsets in F whose union is equal to U . Here the search space is F. HITTING SET: given a collection F of subsets in a universal set U , and an integer k, decide if there is a subset S of k elements in U such that S intersects every subset in F. Here the search space is U . RED / BLUE DOMINATING SET: given a graph G whose vertices are colored either red or blue, and an integer k, decide whether there is a subset B of k blue vertices such that every vertex not in B is adjacent to at least one vertex in B. Here the search space is the set of blue vertices. Many graph problems seek a subset of vertices that meet certain given conditions. For these graph problems, the natural search space is the set of all vertices. For certain problems, the polynomial time preprocessing on the input instance can significantly reduce the size of the search space. For example, for finding a vertex cover of k vertices in a graph G of n vertices, a polynomial time preprocessing can reduce the search space size to 2k [7]. In the following, we present a simple algorithm for reducing the search space size for the DOMINATING SET problem (given a graph G and an integer k, decide whether there is a subset D of k vertices such that every vertex not in D is adjacent to at least one vertex in D). Suppose we are looking for a dominating set of k vertices in a graph G of n vertices. Consider any independent set I of k + 1 vertices in G. We say I has the same neighbor set A if every vertex in I has A as its neighbor set. Obviously, if G has a dominating set of k vertices and I has the same neighbor set A, then at least one of the vertices in A must be included in the dominating set. Thus, the set A can be identified as part of the search space U . This suggests the following preprocessing on an instance (G, k) of the DOMINATING SET problem:

III. L OWER BOUNDS VIA STRONG LINEAR FPT- REDUCTIONS In the above discussion of the problems M - WCS[t], we observed that besides the parameter k and the circuit size m, the number n of input variables has played an important role in the computational complexity of the problems. Unless unlikely collapses occur in parameterized complexity theory, the problems M - WCS[t] have a tight computational bound f (k)nΘ(k) p(m), where p is any polynomial and f is any recursive function. These bounds are exponential, but in the number n of variables in the circuits, instead of in the circuit size m. Note that the circuit size m can be of the order 2n . Each instance (C, k) of a circuit satisfiability problem such as M - WCS[t] can be regarded as a search problem, in which we need to select k elements from a search space consisting of a set of n input variables, and assign them value 1 so that the circuit is satisfied. Many well-known NP-hard problems have similar formulations. We list some of them next: WEIGHTED CNF SAT: given a CNF formula F , and an integer k, decide if there is an assignment of weight k that satisfies all clauses in F . Here the search space is the set of Boolean variables in F . SET COVER: given a collection F of subsets in a universal set U , and an integer k, decide whether

DS-Core 1. i = 0; color all vertices in G WHITE; 2. while there is an independent set Ii of k + 1 WHITE vertices with the same neighbor set Ai color all vertices in Ai BLACK; color each vertex not in Ai but adjacent to a vertex in Ai RED; i = i + 1; 3. if i < k, let U be the set of all vertices in G; stop. 4. if i = k, let U = ∪k−1 i=0 Ai , stop. 5. if i > k, stop (G has no dominating set of k vertices).

It is easy to verify the following facts for the above algorithm: (1) all independent sets Ii are disjoint; (2) no two vertices in two different independent sets Ii and Ij , i < j, would share a common neighbor (otherwise, the construction of Ii would have colored a vertex in Ij RED, which in turn would have not been included in Ij ). Therefore, each vertex set Ai must contain at least one vertex in any dominating set of k vertices in the graph G, if such a dominating set exists. Thus, if there are more than k such sets Ai , we can directly conclude the nonexistence of such a dominating set; while if there are exactly k such subsets Ai , we can simply concentrate on the 5

set ∪k−1 i=0 Ai to see if it contains a dominating set of k vertices for the entire graph G. In case there are less than k such subsets Ai , the preprocessing simply does not help and we reset the search space U to be the entire vertex set of G. Finally, note that in the adjacency matrix representation of the graph G, an independent set Ii of k + 1 WHITE vertices with the same neighbor set corresponds to k+1 identical rows in the adjacency matrix, which can be easily identified in polynomial time. Thus, the algorithm DS-Core runs in polynomial time. In this section, we will concentrate on parameterized problems that seek a subset in a search space satisfying certain properties. Each instance of such a parameterized problem is associated with a triple (k, n, m), where k is the parameter, n is the size of the search space, and m is the size of the instance. We will call such an instance a (k, n, m)-instance.

if GF has a dominating set of 2k vertices. A more careful examination of the reduction reveals that in the graph GF , there are 2k disjoint independent sets Ii , i = 1, . . . , 2k, where each Ii has 2k + 1 vertices and has the same neighbor set Ai consisting of O(n2 ) vertices, such that all neighbor sets Ai are also disjoint. Moreover, besides these independent sets, no two vertices in GF share the same neighbor set. Therefore, if we apply the algorithm DS-Core to GF , the constructed search space UF will be exactly the union of these 2k neighbor sets A1 , . . ., A2k . Thus, for the graph GF , the size nF of the search space UF is O(kn2 ). This, obviously, is a fptsl -reduction from the WEIGHTED CNF SAT problem to the DOMINATING SET problem. With these fptsl -reductions, the theorem follows directly from Theorem 2.5 and Theorem 3.1. Each of the problems in Theorem 3.2 can be solved by a trivial algorithm of running time cnk m, where c is an absolute constant, which simply enumerates all possible subsets of k elements in the search space. Much research has tended to seek new approaches to improve this trivial upper bound. One of the common approaches is to apply a more careful branchand-bound search process trying to optimize the manipulation of local structures before each branch [1], [2], [7], [9], [18]. Continuously improved algorithms for these problems have been developed based on improved local structure manipulations. It has even been proposed to automate the manipulation of local structures [19], [24] in order to further improve the computational time needed to solve the problems. Theorem 3.2, however, provides strong evidence that the power of this approach is quite limited in principle. The lower bound f (k)nΩ(k) p(m) for any function f and any polynomial p in Theorem 3.2 indicates that no local structure manipulation running in polynomial time or in time depending only on the target value k will obviate the need for exhaustive enumerations. Weaker lower bounds of nΩ(k) p(m), where p is a polynomial, have been established previously for the parameterized problems p in Theorem 3.2 [8], by proving that for the case k = n/ log n, an algorithm of running time no(k) p(m) for the problems in Theorem 3.2 would imply W [1] = FPT. However, the results in [8] do not exclude the possibility that for small parameter values of k, an algorithm of running time no(k) p(m) may exist. Note that studying the complexity of NP-hard problems for small parameter values has been an interesting topic in research [14], [21]. Moreover, after all, most research in parameterized complexity theory assumes that the parameter values are small. Using Theorem 3.2, we can strengthen the lower bounds in [8] and prove that the lower bounds in [8] hold even when the parameter values are bounded by an arbitrarily small function. Theorem 3.3: Let µ(n) be a non-decreasing and unbounded function. Even if we restrict the parameter values by k ≤ µ(n), none of the problems in Theorem 3.2 can be solved in time no(k) p(m) for any polynomial p, unless W [1] = FPT. Proof: Let Q be any one of the parameterized problems in Theorem 3.2. Suppose that when k ≤ µ(n), the problem

Definition A parameterized problem Q is linearly fptreducible in the strong sense (shortly fptsl -reducible) to a parameterized problem Q0 if there exist a recursive function f , a polynomial p, and an algorithm A of running time f (k)no(k) p(m), such that on each (k, n, m)-instance x of Q, the algorithm A produces a (k 0 , n0 , m0 )-instance x0 of Q0 , where k 0 = O(k), n0 ≤ p(n), m0 ≤ p(m), and x is a yes-instance of Q if and only if x0 is a yes-instance of Q0 . The following theorem follows directly from the definition. Theorem 3.1: Suppose that a parameterized problem Q is fptsl -reducible to a parameterized problem Q0 . If Q0 is solvable in time f (k)no(k) p(m) for a recursive function f and a polynomial p, then Q is solvable in time f 0 (k)no(k) p0 (m) for a recursive function f 0 and a polynomial p0 . Using the fptsl -reduction, we can immediately derive computational lower bounds for a large group of NP-hard parameterized problems, as follows. Theorem 3.2: Unless W [1] = FPT, none of the following parameterized problems can be solved in time f (k)no(k) p(m) for any recursive function f and any polynomial p: WEIGHTED CNF SAT, SET COVER, HITTING SET, RED / BLUE DOMINATING SET, and DOMINATING SET (for DOMINATING SET, we assume the search space is identified by the algorithm DS-Core). Proof: We highlight the fptsl -reductions from the problem MONOTONE WCS[2] to these problems, which are all we need to prove the theorem. Most of these reductions are simple and standard, thus, we will only indicate the relations without providing details: (1) the MONOTONE WCS[2] problem is fptsl -reducible to the problem WEIGHTED CNF SAT; (2) the MONOTONE WCS[2] problem is fptsl -reducible to the problem HITTING SET; (3) the HITTING SET problem is fptsl -reducible to the problem SET COVER; and (4) the MONOTONE WCS[2] problem is fptsl -reducible to the problem RED / BLUE DOMI NATING SET. For DOMINATING SET, we use the reduction given in [13] from the WEIGHTED CNF SAT problem, described as follows. From a CNF formula F of n variables and m clauses, a graph GF of mF = O(n3 + m) edges is constructed such that F has a satisfying assignment of weight k if and only 6

Q is solvable by an algorithm A of running time no(k) p(m), where p is a polynomial. Define the function µ−1 as µ−1 (r) = max{q | µ(q) ≤ r}. Since µ is non-decreasing and unbounded, µ−1 is also a non−1 decreasing and unbounded function. Define f (k) = 2µ (k) . Consider the following algorithm A0 solving Q:

The class SNP introduced by Papadimitriou and Yannakakis [20] contains many well-known NP-hard problems including CNF q- SAT for any integer q ≥ 3, q- COLORABILITY, q- SET COVER, VERTEX COVER, CLIQUE, and INDEPENDENT SET [16]. It is commonly believed that it is unlikely that all problems in SNP are solvable in subexponential time4 . Impagliazzo and Paturi [16] studied the class SNP and identified a group of SNP-complete problems under the SERF-reduction, in the sense that if any of these SNP-complete problems is solvable in subexponential time, then all problems in SNP are solvable in subexponential time. Theorem 4.2: If WEIGHTED ANTIMONOTONE CNF 2- SAT is solvable in time f (k)no(k) for a function f , then all problems in SNP are solvable in subexponential time. Proof: Recall that a vertex cover of a graph G is a set C of vertices such that every edge in G has at least one end in C. Given a graph G of n vertices, we can construct an instance FG of n variables for MONOTONE CNF 2- SAT as follows: each vertex vi of G makes a positive literal vi in FG , and each edge [vi , vj ] in G makes a clause (vi , vj ) in FG . It is easy to see that the graph G has a vertex cover of k vertices if and only if FG has a satisfying assignment of weight k. From the hypothesis of the theorem, and by Lemma 4.1, WEIGHTED MONOTONE CNF 2- SAT can be solved in time 2o(n) . Therefore, VERTEX COVER can be solved in time 2o(n) . Since VERTEX COVER is SNP-complete under the SERF-reduction [16], this in consequence implies that all problems in SNP are solvable in subexponential time. A nice property WEIGHTED ANTIMONOTONE CNF 2- SAT has is that the size of its search space (i.e., the number of input variables) is polynomially related to the instance size. Thus, we can partially relax the definition of the fptsl -reduction. In the following definition, we assume k is the parameter and m is the instance size. The theorem following the definition directly follows from the definition.

For a given (k, n, m)-instance x of the problem Q, if k ≥ µ(n), then solve the problem using exhaustive enumeration in time O(2n p0 (m)), where p0 is a polynomial; and if k < µ(n), call the algorithm A to solve the problem. We claim that the algorithm A0 solves the problem Q in its general case in time f (k)no(k) (p0 (m) + p(m)). In fact, in case k ≥ µ(n), we have µ−1 (k) ≥ n therefore f (k) ≥ 2n . Thus, the running time of the algorithm A0 is bounded by O(2n p0 (m)) = O(f (k)p0 (m)) = f (k)no(k) p0 (m). On the other hand, in case k < µ(n), by the hypothesis of the theorem, the algorithm A runs in time no(k) p(m) ≤ f (k)no(k) p(m). Thus, the running time of the algorithm A0 is always bounded by f (k)no(k) (p0 (m) + p(m)), which, according to Theorem 3.2, would imply W [1] = FPT. Based on Corollary 2.3 and the fptsl -reductions in Theorem 3.2, we also derive Theorem 3.4: Unless W [t − 1] = FPT, for any nondecreasing and unbounded function µ(n) = O(n² ), where 0 < ² < 1, none of the problems in Theorem 3.2 can be solved in time no(k) p(m) even if we restrict the parameter values k to be Θ(µ(n)). IV. L OWER BOUNDS VIA LINEAR FPT- REDUCTIONS All problems in Theorem 3.2 are W [2]-complete in parameterized complexity theory. This makes it possible to fptsl reduce the W [2]-complete problem MONOTONE WCS[2] to these problems. However, similar approaches are not applicable in deriving lower bounds for W [1]-complete problems. Note that many well-known NP-hard problems are W [1]complete, including WEIGHTED CNF q- SAT for any constant q ≥ 2, CLIQUE, and INDEPENDENT SET. In this section, we present new techniques to derive computational lower bounds for some W [1]-complete problems, including the above three. The parameterized problem WEIGHTED MONOTONE CNF 2SAT (resp. WEIGHTED ANTIMONOTONE CNF 2- SAT) is: given an integer k and a CNF formula F , in which all literals are positive (resp. negative) and each clause contains at most 2 literals, determine whether there is a satisfying assignment of weight k to F . In the following, we will denote by n the number of input variables in an instance of WEIGHTED MONOTONE CNF 2- SAT or WEIGHTED ANTIMONOTONE CNF 2- SAT, which is polynomially related to the size m of the instance. The proof of the following lemma is given in the Appendix. Lemma 4.1: If the problem WEIGHTED ANTIMONOTONE CNF 2- SAT is solvable in time f (k)no(k) for some recursive function f , then the problem WEIGHTED MONOTONE CNF 2SAT can be solved in time 2o(n) .

Definition A parameterized problem Q is linearly fptreducible (shortly fptl -reducible) to a parameterized problem Q0 if there exist a recursive function f , a polynomial p, and an algorithm A of running time f (k)mo(k) , such that on each instance x of Q, the algorithm A produces an instance x0 of Q0 , where k 0 = O(k), m0 ≤ p(m), and x is a yes-instance of Q if and only if x0 is a yes-instance of Q0 . Theorem 4.3: Suppose a parameterized problem Q is fptl reducible to a parameterized problem Q0 . If Q0 is solvable in time f (k)mo(k) for a recursive function f , then Q is solvable in time f 0 (k)mo(k) for a recursive function f 0 . Using the fptl -reduction, we derive computational lower bounds for CLIQUE and other problems, as follows. Theorem 4.4: Unless all problems in SNP are solvable in subexponential time, the following parameterized problems cannot be solved in time f (k)mo(k) for any recursive function 4 A recent result showed the equivalence between the statement that all SNP problems are solvable in subexponential time, and the collapse of a parameterized class called Mini[1] to FPT [12].

7

f : WEIGHTED CNF q- SAT for any integer q ≥ 2, CLIQUE, and INDEPENDENT SET. Proof: The fptl -reductions from WEIGHTED ANTIMONO TONE CNF 2- SAT to these problems are standard. Based on the fptl -reduction, we can also derive stronger lower bounds for the problems in Theorem 3.2, only in terms of parameter k and instance size m. However, the assumption is also stronger 5 . Theorem 4.5: Unless all problems in SNP are solvable in subexponential time, the following problems cannot be solved in time f (k)mo(k) for any recursive function f : WEIGHTED CNF SAT, DOMINATING SET, SET COVER, HITTING SET, and RED / BLUE DOMINATING SET. Proof: The fptl -reductions from WEIGHTED ANTIMONO TONE CNF 2- SAT to WEIGHTED CNF SAT is straightforward. The fptl -reduction from WEIGHTED CNF SAT to DOMINATING SET has been described in the proof of Theorem 3.2 (which actually is an fptsl -reduction). Now DOMINATING SET is fptl reducible to MONOTONE WCS[2] in an obvious way. Then the fptl -reductions from MONOTONE WCS[2] to the other problems are the same as the fptsl -reductions described in Theorem 3.2.

string s. The distance D(s1 , s2 ) between two strings s1 and s2 is defined as follows. If |s1 | = |s2 |, then D(s1 , s2 ) is the Hamming distance between s1 and s2 , and if |s1 | ≤ |s2 |, then D(s1 , s2 ) is the minimum of D(s1 , s02 ) over all substrings s02 of length |s1 | in s2 . DISTINGUISHING SUBSTRING SELECTION ( DSSP): given a tuple (n, Sb , Sg , db , dg ), where n, db , and dg are integers, db ≤ dg , Sb = {b1 , . . . , bnb } is the set of (bad) strings, |bi | ≥ n, and Sg = {g1 , . . . , gng } is the set of (good) strings, |gj | = n, find a string s of length n such that D(s, bi ) ≤ db for all 1 ≤ i ≤ nb , and D(s, gj ) ≥ dg for all 1 ≤ j ≤ ng . The DSSP problem is NP-hard [10] and has important applications in biology such as in drug generic design [11]. Recently, Deng et al. [10], [11] developed a PTAS AD of 6 running time O(m(nb + ng )O(1/² ) ) for the DSSP problem, where m is the size of the instance x. Obviously, this PTAS in its current form is not practical, even for very large relative error bound such as ² = 70%. Moreover, Gramm, Guo, and Niedermeier [15] showed that the parameterized version of the DSSP problem is W [1]-hard, thus excluding the possibility that the DSSP problem has a PTAS whose running time takes the form f (1/²)mc , where c is a constant independent of 1/². However, this still does not exclude the possibility that the problem DSSP has a PTAS of running time of the form O((nb + ng )µ(1/²) p(m)), where p(m) is a polynomial of the instance size m and µ(1/²) is a small function of 1/² (in fact, this is what researchers hope for after developing the first PTAS for an NP-hard optimization problem). Using the technique of fptl -reductions, we develop in this section a much stronger computational lower bound for the DSSP problem. We first consider the parameterized version of the DSSP problem. Each instance of the parameterized DSSP problem is a pair (x, r), where x = (n, Sb , Sg , db , dg ) is an instance of the original DSSP, and r = db + dg . Lemma 5.1: The parameterized DOMINATING SET problem is fptl -reducible to the parameterized DSSP problem. Proof: For a given instance (G, k) of the parameterized DOMINATING SET problem, we show how to construct an instance (xG , r) for the parameterized DSSP problem. Suppose that the graph G has n vertices v1 , . . ., vn . Denote by vec(vi ) the binary string of length n in which all bits are 0 except the i-th bit is 1. The instance xG = (n0 , Sb , Sg , db , dg ) for DSSP is constructed as follows: n0 = n + 5, Sg consists of a single string g0 = 0n+5 , db = k − 1, and dg = k + 3 (thus the parameter r = 2k + 2). The bad string set Sb = {b1 , . . . , bn } consists of n strings, where bi corresponds to the vertex vi in G. Suppose the neighbors of the vertex vi in G are vi1 , . . ., vir , then the string bi takes the form

Entirely similar to Theorem 3.3 and Theorem 3.4, we can prove that tight lower bounds hold for the problems in Theorem 4.4 and Theorem 4.5 even for very specific parameter values. Theorem 4.6: Let µ(n) be a non-decreasing and unbounded function. Even if we restrict the parameter values by k ≤ µ(n), none of the problems in Theorem 4.4 and Theorem 4.5 can be solved in time mo(k) , unless all problems in SNP are solvable in subexponential time. Theorem 4.7: Unless all problems in SNP are solvable in subexponential time, for any non-decreasing and unbounded function µ(n) = O(n² ), where 0 < ² < 1, none of the problems in Theorem 4.4 and Theorem 4.5 can be solved in time mo(k) even if we restrict the parameter values k to be Θ(µ(n)). V. L OWER BOUNDS ON APPROXIMATION SCHEMES In this section, we show another application of the fptl reduction in deriving lower bounds on the computational complexity of polynomial time approximation schemes for NPhard problems. Developing polynomial time approximation schemes has been one of the most popular approaches for tackling NP-hard problems [3]. A polynomial time approximation scheme (PTAS) for a problem Q is an approximation algorithm for Q that for any given constant ² > 0, runs in polynomial time and produces a solution with the relative error bounded by ² (we refer the readers to [3] for more detailed definitions). We pick the DISTINGUISHING SUBSTRING SELECTION problem as an example to illustrate our techniques. Consider all strings in a fixed alphabet. Denote by |s| the length of the

vec(vi ) · 02220 · vec(vi ) · 00000 · vec(vi1 ) · 02220 · vec(vi1 ) · ·00000 · · · · · 00000 · vec(vir ) · 02220 · vec(vir ) where the dots “·” stand for string concatenations. It is easy to see that the size of xG is bounded by a polynomial of the size of the graph G.

5 It

can be shown that if W [1] = FPT then all problems in SNP are solvable in subexponential time.

8

We prove that (G, k) is a yes-instance for DOMINATING SET if and only if (xG , r) is a yes-instance for DSSP. Suppose the graph G has a dominating set H of k vertices. Let vec(H) be the binary string of length n whose h-th bit is 1 if and only if vh ∈ H. Now consider the string s = vec(H) · 02220. Clearly D(s, g0 ) = k + 3. For each bad string bi , since H is a dominating set, either vi ∈ H or a vertex vj ∈ H is a neighbor of vi . If vi ∈ H then the substring b0i = vec(vi ) · 02220 in bi satisfies D(s, b0i ) = k −1, and if a vertex vj ∈ H is a neighbor of vi , then the substring b0i = vec(vj ) · 02220 in bi satisfies D(s, b0i ) = k − 1. This verifies that D(s, bi ) = k − 1 for all 1 ≤ i ≤ n, thus, (xG , r) is a yes-instance of DSSP. Conversely, suppose (xG , r) has a solution s, which is a binary string of length n+5. Since g0 = 0n+5 and D(s, g0 ) ≥ k +3, s has at least k + 3 “non-0” bits. On the other hand, it is easy to see that each substring of length n+5 in any bad string bi contains at most 4 “non-0” bits. Since D(s, bi ) ≤ k − 1 for each bad string bi , the string s should not contain more than k + 3 “non-0” bits. Thus, the string s has exactly k + 3 “non-0” bits. Now consider any substring b0i of length n + 5 in a bad string bi such that D(s, b0i ) ≤ k − 1. The substring b0i must contain “222”: otherwise b0i has at most three “non0” bits so D(s, b0i ) ≤ k − 1 would not be possible. If the substring“222” in b0i does not match three “2”’s in s, then s has at least k “non-0” bits in other places while b0i has only one “non-0” bit in other place, so D(s, b0i ) ≤ k − 1 would not be possible. Thus, the string s must contain the substring “222”, which matches the substring “222” in b0i . Finally, observe that we can always assume that the string s ends with “02220” – otherwise we simply cyclically shift the string s to move the substring “02220” to the end. Note if D(s, b0i ) ≤ k − 1 and b0i is a substring in a segment “00000 · vec(vj ) · 02220 · vec(vj ) · 00000” in the bad string bi , then after shifting s, we must have D(s, b00i ) ≤ k − 1, where b00i = vec(vj ) · 02220. Therefore, if s is a solution to the instance (xG , r), then so is the string after the cyclic shifting. Thus, the string s can be assumed to have the form s0 ·02220, where s0 is a string of length n, with exactly k “non-0” bits. Suppose that the j1 -th, j2 -th, . . ., and jk -th bits of s0 are “non0”. We claim that the vertex set Hs = {vj1 , . . . , vjk } makes a dominating set of k vertices for the graph G. In fact, for any bad string bi , let b0i be a substring of length n + 5 in bi such that D(s, b0i ) ≤ k − 1. According to the above discussion, b0i must be of the form vec(vj ) · 02220, where either vj = vi or vj is a neighbor of vi . The only “non-0” bit in vec(vj ) is the j-th bit, and j must be among {j1 , . . . , jk } – otherwise D(vec(vj ), s0 ) is at least k + 1. Therefore, if vi = vj then vi ∈ Hs , and if vj is a neighbor of vi , then vi is adjacent to the vertex vj in Hs . This proves that Hs is a dominating set of k vertices in G, and that (G, k) is a yes-instance for DOMINATING SET. From Lemma 5.1, we derive immediately Corollary 5.2: The DSSP problem is W [2]-hard. Corollary 5.2 improves a result in [15] that the parameterized DSSP problem is W [1]-hard. The following corollary follows directly from Lemma 5.1,

Theorem 4.3, and Theorem 4.5. Corollary 5.3: Unless all problems in SNP are solvable in subexponential time, the parameterized DSSP problem cannot be solved in time f (k)mo(k) for any recursive function f . Finally, we show how to use Corollary 5.3 to derive a computational lower bound on any PTAS for the DSSP problem. According to [10], [11], an algorithm A0 is a PTAS for the DSSP problem if for a given instance x = (n, Sb , Sg , db , dg ) of DSSP and for a fixed constant ² > 0, the algorithm A0 runs in polynomial time, and in case x is a yes-instance, returns a string s of length n such that for each bad string bi ∈ Sb , D(s, bi ) ≤ db (1 + ²), and for each good string gj ∈ Sg , D(s, gj ) ≥ dg (1 − ²). Theorem 5.4: Unless all problems in SNP are solvable in subexponential time, the DSSP problem has no PTAS of running time f (1/²)mo(1/²) for any recursive function f . Proof: Suppose to the contrary that the DSSP problem has a PTAS A0 of running time f (1/²)mo(1/²) for some recursive function f . Now for an instance (x, k) of the parameterized DSSP problem, where x = (n, Sb , Sg , db , dg ), call the PTAS algorithm A0 on x and ² = 1/(2k) = 1/(2db + 2dg ). First observe that if a string s satisfies the conditions D(s, bi ) ≤ db (1 + ²) for all bad strings bi D(s, gj ) ≥ dg (1 − ²) for all good strings gj

(1) (2)

for ² = 1/(2db + 2dg ), then D(s, bi ) ≤ =

db (1 + ²) = db (1 + 1/(2db + 2dg )) db + db /(2db + 2dg )

Since both D(s, bi ) and db are integers, and db /(2db + 2dg ) < 1, we get D(s, bi ) ≤ db . Similarly, we get D(s, gj ) ≥ dg . Thus, if the algorithm A0 returns a string s satisfying the conditions (1) and (2), then (x, k) is a yes-instance for DSSP. On the other hand, if (x, k) is a yes-instance, then by our assumption, A0 must return a string s satisfying the conditions (1) and (2). Thus, (x, k) is a yes-instance if and only if the algorithm A0 returns a string s satisfying the conditions (1) and (2). By the assumption, the running time of the algorithm A0 is bounded by f (1/²)mo(1/²) = f (2k)mo(k) . This shows that the parameterized DSSP problem can be solved in time f (2k)mo(k) . Now the theorem follows from Corollary 5.3. Compared to the result in [15], Theorem 5.4 is much stronger. The result in [15] proved that if the DSSP problem has a PTAS of running time f (1/²)mµ(1/²) , then µ cannot be a constant function, while Theorem 5.4 shows that µ cannot even be any sublinear function of 1/². This essentially excludes the possibility that the DSSP problem has a practically efficient PTAS even for moderate values of the error bound ². To the authors’ knowledge, this is the first time a specific lower bound has been derived on the running time of a PTAS for an NPhard problem. Theorem 5.4 also demonstrates the usefulness of the linear fpt-reduction. In most cases, computational lower bounds and inapproximability of optimization problems are derived based on approximation ratio-preserving reductions [3], by which if 9

a problem Q1 is reduced to another problem Q2 , then Q2 is at least as hard as Q1 . In particular, if Q1 is reduced to Q2 under a approximation ratio-preserving reduction, then the approximability of Q2 is at least as difficult as that of Q1 . Therefore, the intractability results for an “easier” problem in general cannot be derived using such a reduction from a “harder” problem. On the other hand, our computational lower bound on the DSSP problem was obtained by a reduction from the DOMINATING SET problem. It is well-known that DOMI NATING SET has no polynomial time approximation algorithms of a constant ratio [3], while the DSSP problem has PTAS. Thus, from the viewpoint of approximability, DOMINATING SET is much harder than DSSP, and our linear fpt-reduction reduces a harder problem to an easier problem. This hints that our approach for deriving computational lower bounds and the linear fpt-reduction cannot be simply replaced by the standard approaches based on approximation ratio-preserving reductions.

[4] R. B EIGEL, Finding maximum independent sets in sparse and general graphs, Proc. 10th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA’99), pp. 856-857, (1999). [5] L. C AI AND D. J UEDES, On the existence of subexponential parameterized algorithms, Journal of Computer and System Sciences, to appear. [6] J. C HEETHAM , F. D EHNE , A. R AU -C HAPLIN , U. S TEGE , AND P. TAILLON, Solving large FPT problems on coarse grained parallel machines, Journal of Computer and System Sciences, to appear. [7] J. C HEN , I. A. K ANJ , AND W. J IA, Vertex cover: further observations and further improvements, Journal of Algorithms 41, pp. 280-301, (2001). [8] J. C HEN , B. C HOR , M. F ELLOWS , X. H UANG , D. J UEDES , I. K ANJ , AND G. X IA , Tight lower bounds for parameterized NP-hard problems, Submitted, (2003). [9] D. C OPPERSMITH AND S. W INOGRAD, Matrix multiplication via arithmetic progression, Journal of Symbolic Computation 9, pp. 251280, (1990). [10] X. D ENG , G. L I , Z. L I , B. M A , AND L. WANG, A PTAS for distinguishing (sub)string selection, Lecture Notes in Computer Science 2380 (ICALP’02), pp. 740-751, (2002). [11] X. D ENG , G. L I , Z. L I , B. M A , AND L. WANG, Genetic design of drugs without side-effects, SIAM J. Comput. 32, pp. 1073-1090, (2003). [12] R. D OWNEY, V. E STIVILL -C ASTRO , M. F ELLOWS , E. P RIETO RODRIGUEZ , AND F. ROSAMOND, Cutting up is hard to do: the parameterized complexity of k-cut and related problems, Electronic Notes in Theoretical Computer Science 78, pp. 205-218, (2003). [13] R.G. D OWNEY AND M.R. F ELLOWS, Parameterized Complexity, Springer-Verlag, 1999. [14] U. F EIGE AND J. K ILIAN, On limited versus polynomial nondeterminism, Chicago Journal of Theoretical Computer Science, (1997). [15] J. G RAMM , J. G UO , AND R. N IEDERMEIER, On exact and approximation algorithms for distinguishing substring selection, Lecture Notes in Computer Science 2751 (FCT’2003), pp. 195-209, (2003). [16] R. I MPAGLIAZZO AND R. PATURI, Which problems have strongly exponential complexity? Journal of Computer and System Sciences 63, pp. 512-530, (2001). [17] T. J IAN, An O(20.304n ) algorithm for solving maximum independent set problem, IEEE Trans. Comput. 35, pp. 847-851, (1986). ˘ AND S. P OLJAK, On the complexity of the subgraph [18] J. N E S˘ ET RIL problem, Commentationes Mathematicae Universitatis Carolinae 26 (2), pp. 415-419, (1985). [19] R. N IEDERMEIER AND P. ROSSMANITH, Upper bounds for vertex cover further improved, Lecture Notes in Computer Science 1563 (STACS’99), pp. 561-570, (1999). [20] C. H. PAPADIMITRIOU AND M. YANNAKAKIS, Optimization, approximation, and complexity classes, Journal of Computer and System Sciences 43, pp. 425-440, (1991). [21] C. H. PAPADIMITRIOU AND M. YANNAKAKIS, On limited nondeterminism and the complexity of VC dimension, Journal of Computer and System Sciences 53, pp. 161-170, (1996). [22] C. H. PAPADIMITRIOU AND M. YANNAKAKIS, On the complexity of database queries, Journal of Computer and System Sciences 58, pp. 407427, (1999). [23] J. M. ROBSON, Algorithms for maximum independent sets, J. Algorithms 7, pp. 425-440, (1986). [24] J. M. ROBSON, Finding a maximum independent set in time O(2n/4 )? LaBRI, Universite BordeauxI, 1251-01, (2001). [25] R. E. TARJAN AND A. E. T ROJANOWSKI, Finding a maximum independent set, SIAM J. Comput. 6, pp. 537-546, (1977).

VI. C ONCLUSION In this paper, based on parameterized complexity theory, we developed new techniques for deriving computational lower bounds for well-known NP-hard problems. We started by establishing the computational lower bounds for the generic parameterized problems WCS[t] for t > 1 and WEIGHTED ANTIMONOTONE CNF 2- SAT. We showed that for any integer t > 1, an f (k)no(k) p(m)-time algorithm for WCS[t] for any recursive function f and any polynomial p would collapse the (t − 1)st level W [t − 1] to the bottom level FPT in the fixed-parameter intractability hierarchy, the W-hierarchy, and that an f (k)mo(k) -time algorithm for WEIGHTED AN TIMONOTONE CNF 2- SAT would imply subexponential time algorithms for all problems in SNP. Based on these generic results, we introduced the concept of linear fpt-reductions, and used it to derive tight computational lower bounds for many well-known NP-hard problems. Obviously, the list of the problems we have studied is far from being exhaustive. This new technique should serve as a very powerful tool for deriving strong computational lower bounds for other intractable problems. Moreover, we also demonstrated how our techniques can be used to derive strong computational lower bounds on polynomial time approximation schemes for NPhard problems. This seems to open a new direction for the study of computational lower bounds on the approximability of NP-hard optimization problems. R EFERENCES [1] K. A. A BRAHAMSON , R. G. D OWNEY, AND M. R. F ELLOWS, Fixedparameter tractability and completeness IV: on completeness for W [P ] and PSPACE analogs, Annals of Pure and Applied Logic 73, pp. 235276, (1995). [2] J. A LBER , H. L. B ODLAENDER , H. F ERNAU , T. K LOKS , R. N IEDER MEIER , Fixed parameter algorithms for dominating set and related problems on planar graphs, Algorithmica 33, pp. 461-493, (2002). [3] G. AUSIELLO , P. C RESCENZI , G. G AMBOSI , V. K ANN , A. M ARCHETTI -S PACCAMELA , AND M. P ROTASI, Complexity and Approximation – Combinatorial Optimization problems and their Approximability Properties, Springer, 1999.

10

the integer such that bini (ji ) = τi . Then according to the construction of the circuit C 0 , by setting zi,ji = 1 and all other variables in Bi0 to 0, we can satisfy all level-1 OR gates in C 0 whose corresponding level-1 OR gates in C are satisfied by the assignment τi . Doing this for all blocks Bi , 1 ≤ i ≤ k, gives a weight-k assignment τ 0 to the circuit C 0 that satisfies all level-1 OR gates in C 0 whose corresponding level-1 OR gates in C are satisfied by τ . Since τ satisfies the circuit C, the weight-k assignment τ 0 satisfies the circuit C 0 . Conversely, suppose that the circuit C 0 is satisfied by a weight-k assignment τ 0 . Because of the enforcement circuitry in C 0 , τ 0 assigns value 1 to exactly one variable in each block Bi0 . Now suppose that in block Bi0 , τ 0 assigns value 1 to the variable zi,ji . Then we set an assignment τi to the block Bi in C such that τi = bini (ji ). By the construction of the circuit C 0 , the level-1 OR gates satisfied by the variable zi,ji = 1 are all satisfied by the assignment τi . Therefore, if we make an assignment τ to the circuit C such that the restriction of τ to block Bi is τi for all i, then the assignment τ will satisfy all level-1 OR gates in C whose corresponding level-1 OR gates in C 0 are satisfied by τ 0 . Since τ 0 satisfies the circuit C 0 , we conclude that the circuit C is satisfiable. This completes the proof that when t is even, the circuit C is a yes-instance for SAT[t] if and only if the constructed pair (C 0 , k) is a yes-instance of M - WCS[t]. Case 2. t is odd. Then all level-1 gates in the Πt -circuit C are AND gates. Let xq be an input variable in C and be the h-th variable in block Bi . If the positive literal xq is an input to a level-1 AND gate g1 in C, then all negative literals z i,j in block Bi0 such that the h-th bit in bini (j) is 0 are inputs to gate g1 in C 0 . If the negative literal xq is an input to a level-1 AND gate g2 in C, then all negative literals z i,j in block Bi0 such that the h-th bit in bini (j) is 1 are inputs to gate g2 in C 0. An enforcement circuitry is added to C 0 to ensure that every satisfying assignment to C 0 assigns value 1 to at most one variable in each block Bi0 . This can be achieved as follows. For every two negative literals z i,j and z i,h in Bi , add an OR gate gj,h . Connect z i,j and z i,h to gi,h and connect gi,h to the output AND gate of C 0 . This completes the construction of the circuit C 0 . The circuit C 0 is an antimonotone Πt -circuit (again the enforcement circuitry does not increase the depth of C 0 ). Thus, (C 0 , k) is an instance of the problem ANTIMONOTONE WCS[t]. We verify that the circuit C is satisfiable if and only if the circuit C 0 has a satisfying assignment of weight k. Suppose that the circuit C is satisfied by an assignment τ . Let τi be the restriction of τ to block Bi , 1 ≤ i ≤ k. Let ji be the integer such that bini (ji ) = τi . Consider the weight-k assignment τ 0 to C 0 that for each i assigns zi,ji = 1 and all other variables in Bi0 to 0. We show that τ 0 satisfies the circuit C 0 . Let g1 be a level-1 AND gate in C that is satisfied by the assignment τ . Since C 0 is antimonotone, all inputs to g1 are negative literals. Since all negative literals except z i,ji in block Bi0 have value 1, we only have to prove that no z i,ji from any block Bi0 is an input to g1 . Assume to the contrary that z i,ji in block

A PPENDIX . P ROOFS FOR L EMMAS 2.1 AND 4.1 Lemma 2.1: Denote by n the number of variables and m the size of a Πt -circuit. Let t > 1 be an integer. For any threevariable non-decreasing function φ(k, n, m) if the problem M WCS [t] is solvable in time φ(k, n, m), then for any integer function r(n) ≤ n, the problem SAT [t] can be solved in time φ(k, n0 , m0 ) + O(m0 ), where k = dn/r(n)e, n0 ≤ 2r(n) k, and m0 ≤ 2m + 22r(n)+1 k. Proof: We first show how to construct, from a given instance C of SAT[t], an instance (C 0 , k) for MONOTONE WCS [t] when t is even, and for ANTIMONOTONE WCS[t] when t is odd, such that C is a yes-instance of SAT[t] if and only if (C 0 , k) is a yes-instance of M - WCS[t]. Let C be a Πt -circuit, and x1 , x2 , . . . , xn the input variables to C. Let k = dn/r(n)e, where r(n) is the given integer function such that r(n) ≤ n. We divide the n input variables x1 , . . . , xn into k blocks B1 , . . . , Bk , where block Bi consists of input variables x(i−1)r(n)+1 , . . . , xir(n) , for i = 1, . . . , k − 1, and block Bk consists of input variables x(k−1)r(n)+1 , . . . , xn . Denote by the size |Bi | of block Bi the number of variables in Bi . Note |Bi | = r(n), for 1 ≤ i ≤ k − 1, and |Bk | = r0 ≤ r(n). For a non-negative integer j, we will denote by bini (j) the length-|Bi | binary representation of j, which can also be interpreted as an assignment to the variables in block Bi . We construct a new set of input variables that consists of k blocks B10 , . . . , Bk0 , where for i = 1, . . . , k − 1, block Bi0 consists of s = 2r(n) variables zi,0 , zi,1 , . . ., zi,s−1 , and block 0 Bk0 consists of s0 = 2r variables zk,0 , zk,1 , . . ., zk,s0 −1 . Now the Πt -circuit C 0 is constructed from the Πt -circuit C by replacing the input gates in C by the new input variables in blocks B10 , . . . , Bk0 . We consider two different cases. Case 1. t is even. Then all level-1 gates in the Πt -circuit C are OR gates. Let xq be an input variable in C. Suppose that xq is the h-th variable in block Bi . If the positive literal xq is an input to a level-1 OR gate g1 in C, then all positive literals zi,j in block Bi0 such that the h-th bit in bini (j) is 1 are connected to gate g1 in the circuit C 0 . If the negative literal xq is an input to a level-1 OR gate g2 in C, then all positive literals zi,j in block Bi0 such that the h-th bit in bini (j) is 0 are connected to gate g2 in the circuit C 0 . We also add an “enforcement” circuitry to the circuit C 0 to ensure that every satisfying assignment to C 0 assigns value 1 to at least one variable in each block Bi0 . This can be achieved by adding an OR gate, for each i = 1, . . ., k, whose inputs are connected to all positive literals in block Bi0 and whose output is an input to the output gate of the circuit C 0 . This completes the construction of the circuit C 0 . It is easy to see that the circuit C 0 is a monotone Πt -circuit (note that t > 1 and hence the enforcement circuitry does not increase the depth of the circuit C 0 ). Thus, (C 0 , k) is an instance of MONOTONE WCS[t]. We verify that the circuit C is satisfiable if and only if the circuit C 0 has a satisfying assignment of weight k. Suppose that the circuit C is satisfied by an assignment τ . Let τi be the restriction of τ to block Bi , 1 ≤ i ≤ k. Let ji be 1

Bi0 is an input to g1 . Then by the construction of the circuit C 0 , there is a variable xq that is the h-th variable in block Bi such that either xq is an input to g1 in C and the h-th bit of bini (ji ) is 0, or xq is an input to g1 in C and the hth bit of bini (ji ) is 1. However, by our construction of the index ji from the assignment τ , if the h-th bit of bini (ji ) is 0 then τ assigns xq = 0, and if the h-th bit of bini (ji ) is 1 then τ assigns xq = 1. In either case, τ would not satisfy the gate g1 , contradicting our assumption. Thus, for all i, no z i,ji is an input to the gate g1 , and the assignment τ 0 satisfies the gate g1 . Since g1 is an arbitrary level-1 AND gate in C 0 , we conclude that the assignment τ 0 satisfies all level-1 AND gates in C 0 whose corresponding gates in C are satisfied by the assignment τ . Since τ satisfies the circuit C, the weight-k assignment τ 0 satisfies the circuit C 0 . Conversely, suppose that the circuit C 0 is satisfied by a weight-k assignment τ 0 . Because of the enforcement circuitry in C 0 , the assignment τ 0 assigns value 1 to exactly one variable in each block Bi0 . Suppose that in block Bi0 , τ 0 assigns value 1 to the variable zi,ji . Then we set an assignment τi = bini (ji ) to block Bi in C. Let τ be the truth assignment whose restriction on block Bi0 is τi . We prove that τ satisfies the circuit C. In effect, if a level-1 AND gate g2 in C 0 is satisfied by the assignment τ 0 , then no negative literal z i,ji is an input to g2 . Suppose that g2 is not satisfied by τ in C, then either a positive literal xq is an input to g2 and τ assigns xq = 0, or a negative literal xq is an input to g2 and τ assigns xq = 1. Let xq be the h-th variable in block Bi . If τ assigns xq = 0 then the h-th bit in bini (ji ) is 0. Thus, xq cannot be an input to g2 in C because otherwise by our construction the negative literal z i,ji would be an input to g2 in C 0 . On the other hand, if τ assigns xq = 1 then the h-th bit in bini (ji ) is 1, thus, xq cannot be an input to g2 in C because otherwise the negative literal z i,ji would be an input to g2 in C 0 . This contradiction shows that the gate g2 must be satisfied by the assignment τ . Since g2 is an arbitrary level-1 AND gate in C 0 , we conclude that the assignment τ satisfies all level-1 AND gates in C whose corresponding level-1 AND gates in C 0 are satisfied by the assignment τ 0 . Since τ 0 satisfies the circuit C 0 , the assignment τ satisfies the circuit C. This completes the proof that when t is odd, the Πt -circuit C is a yes-instance of SAT[t] if and only if the pair (C 0 , k) is a yes-instance of M - WCS[t]. Summarizing the above discussion, we conclude that for any t > 1, from an instance C of the problem SAT[t], which is a Πt -circuit of n input variables and size m, we can construct an instance (C 0 , k) of the problem M - WCS[t] such that C is a yes-instance of SAT[t] if and only if (C 0 , k) is a yes-instance of M - WCS[t]. Here k = dn/r(n)e, C 0 is a Πt -circuit with n0 input variables and size m0 , with n0 ≤ k2r(n) , and m0 ≤ m + n0 + k + k22r(n) ≤ 2m + k22r(n)+1 (where the terms k + k22r(n) is an upper bound on the size of the enforcement circuitry). Moreover, the circuit C 0 can be constructed from the circuit C in time O(n0 + m0 ) = O(m0 ). By the hypothesis of the theorem, there is an algorithm of running time φ(k, n0 , m0 ) that determines if (C 0 , k) is a yes-

instance of the M - WCS[t] problem. This, plus the construction of (C 0 , k) from the Πt -circuit C gives an algorithm A of running time φ(k, n0 , m0 ) + O(m0 ) that solves the problem SAT[t]. This completes the proof of the lemma. Lemma 4.2: If the problem WEIGHTED ANTIMONOTONE CNF 2- SAT is solvable in time f (k)no(k) for some recursive function f , then the problem WEIGHTED MONOTONE CNF 2SAT can be solved in time 2o(n) . Proof: Suppose that WEIGHTED ANTIMONOTONE CNF 2- SAT can be solved by an algorithm A0 of running time f (k)nk/λ(k) , where f is a recursive function and λ is a non– decreasing and unbounded function. Without loss of generality, we also assume that f is non-decreasing, unbounded, and satisfies f (k) > 2k . Define f −1 (h) = max{q | f (q) ≤ h}. Then f −1 is a non-decreasing and unbounded function satisfying f −1 (h) ≤ log h and f (f −1 (h)) ≤ h. We first show how to construct a group of instances for WEIGHTED ANTIMONOTONE CNF 2- SAT from an instance of WEIGHTED MONOTONE CNF 2- SAT. Let (F, k) be an instance of WEIGHTED MONOTONE CNF 2- SAT, and let n be the number of Boolean variables in the formula F . Divide the n Boolean variables in F into r = bf −1 (n)c subsets B1 , . . ., Br , each containing at most s = dn/re variables. Let π be a partition of the parameter k into r integers k1 , . . ., kr , where 0 ≤ ki ≤ s and k = k1 + · · · , kr . We say that a satisfying assignment τ of weight k for F is under the partition π if τ assigns value 1 to exactly ki variables in the set Bi for all i. Obviously, F has a satisfying assignment of weight k if and only if F has a satisfying assignment of weight k under a partition of the parameter k. Fix a partition π of the parameter k: k = k1 + · · · + kr . We construct an instance (Fπ , r) for WEIGHTED ANTIMONOTONE CNF 2- SAT as follows. For each subset Bi,j of ki variables in the set Bi , if for each clause (xq , x0q ) in F where both xq and x0q are in Bi , at least one of xq and x0q is in Bi,j , then make the subset Bi,j a Boolean variable in Fπ . For each pair of Boolean variables Bi,j and Bi,q in Fπ from the same set Bi in F , add a clause (Bi,j , Bi,q ) to Fπ . For each pair of Boolean variables Bi,j and Bh,q in Fπ from two different sets Bi and Bh in F , if there exist a variable xq ∈ Bi and a variable x0q ∈ Bh such that xq 6∈ Bi,j , x0q 6∈ Bh,q but (xq , x0q ) is a clause in F , add a clause (Bi,j , Bh,q ) to Fπ . This completes the construction of the CNF formula Fπ . Obviously, (Fπ , r) is an instance of the WEIGHTED ANTIMONOTONE CNF 2- SAT problem. The number nπ of Boolean variables in Fπ is bounded by r2s . We verify that the CNF formula F has a satisfying assignment of weight k under the partition π if and only if the CNF formula Fπ has a satisfying assignment of weight r. Let τ be a satisfying assignment of weight k under the partition π for F . Let C be the set of variables in F that are assigned value 1 by τ , and Ci = C ∩ Bi . Then Ci has ki variables. Note that for any clause (xq , x0q ) in F such that both xq and x0q are in Bi , at least one of xq and x0q is in Ci – otherwise the clause (xq , x0q ) would not be satisfied by the assignment τ . Thus, each subset Ci is a Boolean variable in Fπ . Now in the CNF formula Fπ , by assigning value 1 to all Ci , 1 ≤ i ≤ r, and 2

value 0 to all other Boolean variables, we get an assignment τπ of weight r for Fπ . For each clause of the form (Bi,j , Bi,q ) in Fπ , where Bi,j and Bi,q are from the same set Bi , since only one Boolean variable in Fπ from the set Bi is assigned value 1 by τπ , the clause is satisfied by the assignment τπ . For the two Boolean variables Ci and Ch , i 6= h, which both get assigned the value 1 by the assignment τπ , since no variable in C −(Ci ∪Ch ) is in Bi ∪Bh , each clause (xq , x0q ) in F such that xq ∈ Bi and x0q ∈ Bh must have either xq ∈ Ci or x0q ∈ Ch (otherwise the clause (xq , x0q ) would not be satisfied by τ ). Thus, (Ci , Ch ) is not a clause in Fπ , and in consequence, the clauses of the form (Bi,j , Bh,q ) in Fπ , i 6= h, where Bi,j and Bh,q are from different sets Bi and Bh , are also all satisfied by the assignment τπ . This shows that Fπ is satisfied by the assignment τπ of weight r. Conversely, let τπ be a satisfying assignment of weight r for Fπ . Because (Bi,j , Bi,q ) is a clause in Fπ for each pair Bi,j and Bi,q of Boolean variables from the same set Bi in F , at most one Boolean variable Bi,j from each set Bi , i = 1, . . . , r, can be assigned value 1 by the assignment τπ . Since the weight of τπ is r, we conclude that exactly one Boolean variable Bi,ji from each set Bi is assigned value 1 by τπ . Each Bi,ji of these subsets contains exactly ki variables in Bi . Let C = ∪ri=1 Bi,ji , then C has exactly k variables in F . If we assign all variables in C value 1 and all other variables value 0, we get an assignment τ of weight k for the formula F . We show that τ is a satisfying assignment for F . For each clause (xq , x0q ) in F such that both xq and x0q are in the same set Bi , by the construction of the variables in Fπ , at least one of xq and x0q is in Bi,ji , and hence in C. Thus, all clauses of the form (xq , x0q ) where both xq and x0q are in Bi are satisfied by the assignment τ . For each clause (xq , x0q ) in F such that xq ∈ Bi and x0q ∈ Bh , i 6= h, because (Bi,ji , Bh,jh ) is not a clause in Fπ (otherwise, τπ would not satisfy Fπ ), we must have either xq ∈ Bi,ji or x0q ∈ Bh,jh , i.e., at least one of xq and x0q must be in C. It follows that the clause (xq , x0q ) is again satisfied by the assignment τ . This proves that the assignment τ is a satisfying assignment for the formula F . By the hypothesis of the lemma, the algorithm A0 can determine if (Fπ , r) is a yes-instance for the WEIGHTED r/λ(r) . ANTIMONOTONE CNF 2- SAT problem in time f (r)nπ −1 Now f (r) ≤ f (f (n)) ≤ n, and nr/λ(r) π

This suggests the following algorithm to solve the WEIGHTED MONOTONE CNF 2- SAT problem: Given an instance (F, k) of WEIGHTED MONOTONE CNF 2- SAT, for each possible partition π of the parameter k: k = k1 + · · · + kr , where 0 ≤ ki ≤ s are integers, construct the instance (Fπ , r) for WEIGHTED ANTIMONOTONE CNF 2- SAT, and call the algorithm A0 on (Fπ , r) to determine whether the formula F has a satisfying assignment of weight k under the partition π. This algorithm runs in time 2o(n) and solves the WEIGHTED MONOTONE CNF 2- SAT problem.

≤ (r2s )r/λ(r) ≤ (2r2n/r )r/λ(r) = (2r)r/λ(r) 2n/λ(r) = (2r)r/λ(r) 2n/λ(bf

−1

(n)c)

= 2o(n)

here we have used the facts r = bf −1 (n)c ≤ log n, and hence (2r)r/λ(r) = 2o(n) , and λ(bf −1 (n)c) is a non-decreasing and unbounded function. Also, it is easy to see that the instance (Fπ , r) can be constructed from the instance (F, k) in time O(n2π ) = O((r2s )2 ) = 2o(n) . Therefore, the algorithm A0 , plus the construction of (Fπ , r) from (F, k), can be used to test in time 2o(n) whether the formula F has a satisfying assignment of weight k under the partition π. Note that the total number of partitions of the parameter k into r integers ki , 0 ≤ ki ≤ s, such that k = k1 + · · · + kr is bounded by sr = (dn/re)r ≤ (n + 1)log n = 2o(n) . 3