Maximum Weighted Matching with Interference Constraints Gaurav Sharma, Ness B. Shroff Center for Wireless Systems and Applications School of Electrical & Computer Engineering Purdue University West Lafayette, IN 47907, USA {gsharma, shroff}@purdue.edu Ravi R. Mazumdar Dept. of Electrical & Computer Engineering University of Waterloo Waterloo, ON N2L3G1
[email protected]
Abstract In this paper, we study the problem of utility maximization in multi-hop wireless systems. To study the effect of wireless interference constraints on the utility maximization problem, we introduce a new class of weighted matching problems, namely Maximum Weighted K-Valid Matching problems (MWKVMPs). For K = 1, MWKVMP corresponds to the well studied Maximum Weighted Matching problem (MWMP) in the literature, which can be solved in polynomial time. We prove several interesting results concerning the hardness of these problems for K ≥ 2. In particular, we show that MWKVMP does not even belong to APX; where APX denotes the class of problems to which a constant factor approximation can be obtained in polynomial time. Our results have strong implications concerning the hardness of scheduling transmissions in multi-hop wireless systems.
1. Introduction In this paper, we study the problem of “utility” maximization in multi-hop wireless systems. More precisely, consider a set of N wireless nodes, communicating over a set L of L bi-directional wireless links. In general, depending on the placement of the stations and the peak power constraints at the stations, L can be much smaller than N (N − 1)/2. Let us say that when a set M of links
is activated! at a given time, one obtains an aggregate “utility” of e∈M UM (e). The goal is to choose the set of links M so as to maximize the aggregate utility. Note that the utility here is a general quantity. The utility UM (e) could be a function of the data rates that link e can support, or it could be a function of the queue lengths at link e. Such choices are motivated by the goal of providing throughput guarantees in multi-hop wireless systems (see, for example, [8, 2, 11]). Note the dependence of UM (e) over the set of chosen links M . This dependence arises because of mutual interference between the set of wireless links that are activated at the same time. In general, this dependence can be quite complex. To make the above utility maximization problem analytically tractable, several simplistic interference models have been considered in the literature. The most widely studied interference model, perhaps due to its simplicity, is the nodeexclusive interference model (see [7, 1, 3, 8, 2]). In the node-exclusive interference model, the data rate that can be supported over a link (or the utility obtained by scheduling a link) is assumed to be fixed (in both directions). The only constraint is in terms of the choice of the set M , namely, the set M must be a “matching”. Consider an undirected graph G = (V, E), where V denotes the set of vertices and E denotes the set of edges. A matching is a subset M of E such that e1 ∩ e2 = φ for all e1 , e2 ∈ M with e1 %= e2 . Thus, in the case of node-exclusive interference model, the general utility maximization problem corresponds to the
Maximum Weighted Matching problem (MWMP), defined as follows: Let w : E → R be a “weight” function. For any subset M of E!define the weight function w(M ) of M by w(M ) = e∈M w(e). The MWMP consists of finding a matching of maximum weight. The maximum weighted matching problem (MWMP) can be solved in polynomial time [5]. The unweighted version of MWMP is the classical maximum matching problem (MMP). Since the only constraint imposed by the nodeexclusive interference model is that the chosen set of links M must constitute a matching, even those nodes that are within one hop of a chosen transmitter-receiver pair can potentially transmit. Such an interference model is appropriate only for a limited class of multihop wireless systems (e.g., Bluetooth networks [9] and FH-CDMA networks [1, 7]) in which nodes that are within two hops from each other can transmit over different channels or use orthogonal codes to limit interference. Whereas, in a majority of multi-hop systems, all nodes share the common spectrum, and the codes, if at all they are used, are not completely orthogonal. In such scenarios, under the constraints imposed by the nodeexclusive interference model, the data rate that can be supported at the chosen transmitter-receiver pairs would likely be small due to the interference from other simultaneous transmissions in the system. One possible way of reducing the interference is to prevent the nodes within the K-hop neighborhood of a chosen transmitter-receiver pair from transmitting or receiving, where K ≥ 1 is an integer. This motivates one to study MWKVMP and MKVMP, defined as follows: For v1 , v2 in V , let dS (v1 , v2 ) denote the length (in terms the of number of edges) of the shortest path between v1 and v2 . We now define a function d : (E, E) → N1 as follows: For eu = u1 u2 , ev = v1 v2 ∈ E, let d(eu , ev ) = min dS (ui , vj ). i,j∈{1,2}
The above interference model is most appropriate for wireless systems containing single-channel radios with omni-directional antennas and no interference cancellation or multi-user detection capabilities. Also, it assumes that the communication is bi-directional, i.e., both transmitter and receiver send messages to each other, like in IEEE 802.11 networks, involving a twoway or a four-way handshake. In scenarios where the communication is uni-directional, it might be sufficient to ensure that no node within K hops of the receiver is 1N
denotes the set of non-negative integers.
transmitting at the same time. Due to space limitations, we only consider the bi-directional case in this paper. For an integer K ≥ 1, we call a matching a “Kvalid matching” if for all e1 , e2 ∈ M with e1 %= e2 , we have d(e1 , e2 ) ≥ K. Observe that any matching is also a 1-valid matching, but may not be a K-valid matching for K ≥ 2. Now, the maximum K-valid matching problem (MKVMP) is to find a K-valid matching of maximum cardinality; the maximum weighted K-valid matching problem (MWKVMP) is defined in a similar way. Observe that for K = 1, MKVMP (respectively, MWKVMP) and MMP (respectively, MWMP) are the same. Observe that no link within K − 1 hops of a chosen link can be part of a K-valid matching. Thus, by increasing K, one can potentially support higher and higher rates (higher rates usually imply higher utilities) at each chosen link. However, there is a trade-off in that the number of simultaneous transmissions that can be supported across the system decrease with an increase in K. Thus, one would expect that there should be some optimal value of K, not necessarily 1, that maximizes the aggregate utility. This optimal value of K would, in general, depend on the topology of the system, which, in turn, depends on the density and placement of the nodes as well as on the peak power constraints at the nodes. Since the optimal value of K might be greater than 1 in some cases, it is of interest to study MWKVMP for K ≥ 2. The rest of the paper is organized as follows. We rigorously define the decision problems corresponding to MWKVMP and MKVMP, and show that they are NPComplete, in the next section. We investigate the possibility of obtaining constant factor polynomial time approximation algorithms for MWKVMP and MKVMP in Section 3. We end this paper with some concluding remarks in Section 4.
2. MKVMP and MWKVMP for K ≥ 2 are NP-Complete We now consider the decision problems corresponding to MKVMP and MWKVMP. With a slight abuse of notation, we call these decision problems MKVMP and MWKVMP, respectively, as well; which problem is being referred to should be clear from the context. We have the following definitions: Definition 1. M KV M P = {< G, m G is a graph with a K-valid matching of size m}. Definition 2. M W KV M P
=
{
: >:
G is a graph with a K-valid matching of size m and total weight WM }.
v1,bs
v2,bs
v3,bs
We start by showing that MWKVMP ∈ NP; which implies that MKVMP ∈ NP.
v1,fs
v2,fs
v3,fs
Theorem 1. The decision problem MWKVMP ∈ NP. Proof. Given a certificate in the form of a list of edges, it can easily be verified in polynomial time whether that list corresponds to a set of m edges that are at a distance of K or more from each other and have a total weight of WM or not. Thus, whether the set of edges constitute a K-valid matching of size m with a total weight of WM can be verified in polynomial time. Hence, MWKVMP ∈ NP. Next, we show that MKVMP is NP-Hard; which implies that the decision problem MWKVMP is NP-Hard as well. Theorem 2. The decision problem MKVMP is NPHard. Proof. Due to space constraints, we only provide a proof for K = 2. The proof for K ≥ 3 can be found in [10]. We first need to introduce some standard terminology. A literal in a boolean formula Φ is an occurrence of a variable or its negation. A truth assignment for a boolean formula is the set of values for the variables of Φ, and a satisfying assignment is a truth assignment that causes it to evaluate to 1. A formula with a satisfying assignment is called a satisfiable formula. A boolean formula is in conjunctive normal form, or CNF, if it is expressed as an AND of clauses, each of which is the OR of one or more literals. A boolean formula is in 3conjunctive normal form, or 3-CNF, if each clause has exactly three distinct literals. Example 1. The boolean formula Φ = (x1 ∨x2 ∨x3 )∧(¬x1 ∨x2 ∨x3 )∧(x1 ∨¬x2 ∨¬x3 ), where ∧, ∨, ¬ denote the MIN, MAX, and COMPLEMENT operations, respectively, is in 3-CNF. We are now ready to define the 3-CNF-SAT problem: Definition 3. 3 − CN F − SAT = {< Φ >: Φ is a satisfiable boolean formula in 3-CNF form}. It is well known that the 3-CNF-SAT is NPComplete. In what follows, we prove that 3-CNF-SAT ≤P 2 MKVMP for K = 2, thus showing MKVMP for K = 2 is NP-Hard. 2 The
symbol denotes polynomial time reduction.
Figure 1. The figure shows the triple pair of vertices corresponding to a clause, along with the associated edges.
x1
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Figure 2. The figure shows the graph constructed for the boolean formula Φ of Example 1.
The reduction algorithm begins with an instant of 3CNF-SAT. Let Φ = C1 ∧ C2 ∧ · · · ∧ Cm be a boolean formula in 3-CNF form with m clauses and n variables. The goal is to construct a graph G such that G has a 2valid matching of size m if and only if the formula Φ is satisfiable. The graph G = (V, E) is constructed as follows. For each clause Cs in Φ, we place a triple pair of vertices into V . Each pair of vertices, corresponds to a literal in Cr . Denote the literals of a clause Cs by l1s ,l2s , and l3s . The pair of vertices corresponding to the literal lis s s will be denoted by vi,f and vi,b , respectively. For each s , s ∈ {1, 2, ..., m}, we connect the set of vertices vi,f i = 1, 2, 3, with each other (see Figure 1). For each s with s ∈ {1, 2, ..., m} and i ∈ {1, 2, 3}, we connect vi,f s vi,b . The set of vertices corresponding to a clause, along with the edges connecting them with each other, would henceforth be referred to as a subgraph corresponding to the clause. Next, if r %= s and there exists i and j such that lir = s r s and vj,f with an edge. The ¬lj , then we connect vi,f graph constructed using this procedure for the boolean formula Φ of Example 1 is shown in Figure 2. The graph G can easily be constructed in polynomial (in m and n) time. Thus, we only need to show that this transformation is a reduction, i.e., the formula Φ has a
satisfying assignment if and only if the graph G has a 2-valid matching of size m. First, suppose the formula Φ is satisfiable. Then, for each s ∈ {1, 2, ..., m}, there exists a corresponding is ∈ {1, 2, 3} such that liss = 1 in the satisfying assignment. We now claim that {viss ,f viss ,b }s={1,2,...,m} is a 2-valid matching of size m. That it is a matching of size m is obvious, we therefore only need to prove that it is 2valid. Suppose not, then from the construction of G it is clear that for some s1 , s2 ∈ {1, 2, ..., m}, we must have an edge connecting viss1 ,f with viss2 ,f , which then 1 2 implies that liss1 ,f = ¬liss1 ,f . Thus, one of them must 1 1 be 0, contradicting our initial assumption that liss = 1 for s ∈ {1, 2, ..., m}. This proves our claim that if Φ is satisfiable then the graph G has a 2-valid matching of size m. Now, suppose the graph G has a 2-valid matching of size m. First, observe that at most one edge of the subgraph corresponding to a clause can be part of any 2-valid matching. Now suppose an edge of the form s s vj,f is part of the matching. We can then replace vi,f s s s s vi,b or vj,f vj,b , and we such an edge with the edge vi,f will still have a 2-valid matching. Also, an edge of the s r vj,f for r %= s can never be part of a 2-valid form vi,f matching of size m. For suppose it does, then no other edge in the subgraphs corresponding to clauses Cr and Cs can be part of the matching. Also, no edge of the form viso ,f vjko ,f or viro ,f vjko ,f , for io , jo ∈ {1, 2, 3} and k %= r, s can be part of the 2-valid matching of size m. Thus, the remaining m − 1 edges must either be part of the m − 2 remaining subgraphs or connect one such subgraph with another. This is, however, not possible, since there can at most be m − 2 such edges. From the above discussion, it follows that given a 2valid matching of size m, one can always construct a 2-valid matching of the form {viss ,f viss ,b }s={1,2,...,m} . We now claim that by setting liss = 1 for s ∈ {1, 2, ..., m}, we obtain a satisfying assignment for the boolean formula Φ. For if not, then there must be s1 , s2 ∈ {1, 2, ..., m} such that s1 %= s2 and liss1 = ¬liss2 . But then, there must be an edge be1 2 tween viss1 ,f and viss2 ,f ; contradicting our initial hypoth1 2 esis that {viss ,f viss ,b }s={1,2,...,m} is a 2-valid matching of size m.
3
Approximation MWKVMP
Algorithms
for
We now show that MKVMP for K ≥ 2 does not admit any constant factor polynomial time approximation algorithm in case of arbitrary graphs. In fact, one can show the following (see [10] for a proof): Theorem 3. Let η be such that (|V | + K|E|)η = Θ(|V |). Then, MKVMP for K ≥ 2 is not approximable within |V |η/2−" for any # > 0, unless NP = P. Further, it is not approximable within |V |η−" for any # > 0, unless NP = ZPP. Note that the complexity class ZPP denotes the class of Zero-error Probabilistic Polynomial time problems. We refer the reader to the original work of Gill [6] for a rigorous definition of the complexity class ZPP. The following result is a direct consequence of Theorem 3: Corollary 1. Let η be such that (|V | + K|E|)η = Θ(|V |). Then, MWKVMP for K ≥ 2 is not approximable within |V |η/2−" for any # > 0, unless NP = P. Further, it is not approximable within |V |η−" for any # > 0, unless NP = ZPP. Since K = O(V ) and E = O(V 2 ), the following result follows from Theorem 3 and Corollary 1: Corollary 2. MWKVMP and MKVMP for K ≥ 2 are not approximable within |V |1/6−" for any # > 0, unless NP = P. Further, they are not approximable within |V |1/3−" for any # > 0, unless NP = ZPP. Theorem 3 shows that in case of arbitrary graphs one cannot in polynomial time always obtain a K-valid matching with weight arbitrarily close to the weight of the optimal K-valid matching. However, it is clear that one can always obtain a matching with weight no smaller than 1/|E| times the weight of the optimal Kvalid matching in polynomial time (e.g., by using the obvious greedy approach [10]). The next result (see [10] for a proof) shows that it is possible to do better than this: Theorem 4.# MWKVMP is approximable within " |E| Θ (log |E|)2 .
The following Corollary is an immediate consequence of Theorem 4: Next, we investigate whether MWKVMP admits any constant factor polynomial time approximation algorithm.
Corollary 3. MKVMP " # Θ (log|E| |E|)2 .
is
approximable
within
4. Concluding Remarks We studied the problem of utility maximization in multi-hop wireless systems. To study the effect of wireless interference constraints on the utility maximization problem, we introduced a new class of weighted matching problems, namely Maximum Weighted KValid Matching problems (MWKVMPs). For K = 1, MWKVMP corresponds to the well studied Maximum Weighted Matching problem (MWMP), which can be solved in polynomial time. Interestingly, and perhaps surprisingly, for K ≥ 2, MWKVMP is NP-Hard. We further showed that it does not even belong to APX for K ≥ 2. The utility maximization problem we studied is closely related to the problem of throughput maximization in multi-hop wireless systems. In particular, an approximation algorithm for MWKVMP can be used to construct a scheme that guarantees a certain fraction of the maximum achievable throughput in multi-hop wireless systems. Making this connection precise, is the focus of our ongoing research. The previous works on multi-hop wireless systems [8, 3, 2, 11] have mainly focused on characterizing the achievable throughput region in the case of maximal scheduling policy [3] or one of its variants. The contributions of our work are threefold: First, we introduced a new class of interference models, characterized by the parameter K. By varying the parameter K, one can relax or strengthen the interference constraints. This allows one to study a wide variety of interference scenarios at once, which was not possible before. Second, we introduced a new class of weighted matching problems that we termed Maximum Weighted K-valid Matching Problems (MWKVMPs). For each K, MWKVMP is the weighted matching problem corresponding to the interference model with parameter K. Although, these problems are motivated by their application to scheduling in multi-hop wireless systems, they are quite general, and might find applications in other areas. Third, we provide results concerning the hardness and approximability of MWKVMP for different values of K. Several works (see, for example, [4]) model the wireless interference constraints in the form of a “contention matrix”. As a direct consequence of our results, one can obtain results concerning the hardness and approximability of maximum weighted problems, with additional constraints determined by the contention matrix. There are several avenues for future research in this area. One possible direction for research is to devise approximation algorithms for MWKVMP with provably good performance in case of certain specific graphs.
This direction is pursued in [10], where a polynomial time approximation scheme (PTAS) for MWKVMP is developed in case of geometric graphs. This is, however, a centralized scheme. Developing a decentralized PTAS for MWKVMP with provably good performance in case of geometric graphs is still an open problem. Also, it would be interesting to determine for specific instances of multi-hop wireless systems, the value of K that results in the maximum aggregate utility.
5. Acknowledgements We would like to thank the anonymous reviewers for their useful comments.
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