Exploiting Problem Decomposition in Multi-objective Constraint Optimization Radu Marinescu Cork Constraint Computation Centre University College Cork, Ireland
[email protected]
Abstract. Multi-objective optimization is concerned with problems involving multiple measures of performance which should be optimized simultaneously. In this paper, we extend AND/OR Branch-and-Bound (AOBB), a well known search algorithm, from mono-objective to multi-objective optimization. The new algorithm MO-AOBB exploits efficiently the problem structure by traversing an AND/OR search tree and uses static and dynamic mini-bucket heuristics to guide the search. We show that MO-AOBB improves dramatically over the traditional OR search approach, on various benchmarks for multi-objective optimization.
1 Introduction A Constraint Optimization Problem (COP) is the minimization (or maximization) of an objective function subject to a set of constraints on the possible values of a set of independent decision variables. Many real-world problems involve multiple measures of performance or objectives which should be optimized simultaneously. In contrast with single function optimization, the simultaneous optimization of multiple, possibly conflicting, objective functions does not in general admit a single, perfect solution. Instead, Multi-objective Constraint Optimization Problems (MO-COP) tend to be characterized by a set of alternatives which must be considered equivalent in the absence of information concerning the relevance of each objective relative to the others. Therefore, solving a MO-COP is to find the efficient frontier, namely the set of equivalent or non-dominated costs of the set of feasible solutions. Most complete algorithms for solving MO-COPs typically fall within one of the following two categories: search and inference. Search-based algorithms (e.g. depth-first Branch-and-Bound) transform a problem into a set of subproblems by selecting a variable and considering the assignment of each of its domain values. The subproblems are solved in sequence applying recursively the same transformation rule. These algorithms are not sensitive to the problem structure, have a time complexity which is exponential in the number of variables, but can operate in polynomial space. Inference-based algorithms (e.g., variable elimination) are known to be good at exploiting the structural information encoded in the problem. These methods apply a sequence of transformations that reduce the problem size, while preserving the solution space of the problem. Their time and space complexity is exponential in a topological parameter called treewidth (always less than or equal to the number of variables). Due to their high space requirements, when the treewidth is large, the latter methods are often impractical.
The AND/OR search space for COPs [1] is a relatively new framework for search that is sensitive to the problem structure, often resulting in substantially improved performance. It is based on a pseudo tree that captures conditional independencies in the problem, resulting in a search tree exponential in the depth of the pseudo tree, rather than in the number of variables. The AND/OR Branch-and-Bound (AOBB) is a recent search algorithm introduced in [2] that traverses depth-first an AND/OR search tree associated with a single objective COP instance. In this paper, we extend the AND/OR search tree from mono-objective to multiobjective optimization. We then present a multi-objective depth-first AND/OR Branchand-Bound algorithm (MO-AOBB) for finding the efficient frontier of a MO-COP instance. The efficiency of the proposed algorithm also depends on the accuracy of its guiding heuristic function. We investigate a class of partitioning-based heuristics which is based on the multi-objective mini-bucket elimination introduced recently in [3]. The mini-bucket heuristics can be either pre-compiled or generated dynamically at each node in the search tree. We evaluate the impact of our advancement on several benchmarks for MO-COPs, including risk-conscious combinatorial auctions, MAX-SAT-ONE problem instances, bi-objective weighted vertex cover problems, as well as a set of MO-COP instances derived from real-world mono-objective COP instances. Our results show conclusively that the new multi-objective AND/OR Branch-and-Bound algorithm improves dramatically over state-of-the-art search algorithms traversing the traditional OR space, in many cases by several orders of magnitude. The paper is organized as follows. Section 2 provides background on COPs, AND/OR search trees for COPs as well as on MO-COPs. In Section 3 we extend the AND/OR search tree from mono-objective to multi-objective optimization. Section 4 presents the multi-objective AND/OR Branch-and-Bound search algorithm for finding the efficient frontier of a MO-COP instance. Section 5 is dedicated to our empirical evaluation, while Section 6 provides concluding remarks and directions of future research.
2 Background 2.1
Mono-objective Constraint Optimization Problems
A Constraint Optimization Problem (COP) is a triple P = hX, D, Fi, where X = {X1 , ..., Xn } is a set of variables, D = {D1 , ..., Dn } is a set of finite domains and F = {f1 , ..., fr } is a set of cost functions. Cost functions can be either soft or hard (constraints). Without loss of generality we assume that hard constraints are represented as (bi-valued) cost functions. Allowed and forbidden tuples have cost 0 and ∞, respectively. The scope of function fi , denoted scope(fi ) ⊆ X, is the set of arguments of fi . The goal is to find a complete value assignment to theP variables, x = (x1 , ..., xn ), that r minimizes the objective function defined by F(X) = i=1 fi (Yi ), where Yi ⊆ X. Given a COP instance, its primal graph G has a node for each variable and an edge connects any two nodes whose variables appear in the scope of the same function. The induced graph of G relative to an ordering d of its variables is obtained by processing the nodes in reverse order of d. For each node all its earlier neighbors are connected, including neighbors connected by previously added edges. The width of a node is the number of edges connecting it to nodes lower in the ordering. The treewidth of a graph along d, denoted w∗ (d), is the maximum width of nodes in the induced graph.
2.2
AND/OR Search Trees for COP
The AND/OR search space [1] is a unifying framework for advanced algorithmic schemes for graphical models, including constraint networks and cost networks. Its main virtue consists in exploiting independencies between variables during search, which can provide exponential speedups over traditional structure-blind search methods (called here OR search). The search space is defined using a backbone pseudo tree [4]. Definition 1 (pseudo tree). Given an undirected graph G = (V, E), a directed rooted tree T = (V, E ′ ) defined on all its nodes is called a pseudo tree if any edge of G that is not included in E ′ is a back-arc in T , namely it connects a node to an ancestor in T . Given a COP instance P = hX, D, Fi, its primal graph G and a pseudo tree T of G, the associated AND/OR search tree ST (P) has alternating levels of OR and AND nodes. The OR nodes are labeled Xi and correspond to the variables. The AND nodes are labeled hXi , xi i (or just xi ) and correspond to value assignments of the variables. The structure of the AND/OR search tree is based on the underlying pseudo tree T . The root of the AND/OR search tree is an OR node labeled with the root of T . The children of an OR node Xi are AND nodes labeled with assignments hXi , xi i that are consistent with the assignments along the path from the root. The children of an AND node hXi , xi i are OR nodes labeled with the children of variable Xi in T . A solution tree T of an AND/OR search tree ST (P) is an AND/OR subtree such that: (1) it contains the root of ST (P), s; (2) if a non-terminal AND node n ∈ ST (P) is in T then all of its children are in T ; (3) if a non-terminal OR node n ∈ ST (P) is in T then exactly one of its children is in T ; It was shown [4, 5, 1] that given a COP instance and a pseudo tree T of depth m, the size of the AND/OR search tree based on T is O(n · k m ), where k bounds the domains of variables. Moreover, a COP having treewidth w∗ has a pseudo tree of depth at most ∗ w∗ · log n, and therefore it has an AND/OR search tree of size O(n · k w ·log n ). 2.3
Multi-objective Constraint Optimization Problems
Let us consider now problems with p objectives. A cost vector (also called a p-vector) v = (v1 , ..., vp ) is a vector of p components where each vj ∈ R represents the cost with respect to objective j. A cost vector which has all components equal to 0 is denoted by {0}. A multi-objective function f is a function that associates a cost vector to each assignment of its scope. Let v and u be two cost vectors. We say that v dominates u, denoted by v ≤ u, if ∀j, vj ≤ uj . We say that v < u if v ≤ u and v 6= u. The sum of cost vectors is the usual point-wise sum. Let α be a set of cost vectors. We define its non-dominated closure as k α k= {u ∈ α | ∀v ∈ α, v ≮ u}. A set of cost vectors closed under non-domination is called frontier. Let α and β be two frontiers. We say that α dominates β, denoted by α ≤ β, if ∀v ∈ β, ∃u ∈ α s.t. u ≤ v. We say that α < β, if α ≤ β and α 6= β. The sum of frontiers is α + β =k {w = u + v | u ∈ α, v ∈ β} k. A Multi-objective Constraint Optimization Problem (MO-COP) is a triple M = hX, D, Fi, where X = {X1 , ..., Xn }, D = {D1 , ..., Dn } and F = {f1 , ..., fr } are variables, domains and multi-objective functions, respectively. The sum of functions in
(a) Bi-objective functions
(b) Primal graph
Fig. 1. An example of a MO-COP instance with 2 objectives.
F defines the objective function F. The only difference with respect to mono-objective COP is that multi-objective functions are used. A solution is a complete assignment x = (x1 , ..., xn ). A solution x is efficient or Pareto optimal if ∀x′ , F(x′ ) ≮ F(x). The efficient frontier is the set E = {F(x) | ∀x′ , F(x′ ) ≮ F(x)}. The optimization task is to compute E, and for each v ∈ E, one (of the possibly many) Pareto optimal assignment with cost v 1 . As in the mono-objective case, each MO-COP instance has an associated primal graph, which is computed in the same way: nodes correspond to the variables and an arc connects any pair of nodes whose variables belong to the scope of the same function. Example 1. Figure 1 shows a MO-COP instance with 2 objectives, 5 bi-valued variables and 8 cost functions. The first objective is defined by functions f1 , f2 and f3 , while functions g0 , g1 , g2 , g3 and g4 form the second objective, respectively. The corresponding bi-objective functions are given in Figure 1(a). For example, the cost vector associated with f1 (X0 = 0, X1 = 0, X2 = 0) has two components, 5 and 0, which represent the costs with respect to the first and second objective, respectively. The primal graph is shown in Figure 1(b). The problem has three solutions: (00000), (01000) and (01100) with non-dominated costs (7, 0), (4, 2) and (3, 5), respectively.
3 Weighted AND/OR Search Trees for MO-COP In this section we extend the AND/OR search tree from mono-objective to multi-objective optimization. Given a MO-COP instance M = hX, D, Fi and its primal graph G, the AND/OR search tree ST (M) of M is driven by a pseudo tree T of G, as in the monoobjective case. The arcs from nodes OR nodes Xi to AND nodes hXi , xi i in ST (M) are annotated by weights derived from the multi-objective cost functions in F. Each node n in the weighted search tree is associated with a value v(n) which stands for the answer to the optimization query restricted to the conditioned subproblem below n. Definition 2 (weight). The weight w(n, n′ ) of the arc from the OR node n labeled Xi to the AND node n′ labeled hXi , xi i is a cost vector defined as the sum of all the multiobjective cost functions whose scope includes Xi and is fully assigned along the path from the root to hXi , xi i, evaluated at the values along the path. 1
In some cases, E may be too large and must be approximated. In this paper we do not consider this situation.
(a) Pseudo tree
(b) AND/OR search tree
Fig. 2. AND/OR search tree for the MO-COP instance from Fig. 1.
Definition 3 (value). The value v(n) of a node n in a weighted AND/OR search tree of a MO-COP instance M = hX, D, Fi with p objectives is defined recursively as follows (where succ(n) are the children of n): (1) v(n) = P {0}, if n = hXi , xi i is a terminal AND node; (2) v(n) = n′ ∈succ(n) v(n′ ), if n = hXi , xi i is an internal AND node; (3) v(n) =k {(w(n, n′ ) + v(n′ )) | n′ ∈ succ(n)} k, if n = Xi is an OR node. It is easy to see that the value v(n) of a node in the AND/OR search tree ST (M) is the set of cost vectors representing the efficient frontier of the subproblem rooted at n, subject to the current variable instantiation along the path from the root to n. If n is the root of ST (M), then v(n) is the efficient frontier of the initial problem. Example 2. Figure 2(b) shows the AND/OR search tree for the MO-COP instance from Figure 1, relative to the pseudo tree from Figure 2(a). The numbers on the ORto-AND arcs are the weights corresponding to the function values. For example, the weight of the arc from the OR node X2 to its AND child hX2 , 0i, along the path (X0 , hX0 , 0i, X1 , hX1 , 0i, X2 , hX2 , 0i), is (5, 0) and is obtained by summing the values of the functions f1 (X0 , X1 , X2) and g2 (X2 ), whose scopes contain X2 and are fully instantiated along that path. A solution tree that corresponds to the assignment (X0 = 0, X1 = 1, X2 = 1, X3 = 0, X4 = 0) with cost (3, 5) is highlighted in Figure 2(b). The OR node X2 along the path (X0 , hX0 , 1i, X1 , hX1 , 0i, X2 ) has value (2, 0) which is the non-dominated closure of the frontier below it, namely {(2, 0), (3, 3)}.
4 Multi-objective AND/OR Branch-and-Bound Search Multi-objective problems can be solved using a depth-first Branch-and-Bound schema, which traverses a traditional OR search tree of possible assignments in a depth-first manner and outputs the efficient frontier E of the problem [6, 7]. The algorithm, called MO-BB, handles two frontiers, called lower and upper bound frontiers. The upper bound frontier is the set of cost vectors of the best solutions found so far and is an overestimation of the efficient frontier E. During search, MO-BB computes at each visited
node a lower bound frontier [8] of the current subproblem using a heuristic evaluation function. MO-BB backtracks if the upper bound dominates the lower bound, because it implies that the current partial assignment cannot lead to any new efficient cost vector. In this section, we apply the general principles of AND/OR search and extend MOBB into a Branch-and-Bound algorithm that explores an AND/OR rather than a regular OR search tree for finding the efficient frontier of a MO-COP instance. The proposed algorithm is an extension of the recently introduced single objective AND/OR Branchand-Bound (AOBB) algorithm [2] to the multi-objective optimization case. We start by revisiting the notion of partial solution trees [9] to represent sets of solution trees. Definition 4 (partial solution tree). A partial solution tree T ′ of an AND/OR search tree ST (M) is a subtree which: (1) contains the root node s of ST (M); (2) if n is an OR node in T ′ then it contains one of its AND child nodes in ST (M), and if n is an AND node it contains all its OR children in ST (M). A node of T ′ is a tip node if it has no children in T ′ . A tip node of T ′ is either a terminal node (if it has no children in ST (M)), or a non-terminal node (if it has children in ST (M)). A partial solution tree may be extended (possibly in several ways) to a full solution tree. It represents extension(T ′ ), the set of all full solution trees which can extend it. Clearly, a partial solution tree all of whose tip nodes are terminal is a solution tree. 4.1
Heuristic Lower Bounds on Partial Solution Trees
We next define the notion of heuristic evaluation function of a partial solution tree, which will be used to guide the AND/OR Branch-and-Bound search. Like in OR search, we assume a given heuristic evaluation function h(n) associated with each node in the search tree such that h(n) is an underestimation of the efficient frontier v(n) of the conditioned subproblem below n. Definition 5 (heuristic evaluation function). Given a partial solution tree Tn′ rooted at node n in the AND/OR tree ST (M), the heuristic evaluation function f (Tn′ ), is defined recursively by: (1) if Tn′ consists of a single node n, then f (Tn′ ) = h(n); (2) if ′ n is an OR node having the AND child m in Tn′ , then f (Tn′ ) = w(n, m) + f (Tm ); (3) if Pk ′ ′ ′ n is an AND node having OR children m1 , ..., mk in Tn , then f (Tn ) = i=1 f (Tm ). i Clearly, by definition, f (Tn′ ) is a lower bound frontier that underestimates the efficient frontier of the subproblem represented by Tn′ . During search, the algorithm maintains also an upper bound frontier ub(s) of the efficient frontier v(s), where s is the root of the search tree. Given the current partial solution tree Ts′ and if the upper bound frontier ub(s) dominates the lower bound frontier f (Ts′ ) (i.e., ub(s) ≤ f (Ts′ )), then searching below the current tip node t of Ts′ is guaranteed not to lead to any new efficient cost vector and, therefore, search below t can be safely halted. 4.2
The Branch-and-Bound Algorithm
The depth-first AND/OR Branch-and-Bound search algorithm, MO-AOBB, that traverses an AND/OR search tree is described by Algorithm 1. It takes as input a MO-COP instance M = hX, D, Fi with p objectives as well as a pseudo tree T of its primal graph
Algorithm 1: MO-AOBB 1 2 3 4 5 6 7 8 9 10
Data: A MO-COP instance M = hX, D, Fi with p objectives, pseudo tree T . Result: Efficient frontier E of M. create an OR node s labeled X1 // Create and initialize the root node v(s) ← ∅; OP EN ← {s} while OP EN 6= ∅ do n ← top(OP EN ); remove n from OPEN if n is an OR node, labeled Xi then // EXPAND for xi ∈ DXi do ′ create an AND node n labeled hXi , xi i v(n′ ) ← {0}; h(n′ ) ← heuristic(n′ ) P w(n, n′ ) ← f ∈F,X ∈scope(f ) f (asgn(πn )) i
succ(n) ← succ(n) ∪ {n′ }
11 12 13 14 15 16 17
else if n is an AND node, labeled hXi , xi i then let Ts′ be the current partial solution tree if v(s) f (Ts′ ) then for Xj ∈ childrenT (Xi ) do create an OR node n′ labeled Xj v(n′ ) ← ∅; h(n′ ) ← heuristic(n′ ) succ(n) ← succ(n) ∪ {n′ }
18 19 20 21 22 23
Add succ(n) on top of OP EN while succ(n) == ∅ do let par be the parent of n if n is an OR node, labeled Xi then if Xi == X1 then return v(n)
24
// PROPAGATE
// Search is complete
v(par) ← v(par) + v(n)
25 26
if n is an AND node, labeled hXi , xi i then v(par) ←k v(par) ∪ {(w(par, n) + v(n))} k
27 28
remove n from succ(par) n ← par
and returns the efficient frontier of M. The fringe of the search is maintained by a stack called OPEN. The current node is denoted by n, its parent by par, the current path by πn and its assignment by asgn(πn ). The successors of the current node are denoted by succ(n), while the children of a variable Xi in T are denoted by childrenT (Xi ). Each node n in the search tree maintains its current value v(n), which is updated based on the values of its children. For OR nodes, the current v(n) is an upper bound frontier on the efficient frontier below n. Initially, v(n) is set to ∅ if n is OR, and {0} if n is AND, respectively. Procedure heuristic(n) computes the lower bound h(n) on the efficient frontier of the subproblem below n, conditioned on asgn(πn ). In the EXPAND step, the current node n is expanded in the usual way, depending on whether it is an AND or an OR node (lines 5–17). The successors of an OR node Xi are AND nodes labeled by the domain values xi of Xi . The successors of an AND node hXi , xi i are OR nodes labeled by the children Xj of Xi in the pseudo tree T . The algorithm also computes the heuristic evaluation function f (Ts′ ) of the current partial solution tree Ts′ being explored, based on Definition 5. The search below n is terminated if the current upper bound frontier v(s) dominates f (Ts′ ) (line 13). The node values are updated by the PROPAGATE step (lines 19–28). It is triggered when a node has an empty set of descendants (note that as each successor is evaluated,
it is removed from the set of successors in line 27). This means that all its children have been evaluated, and their values are already determined. If the current node is the root, then the search terminates with its value, namely the efficient frontier (line 23). If n is an OR node, then its parent par is an AND node, and par updates its current value v(par) by summation with the value of n (line 24). An AND node n propagates its value to its parent par in a similar way, by the non-dominated closure (lines 25–26). Finally, the current node n is set to its parent par (line 28), because n was completely evaluated. Search continues either with a propagation step (if conditions are met) or with an expansion step. The algorithm can be easily instrumented to also record the solution trees that correspond to the cost vectors in the efficient frontier. Clearly, Theorem 1. MO-AOBB is sound and complete for MO-COP. In a problem with a single objective function (i.e., p = 1), the algorithm MO-AOBB is equivalent to AOBB. While the time complexity of MO-AOBB is bounded by O(n · exp(m)), the size of the AND/OR search tree, in practice, the running time of MO-AOBB is expected to be much better than the worst-case bound because the pruning mechanism via the heuristic evaluation function will prune unpromising portions of the search space. 4.3
Partitioning-based Lower Bound Heuristics
We now describe briefly two general schemes for generating the heuristic estimates h(n), based on the multi-objective Mini-Bucket approximation. These schemes are parameterized by the Mini-Bucket i-bound, thus allowing for a controllable trade-off between heuristic strength and its computational overhead. Mini-Bucket Elimination (MBE) [10] is an approximation algorithm designed to avoid the high time and space complexity of Bucket Elimination (BE) [11], by partitioning large buckets into smaller subsets, called mini-buckets, each containing at most i (called i-bound) distinct variables. The mini-buckets are then processed separately. The algorithm computes a lower bound (assuming minimization) on the optimal solution to a COP. The complexity is time and space O(exp(i)). More recently, [3] extended MBE(i) from mono-objective to multi-objective optimization problems, yielding Multi-objective Mini-Bucket Elimination (MO-MBE). Given a MO-COP instance, MO-MBE(i) outputs not only a lower bound frontier of the efficient frontier, but also the collection of augmented buckets, which form the basis for the heuristics generated. Static Mini-Bucket Heuristics In the past, [12, 2] showed that the intermediate functions generated by the MBE(i) can be used to compute a heuristic function that underestimates the minimal extension of the current assignment in both OR and AND/OR search trees for COP. Here, we extend the idea to multi-objective optimization. Consider a MO-COP instance M and the ordered set of augmented buckets {B(X1 ), ..., B(Xn )} generated by MO-MBE(i) along a DFS traversal of the pseudo tree T of M. Given a node n in the AND/OR search tree relative to T , the static mini-bucket heuristic function h(n) is computed as follows: (1) if n is an AND node labeled hXj , xj i, then h(n) is the sum of all intermediate functions that were generated in buckets corresponding to descendants of Xj in T and reside in bucket B(Xj ) or the buckets corresponding to the ancestors of Xj in T ; (2) if n is an OR node labeled Xj , then h(n) =k {(w(n, m) + h(m)) | m ∈ succ(n)} k.
Dynamic Mini-Bucket Heuristics Rather than pre-compiling the mini-bucket heuristic information, it is possible to generate it dynamically, during search [2, 7]. Specifically, given the set of buckets {B(X1 ), ..., B(Xn )} ordered along a DFS traversal of T , a node n in the AND/OR search tree and given the current partial assignment asgn(πn ) along the path to n, the dynamic mini-bucket heuristic function h(n) is computed as follows: (1) if n is an AND node labeled hXj , xj i, then h(n) is the sum of the intermediate functions that reside in bucket B(Xj ) and were generated by MO-MBE(i), conditioned on asgn(πn ), in the buckets corresponding to the descendants of Xj in T ; (2) if n is an OR node labeled Xj , then h(n) =k {(w(n, m) + h(m)) | m ∈ succ(n)} k. Given an i-bound, the dynamic mini-bucket heuristic implies a much higher computational overhead compared with the static version. However, the bounds generated dynamically may be far more accurate since some of the variables are assigned and will therefore yield smaller functions and less partitioning.
5 Experiments To evaluate our multi-objective AND/OR Branch-and-Bound approach we have conducted a number of experiments on several MO-COP problem classes such as bi-objective combinatorial auction, bi-objective weighted vertex cover problems, MAX-SAT-ONE problem instances as well as bi-objective Weighted CSPs derived from standard monoobjective instances. We implemented our algorithms in C++ and carried out all experiments on a 2.4GHz dual quad-core with 8GB of RAM running Linux Ubuntu 8.10. We considered two classes of depth-first AND/OR Branch-and-Bound search algorithms guided by static and dynamic mini-bucket heuristics. They are denoted by MO-AOBB+SMB(i) and MO-AOBB+DMB(i), respectively. We compare these algorithms against traditional depth-first OR Branch-and-Bound algorithms with static and dynamic mini-bucket heuristics, denoted by MO-BB+SMB(i) and MO-BB+DMB(i), respectively. The parameter i represents the mini-bucket i-bound and controls the accuracy of the heuristic. For reference, we also ran the multi-objective Russian Doll Search algorithm, denoted by MO-RDS, which was recently introduced in [6]. MORDS explores a traditional OR search tree and extends the well known Russian Doll Search [13] from mono-objective to multi-objective optimization. Algorithms MOBB+DMB(i) and MO-RDS were evaluated extensively in [6, 7] and shown to outperform dramatically state-of-the-art algorithms for multi-objective optimization such as ǫ-constraint based search [14], multi-objective iterative deepening search [3] and multiobjective bucket elimination [3]. The pseudo trees that guide the AND/OR search algorithms were generated using the min-fill heuristic [2]. All competing OR search algorithms used a static variable ordering which was derived from the pseudo tree as well. In all our experiments we report the CPU time in seconds and the number of nodes visited for solving the problems. We also specify the problems’ parameters such as the number of variables (n), maximum domain size (k), the number of cost functions (c), the depth of the pseudo tree (h) and the treewidth of the graph (w∗ ). In addition, we also record the size of the efficient frontier found (|E|). The best performance points are highlighted in each table. A ”-” stands for exceeding the time limit. Bi-objective Weighted Vertex Cover Given a graph G = (V, E), a vertex cover is a subset of vertices S ⊆ V such that ∀(u, v) ∈ E, either u ∈ S or v ∈ S. The minimum
(V, E) (w*, h) |E|
MO-RDS
time
MO-BB+SMB(i) MO-BB+SMB(i) MO-BB+SMB(i) MO-BB+SMB(i) MO-BB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) MO-BB+DMB(i) MO-BB+DMB(i) MO-BB+DMB(i) MO-BB+DMB(i) MO-BB+DMB(i) MO-AOBB+DMB(i) MO-AOBB+DMB(i) MO-AOBB+DMB(i) MO-AOBB+DMB(i) MO-AOBB+DMB(i) i=2 i=4 i=6 i=8 i=10 nodes time nodes time nodes time nodes time nodes time nodes sparse
(60, 100) (10, 19) 14
108.74
12,774,381
(70, 120) (12, 22) 16 1560.06 224,998,226 (80, 140) (14, 26) 17
-
-
(90, 160) (16, 27) 29
-
-
13.27 6.25 41.74 14.80 444.19 103.91 1815.14 169.56
109,182 52,984 253,516 112,891 2,466,147 735,773 6,047,711 704,192
14.10 5.78 10.87 12.60 74.70 83.71 112.88 84.37
18.56 21.89 139.99 158.00 1356.05 1415.91 3116.71 3112.38
356,178 351,295 2,312,847 2,271,245 19,902,899 18,386,161 42,000,631 36,825,713
17.76 24.18 133.06 173.75 1386.05 1455.62 3054.23 3122.18
53,051 49,149 119,060 103,527 747,408 671,259 731,474 533,688
3.39 6.23 7.69 13.44 57.01 85.34 58.93 90.17
49,877 49,081 106,561 103,275 689,606 669,980 568,307 532,051
3.28 6.80 7.15 14.83 52.33 91.02 51.77 100.80
49,317 49,076 104,145 103,246 675,044 698,885 551,728 531,590
3.27 7.13 6.98 16.21 51.40 96.33 46.09 118.75
49,104 49,076 103,529 103,242 672,451 698,860 536,089 531,519
17.43 25.39 130.34 179.67 1306.88 1469.87 3037.54 3129.76
351,828 351,039 2,277,972 2,269,844 19,962,033 17,559,896 41,831,396 34,158,428
17.27 26.17 129.44 184.42 1305.10 1485.19 3031.55 3138.68
351,304 351,027 2,274,436 2,269,771 19,906,542 17,484,973 41,834,376 33,822,995
17.27 26.76 128.74 189.31 1303.90 1499.06 3028.49 3148.01
351,098 351,026 2,271,194 2,269,739 19,874,904 17,414,634 41,689,599 33,801,126
medium (30, 90) (11, 19) 6
0.05
8,314
(40, 120) (14, 23) 10
0.97
113,751
(50, 150) (18, 27) 8
21.56
2,252,573
(60, 180) (21, 32) 296.14 12
21,624,533
353,469 351,081 2,291,297 2,269,975 20,737,676 17,863,968 41,731,445 34,713,221
Table 1. Result for bi-objective weighted vertex cover problems using static and dynamic minibucket heuristics. Time limit 1 hour. Overall best performance points are boxed.
vertex cover is a vertex cover of minimum size. In the weighted version every vertex v has an associated weight w(v) P and the weighted minimum vertex cover is a vertex cover S with minimum F (S) = v∈S w(v). In the bi-objective version, each vertex v has two weights w1 (v) and w2 (v) and the task is to minimize the two associated objective functions simultaneously [3]. We generated random graphs with parameters (V, E, C), where V is the number of vertices, E is the number of edges and C is the maximum weight, as suggested in [3]. Instances were generated by randomly selecting E edges. The two weights associated with each vertex were generated uniformly at random between 0 and C. Table 1 reports the results obtained on two classes of random graphs: sparse (with V ∈ {60, 70, 80, 90} and E ∈ {100, 120, 140, 160}), and medium (with V ∈ {30, 40, 50, 60} and C ∈ {90, 120, 150, 180}), respectively. In both cases we set C = 30. The columns are indexed by the mini-bucket i-bound and the table entries represent an average over 10 random instances generated for each parameter configuration. When looking at the algorithms guided by mini-bucket heuristics, we observe that both MO-AOBB+SMB(i) and MO-AOBB+DMB(i) outperformed dramatically the corresponding OR Branch-and-Bound algorithms MO-BB+SMB(i) and MO-BB+DMB(i), across all reported i-bounds. For example, on sparse graphs of size (60,100), MOAOBB+SMB(10) solved all instances in about 3 seconds, while MO-BB+SMB(10) exceeded the 1 hour time limit. In terms of the quality of the heuristic, we also see that static mini-buckets with a relatively large i-bound represent the best choice. However,
paths - 30 goods - static mini-bucket heuristics [time] 4,000
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Fig. 3. Results for bi-objective combinatorial auctions with 30 goods and increasing number of bids. CPU time and nodes visited using static (top) and dynamic (bottom) mini-bucket heuristics. Time limit 1 hour. Average treewidth w∗ ∈ [8, 53], average pseudo tree depth h ∈ [16, 88].
if larger i-bounds are not possible, dynamic mini-buckets with small i-bounds are preferred, especially on sparse problems. For problems from the medium class, MORDS proved to be cost effective with respect to the mini-bucket based algorithms. Bi-objective Combinatorial Auctions In combinatorial auctions, an auctioneer has a set of goods to sell and the buyers submit a set of bids on indivisible subsets of goods [15]. In risk-conscious auctions, the auctioneer wants also to control the risk of not being paid after a bid has been accepted, because it may cause large losses in revenue [16]. Let M = {1, ..., n} be the set of goods to be auctioned and let B = {B1 , ..., Bm } be the set of bids. A bid Bj is defined by a triple (Sj , pj , rj ), where Sj ⊆ M , pj is the bid price and rj is the probability of failure, respectively. The auctioneer must decide which bids to accept under the constraint that each good is allocated to at most one bid. The first objective is to maximize the auctioneer profit. The second objective is to minimize risk. Assuming independence, after a logarithmic transformation of probabilities, this objective can also be expressed as an additive function [7]. For our purpose, we generated mono-objective auctions from the paths distribution of the CATS suite [15] and randomly added failure probabilities to the bids in the range 0 to 1. Figure 3 summarizes the results obtained on randomly generated auctions with 30 goods and increasing number of bids (each data point represents an average over 10 random instances). These problem instances tend to become highly connected as the number of bids increases. The average treewidth ranged between 8 (for 30 bids) and 53 (for 150 bids), while the average pseudo tree depth was between 16 (for 30
paths 30 goods 80 bids - [w*=25, h=50] - (time)
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Fig. 4. Impact of the i-bound on bi-objective combinatorial auctions with 30 goods and 80 bids using static and dynamic mini-bucket heuristics. CPU time (left) and nodes visited (right). Average treewidth w∗ = 25, average pseudo tree depth h = 50.
bids) and 88 (for 150 bids), respectively. We report on three values of the mini-bucket i-bound, namely i = 2, i = 8 and i = 14, respectively. We see that MO-AOBB+SMB(i) is better than MO-BB+SMB(i) at relatively small i-bounds when the heuristic is weak. This demonstrates the benefit of AND/OR versus classical OR search when the heuristic estimates are relatively weak and the algorithms rely primarily on search rather than pruning via the heuristic evaluation function. For example, on auctions with 60 bids, the average running time of MO-AOBB+SMB(2) was about 17 seconds, while MO-BB+SMB(2) took on average 1,951 seconds. As the ibound increases and the corresponding heuristics are strong enough to prune the search space substantially, the difference between AND/OR and OR search decreases. When focusing on dynamic mini-bucket heuristics, we see that MO-AOBB+DMB(i) is better than MO-BB+DMB(i) at relatively small i-bounds, but the difference is not that prominent as in the static case. This is because these heuristics are far more accurate compared with the pre-compiled ones and the savings in number of nodes caused by traversing the AND/OR search tree do not translate into additional time savings. On this domain, we see that MO-AOBB+SMB(i) with relatively large i-bounds outperformed MO-RDS. However, MO-RDS was superior to MO-AOBB+DMB(i), except for the smallest reported i-bound, in which case MO-AOBB+DMB(2) was competitive. The better performance of MO-RDS relative to the AND/OR algorithms can be explained by a reduced computational overhead for generating its guiding heuristic. Figure 4 plots the running time and number of nodes visited by MO-AOBB+SMB(i) and MO-AOBB+DMB(i) (resp. MO-BB+SMB(i) and MO-BB+DMB(i)), on auctions with 30 goods and 80 bids. Focusing on MO-AOBB+SMB(i), for example, we see that its running time, as a function of i, forms a U-shaped curve. At first (i = 2) it is high, then as the i-bound increases the total time decreases (when i = 12 the time is 14.01), but then as i increases further the time starts to increase again. The same behavior can be observed in the case of MO-AOBB+DMB(i) as well. MAX-SAT-ONE Instances Let F be a Boolean formula in conjunctive normal form (CNF). MAX-SAT is the problem of finding a truth assignment such that the number of satisfied clauses in F is maximized. MAX-ONE is the problem of finding a model for F with a maximum number of variables assigned to true. MAX-SAT-ONE is the
instance (n, c, k) (w*, h) |E|
MO-BB+SMB(i) MO-BB+SMB(i) MO-BB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) MO-BB+DMB(i) MO-BB+DMB(i) MO-BB+DMB(i) MO-AOBB+DMB(i) MO-AOBB+DMB(i) MO-AOBB+DMB(i) i=6 i=8 i=10 nodes time nodes time nodes time nodes 14.20 43,072 7.98 27,929 - 237.72 27,739 152.70 24,457 - 220.02 13,184 134.92 9,903 - 899.74 2,425,636 - 271.66 48,118 295.76 36,233 314.72 33,226 218.24 27,115 237.42 16,524 251.69 13,926 - 1125.50 3,820,213 379.20 1,127,093 - 335.04 92,522 182.50 77,540 176.86 76,623 233.46 26,465 84.97 11,500 78.85 10,583 - 423.19 900,024 312.10 660,848 - 311.94 657,833 269.36 592,225 - 1102.11 94,624 587.73 44,434 721.36 38,521 1040.83 72,361 531.54 23,047 667.06 17,323 - 619.58 1,417,018 80.02 213,427 50.02 140,081 25.14 78,158 177,981,953 187.92 115,004 208.41 113,821 269.87 113,774 67.23 30,086 85.67 28,911 148.03 28,865 74.05 183,418 75.17 198,489 596.16 1,757,792 38.33 106,507 61.72 175,580 - 394.17 65,879 277.48 36,430 342.56 34,197 356.38 48,982 244.55 20,111 310.87 17,880 - 653.68 1,027,972 - 150.80 397,432 61.28 146,520 - 169.36 88,156 230.36 87,401 226.42 85,373 87.62 34,017 147.73 33,262 144.41 31,256 159.71 362,140 66.09 142,424 24.08 62,518 43.91 136,779 28.46 76,451 17.46 48,822 22,063,228 55.05 24,839 100.66 24,434 132.36 23,524 43.19 11,842 88.56 11,435 121.40 10,525
MO-RDS
time aim-50-1-6no-1 (50, 119, 2) (15, 31) 8 aim-50-1-6no-2 (50, 127, 2) (19, 31) 11 aim-50-1-6no-3 (50, 120, 2) (17, 27) 10 aim-50-1-6no-4 (50, 126, 2) (20, 30) 11 aim-50-1-6yes1-1 (50, 127, 2) (18, 29) 155.33 10 aim-50-1-6yes1-2 (50, 126, 2) (16, 34) 9 aim-50-1-6yes1-3 (50, 128, 2) (20, 27) 10 aim-50-1-6yes1-4 (50, 127, 2) (19, 29) 146.56 9
MO-BB+SMB(i) MO-AOBB+SMB(i) MO-BB+DMB(i) MO-AOBB+DMB(i) i=12 time nodes 12.46 36,837 6.49 22,054 180.40 24,244 164.30 9,690 54.78 148,563 35.15 106,091 535.52 32,712 450.64 13,504 32.03 79,194 3.84 13,015 223.34 76,528 125.51 10,488 433.63 778,334 379.00 730,513 1135.44 38,127 1068.47 16,931 68.76 153,541 26.28 68,358 305.64 113,680 183.73 28,771 30.43 75,738 20.11 57,089 380.54 33,463 348.11 17,146 525.84 846,067 42.28 100,060 264.30 85,144 181.32 31,028 18.81 51,603 13.15 38,478 280.12 23,317 267.98 10,318
Table 2. Results for MAX-SAT-ONE instances using static and dynamic mini-bucket heuristics. Time limit 30 minutes. Overall best performance points are highlighted.
problem of maximizing both the number of satisfied clauses and variables assigned to true, as described in [3]. Table 2 shows the results obtained for experiments with 8 CNF instances from the DIMACS benchmark2 . As before, we observe that MO-AOBB+SMB(i) offers the overall best performance, especially for relatively large i-bounds. For example, on the aim50-1-6yes1-1, MO-AOBB+SMB(10) is about 25 times faster than MO-BB+SMB(10), and about 6 times faster than MO-RDS, respectively. When using dynamic mini-bucket heuristics, we see again that MO-AOBB+DMB(i) is competitive only for relatively small i-bounds, due to computational overhead issues. Note also that MO-RDS could solve only two instances, namely aim-50-1-6yes1-1 and aim-50-1-6yes1-4. The relatively worse performance of MO-RDS on this domain can be explained by the quality of its guiding heuristic function which was far inferior to the mini-bucket based ones. Bi-objective Weighted CSPs We considered a set of standard mono-objective Weighted CSP [17] benchmarks from the UCI Graphical Models Repository (available online at: http://graphmod.ics.uci.edu/group/Repository). For our purpose, we converted each of these instances into a bi-objective optimization problem by adding a secondary additive Pn objective function defined by F2 = i=1 g(Xi ), where g(Xi ) is a unary cost function defined on variable Xi with costs generated uniformly at random between 0 and 10. Table 3 shows the results obtained for experiments with 14 WCSP instances using static mini-bucket heuristics. We did not report results with dynamic mini-bucket heuristics because of the prohibitively large computational overhead associated with 2
Available online at: ftp://dimacs.rutgers.edu/pub/challenge/satisfiability/benchmarks/cnf/
instance (n, c, k)
(w*, h) |E|
c432 (432, 648, 2) c499 (499, 748, 2) c880 (881, 1323, 2) s386 (172, 258, 2) s1423 (749, 1125, 2) s1488 (667, 1000, 2) s1494 (661, 991, 2) 90-14-1 (196, 392, 2) 90-16-1 (256, 512, 2) cpcs360b (360, 720, 2) blockmap-05-01 (700, 1400, 2) blockmap-05-02 (855, 1710, 2) blockmap-05-03 (1005, 2010, 2) mm-03-08-03 (1220, 2440, 2)
(27, 45) 31 (23, 55) 24 (27, 67) 53 (19, 44) 13 (24, 54) 119 (47, 67) 10 (48, 69) 14 (22, 66) 12 (24, 82) 8 (20, 27) 76 (18, 48) 1 (20, 53) 2 (22, 59) 2 (20, 57) 1
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MO-BB+SMB(i) MO-BB+SMB(i) MO-BB+SMB(i) MO-BB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) MO-AOBB+SMB(i) i=14 i=16 i=18 i=20 time nodes time nodes time nodes time nodes 38.99 96,805 25.15 59,330 31.21 62,597 39.95 50,753 25040.22 17,482,361 27633.93 20,140,468 5275.72 5,263,053 1611.60 1,266,736 21967.54 16,840,017 24407.10 19,498,060 4290.26 4,620,932 668.43 625,078 - 9088.16 9,723,797 3181.03 3,959,806 14.30 73,572 20.85 73,372 - 3990.74 1,193,420 4284.95 1,195,986 - 2510.72 11,708,708 1751.86 9,392,631 - 2794.69 17,273,392 2924.80 16,089,333 19.45 130,651 7.96 23,559 13.26 17,677 24.92 16,761 84.88 328.61 1,073,957 122.99 446,091 139.18 441,316 127,279 85.17 119,216 113.62 119,216 181.77 119,200 524.64 119,200 3.51 3,483 11.61 3,417 39.74 2,269 52.32 2,110 5.72 5,579 14.52 4,296 54.78 3,484 119.87 3,463 7.59 11,325 15.78 30,324 29.03 26,454 65.96 20,459 10.50 47,783 20.06 7,669 57.27 6,426 105.90 6,257
Table 3. Results for bi-objective WCSPs derived from standard mono-objective WCSPs using static mini-bucket heuristics. Time limit 10 hours. Overall best performance points are boxed.
relatively large i-bounds. We see that MO-AOBB+SMB(i) offers the overall best performance on this domain, outperforming its competitors by several orders of magnitude. For example, MO-AOBB+SMB(16) solves the 90-14-1 instance in about 8 seconds, whereas MO-BB+SMB(16) exceeds the 10 hour time limit. We also note that MO-RDS was not able to solve any of the test instance within the time limit, while MO-BB+SMB(i) could solve only one instance, namely c499. Summary of the Empirical Evaluation Our empirical evaluation on several classes of multi-objective optimization problems demonstrated conclusively that the AND/OR Branch-and-Bound tree search algorithms guided by static mini-bucket heuristics were the best performing algorithms overall. The difference between MO-AOBB+SMB(i) and the OR tree search counterpart MO-BB+SMB(i) was more pronounced at relatively small i-bounds (corresponding to relatively weak heuristic estimates) and amounted to several orders of magnitude in terms of both running time and size of the search space explored. For larger i-bounds, when the heuristic estimates are strong enough to prune the search space substantially, the difference between MO-AOBB+SMB(i) and MOBB+SMB(i) decreased. We also showed that MO-AOBB+SMB(i) was able in many cases to outperform dramatically recent state-of-the-art solvers for multi-objective optimization such as MO-RDS. With dynamic mini-bucket heuristics, MO-AOBB+DMB(i) proved competitive only for relatively small i-bounds due to computational overhead issues. This suggests that these heuristics can be considered when space is restricted. When comparing the AND/OR algorithms guided by mini-bucket heuristics with MORDS, we observed that the former were better on relatively sparse networks while on highly connected networks, the latter provided sometimes a better alternative.
6 Conclusion The paper investigates the impact of AND/OR search spaces for graphical models on multi-objective optimization. We introduced a general multi-objective AND/OR Branch-and-Bound algorithm and specialized it with two schemes for generating heuristic estimates that can guide the search. Our empirical evaluation on several benchmarks for MO-COP showed that the new AND/OR algorithms improved dramatically over traditional OR ones, in many cases by several orders of magnitude. Future work includes the extension of the algorithm to explore a search graph rather than a tree, via caching, as well as to investigate alternative control strategies such as best-first or frontier search in the context of MO-COPs. We also plan to apply the AND/OR search principle to the MO-RDS algorithm. Finally, we can exploit point quad-trees for the representation of the efficient frontier in order to access the data structure more efficiently.
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