Resolution-based lower bounds in MaxSAT

Transcription

1 Constraints (2010) 15: DOI /s Resolution-based lower bounds in MaxSAT Chu Min Li Felip Manyà Nouredine Ould Mohamedou Jordi Planes Published online: 24 July 2010 Springer Science+Business Media, LLC 2010 Abstract The lower bound (LB) implemented in branch and bound MaxSAT solvers is decisive for obtaining a competitive solver. In modern solvers like MaxSatz and MiniMaxSat, the LB relies on the cooperation of the underestimation and inference components. Actually, the underestimation component of some solvers guides the application of the inference component when a conflict is reached and certain structures are found. In this paper we analyze algorithmic and logical aspects of the underestimation components that have been implemented in MaxSatz during its evolution. From an algorithmic point of view, we define novel strategies for selecting Research partially supported by the Generalitat de Catalunya under grant 2009-SGR-1434, the Ministerio de Ciencia e Innovación research projects CONSOLIDER CSD , INGENIO 2010, Acción Integrada HA , TIN C04-04 and TIN C03-01, and the Ministerio de educación program Programa Nacional de Movilidad de Recursos Humanos. C. M. Li Hunan Normal University, Changsha, China C. M. Li N. O. Mohamedou MIS, Université de Picardie Jules Verne, 33 Rue St. Leu, Amiens Cedex 01, France C. M. Li chu-min.li@u-picardie.fr N. O. Mohamedou nouredine.ould@u-picardie.fr F. Manyà (B) Artificial Intelligence Research Institute (IIIA, CSIC), Campus UAB, Bellaterra, Spain felip@iiia.csic.es J. Planes Computer Science Department, Universitat de Lleida, Jaume II, 69, Lleida, Spain jplanes@diei.udl.cat

2 Constraints (2010) 15: unit clauses in UP(the underestimation of LB in UPis the number of independent inconsistent subformulas detected using unit propagation), the extension of UPwith failed literal detection, and a clever heuristic for guiding the application of MaxSAT resolution when UPaugmented with failed literal detection is applied in the presence of cycles structures. From a logical point of view, we prove that the inconsistent subformulas detected by UP are minimally unsatisfiable, but this property does not hold if failed literal detection is added. In the presence of cycle structures in conflicts detected by UP augmented with failed literal detection, we prove that the application of MaxSAT resolution produces smaller inconsistent subformulas and, besides, generates additional clauses that may be used to improve the LB. The conducted empirical investigation indicates that the LB techniques described in this paper lead to better quality LBs. Keywords MaxSAT Lower bound Inference Underestimation 1 Introduction The lower bound (LB) computation method implemented in branch and bound MaxSAT solvers (e.g. [8, 9, 15, 17, 19]) is decisive for obtaining a competitive solver [2]. The LB of modern solvers like MaxSatz [15] and MiniMaxSat [8] has two components: (i) the underestimation component, which detects independent inconsistent subformulas 1 and takes the number of detected subformulas as an underestimation of the LB, and (ii) the inference component, which applies inference rules and, in the best case, makes explicit a contradiction by deriving an empty clause which allows to increment the LB. Both components are applied at each node of the search space, and cooperate rather than work independently. Actually, in all the existing variants of MaxSatz [1, 5, 12, 15, 16] and in MiniMaxSat [8], the underestimation component guides the application of inference rules. More precisely, once a conflict is detected using unit propagation, MaxSAT resolution is only applied if the implication graph created by unit propagation contains certain structures. The identification of such structures has given rise to the definition of a collection of powerful inference rules [9, 15], which can be incorporated into the solvers with a low overhead, and are restrictions of the MaxSAT resolution rule [4, 9]. In this paper we analyze algorithmic and logical aspects of the underestimation components that have been implemented in MaxSatz during its evolution. The first underestimation procedure implemented in MaxSatz was UP[13]: the underestimation returned by UPis the number of independent inconsistent subformulas that can be detected using unit propagation. We present several algorithmic improvements for the original UP: we define different strategies for selecting unit clauses, and show empirically that they yield different performance profiles. From a logical point of view, we prove that the inconsistent subformulas detected by UP are minimally 1 The notion of independent subformulas is formally defined in Section 2.

3 458 Constraints (2010) 15: unsatisfiable. Then, we define an extension of UP, denoted by UP FL, based on adding failed literal detection to the search of conflicts. This way, the solver may detect inconsistent subformulas that are beyond the reach of unit resolution. The empirical investigation provides evidence that UP FL is, for a remarkable number of the tested instances, superior to UP. From a logical point of view, we prove that the inconsistent subformulas detected by UP FL are not necessarily minimally unsatisfiable. Finally, we define a heuristic for guiding the application of MaxSAT resolution when UP FL is applied in the presence of cycles structures (i.e., subformulas of the form l 1 l 2, l 1 l 3, l 2 l 3 ). Our analysis suggests that the best gains are obtained if MaxSAT resolution is applied when the cycle structure is contained in an inconsistent subformula, and that the exhaustive application of MaxSAT resolution to cycle structures is not sufficiently advantageous. From a logical point of view, we prove that, in the presence of cycle structures in conflicts detected by UP FL, the application of MaxSAT resolution produces inconsistent subformulas that are smaller than the inconsistent subformulas detected without applying MaxSAT resolution, and, besides, generates additional clauses that may be used to improve the LB. There are a number of related results in the literature: Shen and Zhang [20] defined a LB computation method which is similar to UP but restricted to Max- 2SAT instances and using a static variable ordering. Darras et al. [5] andhanetal. [16] have developed two versions of UP in which the computation of the LB is made more incremental. Simpler LBs that may be seen as particular cases of UP are surveyed in [10]. An alternative approach for computing underestimations is based on first reducing the MaxSAT instance one wants to solve to an instance of another problem, and then solving a relaxation of the obtained instance. For example, Clone [17, 18] and SR(w) [19] solve the minimum cardinality problem of a deterministic decomposable negation normal form (d-dnnf) compilation of a relaxation of the current MaxSAT instance. This paper, which is an extended version of the results published in [11, 12, 14], is structured as follows. Section 2 contains the definition of basic concepts which are used throughout the text. Section 3 describes UP and its variants, and proves that UP generates minimally unsatisfiable subformulas. Section 4 introduces UP FL, and proves that now the generated subformulas are not necessarily minimally unsatisfiable. Section 5 motivates the benefits of applying MaxSAT resolution to cycle structures contained in an inconsistent subformula detected by UP FL, proves that then the detected inconsistent subformulas are smaller than the inconsistent subformulas detected without applying MaxSAT resolution, and defines a clever heuristic for guiding the application of MaxSAT resolution. Section 6 reports on the empirical investigation conducted for comparing the different strategies defined for selecting unit clauses in UP,andUP FL, and for evaluating the application of MaxSAT resolution to cycle structures in different scenarios. Section 7 presents some concluding remarks. For the sake of clarity, we explain our work for unweighted MaxSAT, but the implementations and experiments include Weighted MaxSAT, Partial MaxSAT, and Weighted Partial MaxSAT. From the problem solving and knowledge representation points of view, MaxSAT formalisms with weights, and considering both hard and soft clauses, are better suited to deal with preferences.

4 Constraints (2010) 15: Preliminaries AmultisetA is a pair (A, f ), wherea is a set and f is a function, called mutiplicity function, from A to the natural numbers. The set A is called the underlying set of elements and, for each element a of A, f (a) is the number of occurrences of a in A. Note that f is a generalization of the notion of characteristic function. The sum of the multiset A = (A, f ) and the multiset B = (A, g), denoted by A B, is the multiset (A, h), whereh(a) = f (a) + g(a) for all a A. The multiset A = (A, f ) is a submultiset of the multiset B = (A, g), denoted by A B, iff f (a) g(a) for all a A. The subtraction of the multiset B = (A, g) from the multiset A = (A, f ), denoted by A \ B, is the multiset (A, h), whereh(a) = max( f (a) g(a), 0) for all a A (notice that h(a) = f (a) g(a) if B A). The submultisets A 1 = (A, f 1 ),...,A k = (A, f k ) of A = (A, f ) are independent iff A 1 A k A. We define CNF formulas as multisets of clauses because, in MaxSAT, duplicated clauses cannot be collapsed into one clause, and we say that the multiset φ is a subformula of a formula φ iff φ is a submultiset of φ. We define weighted CNF formulas as multisets of weighted clauses. A weighted clause is a pair (C i,w i ),where C i is a disjunction of literals and w i, its weight, is a positive number. The weighted clauses (C,w i ), (C,w j ) can be replaced with (C,w i + w j ). A literal l in a (weighted) CNF formula φ is failed if unit propagation derives a contradiction from φ {l} but not from φ. A truth assignment is a mapping that assigns either 1 (for true) or 0 (for false) to each propositional variable. A truth assignment v satisfies a literal x if v(x) = 1 and satisfies a literal x if v(x) = 0. A truth assignment satisfies a clause if it satisfies at least one of its literals. A CNF formula is satisfiable if there is a truth assignment that satisfies all its clauses, is unsatisfiable if it is not satisfied by any truth assignment, and is minimally unsatisfiable if it is unsatisfiable and all its proper subsets are satisfiable. The Unweighted MaxSAT problem for a CNF formula φ is the problem of finding a truth assignment v that maximizes (minimizes) the number of satisfied (unsatisfied) clauses in φ. 2 We say then that φ is a MaxSAT instance and v is a MaxSAT solution for φ. Actually, MaxSAT can be viewed as an infinite collection of instances together with a solution for every instance. MaxSAT instances φ 1 and φ 2 are equivalent if φ 1 and φ 2 have the same number of unsatisfied clauses for every complete assignment of φ 1 and φ 2. A MaxSAT inference rule is sound if it transforms an instance into an equivalent instance. The Weighted MaxSAT problem for a weighted CNF formula φ is the problem of finding an assignment that minimizes the sum of weights of unsatisfied clauses in φ. A Partial MaxSAT instance is a CNF formula in which some clauses are relaxable or soft and the rest are non-relaxable or hard. Solving a Partial MaxSAT instance amounts to finding an assignment that satisfies all the hard clauses and the maximum number of soft clauses. The Weighted Partial MaxSAT problem is the combination of Weighted MaxSAT and Partial MaxSAT. Every soft clause in a Weighted Partial MaxSAT instance has a weight, and solving this instance amounts to finding an assignment that satisfies all the hard clauses and minimizes the sum of weights of unsatisfied soft clauses. 2 In the sequel, we always refer to the minimization version of MaxSAT, also called MinUNSAT.

5 460 Constraints (2010) 15: Lower bound UP In the first subsection, we make precise the concepts involved in lower bound UP, and explain in detail how UP detects inconsistent subformulas from the conflict graphs created by unit propagation. We also prove that the subformulas detected by UPare minimally unsatisfiable. In the second subsection, we define three variants of UPthat we have designed and implemented in MaxSatz. 3.1 Description of lower bound UP We remind that the unit resolution rule establishes that from a unit clause l and a clause l l 1 l k, where l is the complementary literal of l, the clause l 1 l k can be inferred (i.e., l, l D D). A unit refutation is a derivation of the empty clause using the unit resolution rule. Given a CNF formula φ, unit propagation consists of applying unit resolution until a contradiction is found (i.e., until the empty clause is derived) or until a saturation state is reached. Alternatively, unit propagation can be seen as the construction of a directed acyclic graph, called implication graph, that is constructed as follows [6]: 1. Create a node for each unit clause in the input CNF formula φ, labeled with its literal, and associate the unit clause with that node. These will be the indegree zero root nodes of G. 2. While there exists a clause c = l 1 l k 1 l k such that l 1,...,l k 1 label nodes in G and there is no node in G labeled with l k, (a) Add a node labeled l k. In this case, we say that the unit clause l k is derived. (b) Add edges (l i,l k ) for every i (1 i < k). (c) Associate clause c with node l k. 3. Add to G a special node. For any variable p that occurs both positively and negatively in G, add directed edges from p and p to. In this case, we say that a a contradiction has been found. We refer to the complementary literals with directed edges to the special node as conflict literals, to the variable of the conflict literals as conflict variable, and to the rest of literals as non-conflict literals. We refer to the clauses of G implying l and l as conflicting clauses, and to the rest of clauses as non-conflict clauses. In the sequel, when we say that we propagate a unit clause l, whose literal has been added as a node of the implication graph, we mean that we remove the complementary literal (l) from the clauses in which it appears, and we add a node for each resulting unit clause, if not already present, labeled with the literal in the unit clause. A node l 1 is said to be descendant of a node l 2 if there is a directed path from l 2 to l 1.Ifl 1 is a descendant node of l 2,thenl 2 is a predecessor node of l 1. Figure 1 shows a pseudocode of unit propagation. The algorithm applies unit resolution until a contradiction is found or a saturation state is reached. The input is a CNF formula φ, and the output is the implication graph G resulting of applying unit propagation to φ. The time complexity of constructing G is linear in the size of φ [6].

6 Constraints (2010) 15: Fig. 1 Pseudocode of unit propagation Notice that the algorithm of Fig. 1 is non-deterministic: we have not fixed an ordering for propagating unit clauses. For the time being, we assume that unit clauses are stored in a queue, and that unit clauses are propagated following a first-in-firstout policy. We analyze this issue in more detail in Section 3.2, where we show that the shape and contents of the implication graph depend on the ordering in which unit clauses are propagated. Example 1 Let φ ={x 1, x 1 x 2, x 1 x 3, x 2 x 3 x 4, x 5, x 5 x 6, x 5 x 7, x 6 x 7 x 4, x 5 x 8 } be a MaxSAT instance. UnitPropagation(φ) returns the implication graph G of Fig. 2. Black nodes correspond to the unit clauses of φ. The conflict literals of G are x 4 and x 4, and the conflicting clauses of G are x 2 x 3 x 4 and x 6 x 7 x 4. Observe that, in order to derive a unit refutation from the implication graph G created by unit propagation, there is no need to consider the whole graph. Only the nodes of G having a directed path to the conflict are needed. More formally, we just Fig. 2 An example of implication graph

7 462 Constraints (2010) 15: Fig. 3 An example of conflict graph need to consider the nodes of the conflict graph of G, which is the subgraph H of G with the following properties: 1. H contains and exactly one conflict variable. 2. All nodes in H have a path to. 3. Every node l in H other than either corresponds to a starting unit clause or has precisely the nodes l 1, l 2,...,l k as predecessor nodes where l 1 l 2 l k l is a known clause. While an implication graph may or may not contain conflicts, a conflict graph always contains exactly one conflict. Notice that the clauses associated to the set of nodes of a conflict graph form an inconsistent subformula because they are the clauses needed to derive a unit refutation. In the sequel, we refer to such an inconsistent subformula as the subformula detected by unit propagation. Example 2 Let φ be the CNF formula from Example 1. The conflict graph corresponding to the the implication graph of Fig. 2 is shown in Fig. 3. Observe that clauses x 1, x 1 x 2, x 1 x 3 and x 2 x 3 x 4 have a directed path to x 4,andclausesx 5, x 5 x 6, x 5 x 7 and x 6 x 7 x 4 have a directed path to x 4. So, the inconsistent subformula detected by UPis {x 1, x 1 x 2, x 1 x 3, x 2 x 3 x 4, x 5, x 5 x 6, x 5 x 7, x 6 x 7 x 4 }. Given a MaxSAT instance φ, LB UP computes an underestimation of the minimum number of unsatisfied clauses in φ. To this end, UP applies repeatedly unit propagation, following the algorithm in Fig. 4, in order to detect independent Fig. 4 Pseudocode of lower bound UP

8 Constraints (2010) 15: inconsistent subformulas. If U P detects an inconsistent subformula S in φ, then it tries to detect another inconsistent subformula in φ = φ \ S.IfUPdetects an inconsistent subformula S in φ, then it tries to detect another inconsistent subformula in φ \ S, and so on. Once no more contradictions can be found by applying unit propagation, UPreturns the number of detected inconsistent subformulas as lower bound. The correctnessofupfollows from the fact that all the detected subformulas are independent. The time complexity of UP is linear in the size of φ and in the number of detected inconsistent subformulas. We now show that the inconsistent subformulas detected by applying unit propagation are minimally unsatisfiable. This means that the inconsistent subformulas detected by lower bound UPdo not contain redundant clauses. Proposition 1 The inconsistent subformulas detected by unit propagation are minimally unsatisf iable. Proof Let H be the conflict graph created by unit propagation when a contradiction is found in a given CNF formula, and let S be the detected inconsistent subformula. We prove that if we remove an arbitrary clause c of S, thens \{c} is satisfiable. We distinguish two cases: First, assume that we remove a conflicting clause in S, i.e., we remove a node labeling a conflict literal. Then, the literals labeling the remaining nodes in H are not conflicting any more, so that we can assign true to every remaining literal in H. Any clause l 1 l 2 l k 1 l k in S, associated with node l k, becomes satisfied because l k is set to true. So, S is satisfiable after a conflicting clause is removed. Second, assume that we remove a non-conflicting clause c = l 1 l 2 l k 1 l k associated with node l k in H, wherel k is a non-conflict literal. Then, we may set to false l k and all the non-conflict literals labeling the descendant nodes of l k.all clauses that can be reached from l k are satisfied, because they contain l k or l j,where l j is a descendant literal of l k, by construction of H. Observe that at least a conflicting clause contains a literal l j such that l j is a descendant literal of l k. The conflicting clause is satisfied because l j is set to false. Any other clause (that cannot be reached from l k ) can be satisfied by setting to true the literal labeling the corresponding node, because the literals labeling the corresponding nodes are not conflicting any more. In Fig. 3, if we remove the conflicting clause x 2 x 3 x 4 (i.e., if we remove the node x 4 ), we can satisfy all the clauses by assigning to true all the remaining literals in the figure. On the other hand, if we remove the non-conflicting clause x 1 x 2 (i.e., if we remove the node x 2 ), we can set to false x 2 to satisfy x 2 x 3 x 4, and to true the literals labeling all the other nodes to satisfy all the remaining clauses. 3.2 Strategies for selecting unit clauses in UP Unit propagation is performed by repeatedly picking a unit clause l to satisfy it. The satisfaction of the unit clause l implies the removal of all clauses containing l and the removal of l from all the clauses containing it, possibly resulting in new unit clauses. The conflict graph generated by unit propagation is generally not unique. Its contents and shape depend on the ordering in which unit clauses are propagated. In UP,the size and the clause occurrences of the detected inconsistent subformulas, as well as

9 464 Constraints (2010) 15: the number of detected subformulas, depend on the ordering in which unit clauses are propagated. So, the selection of that ordering may be an important aspect of the quality of the lower bound [14]. In this section we start by giving an example that illustrates the impact, in UP,of the ordering used to propagate unit clauses, and then we define three different orderings that we have designed and implemented in MaxSatz, and that are empirically compared in the experimental part. Example 3 Let φ be the MaxSAT instance {x 1, x 2, x 3, x 1 x 2 x 3, x 1 x 4, x 1 x 5, x 4 x 5, x 1 x 2 }. We show that, by varying the ordering in which unit clauses are propagated, we get different implication graphs. Moreover, we show that the quality of UPalso depends on that ordering. IfUPfirst propagates x 3, then propagates x 2, and then propagates x 1, the conflict graph created byupis the leftmost graph in Fig. 5, and the detected inconsistent subformula is {x 1, x 2, x 3, x 1 x 2 x 3 }. Since the rest of clauses in φ are non-unit, UPcannot find another contradiction and the underestimation returned by UP is 1. If UPfirst propagates x 1 (deriving the unit clauses x 4 and x 5 ), and then propagates x 4 (deriving the unit clause x 5 ), the conflict graph created by UPis the middle graph in Fig. 5. The subformula detected by UPis {x 1, x 1 x 4, x 1 x 5, x 4 x 5 }. Once this subformula is detected, UP is able to find another contradiction and detect another inconsistent subformula ({x 2, x 3, x 1 x 2 x 3, x 1 x 2 }). In this case, the underestimation returned by UPis 2. If UP first propagates x 1 (deriving the unit clauses x 4 and x 5 ), and then propagates x 5 (deriving the unit clause x 4 ), the conflict graph created by UP is the rightmost graph in Fig. 5. The subformula detected by UPis the same as the subformula detected for the middle graph: {x 1, x 1 x 4, x 1 x 5, x 4 x 5 }.As before, the underestimation returned by UPis 2. We define three variants of UP taking into account the propagation of unit clauses: UP Q : All unit clauses are stored in a queue, and are propagated in a first-infirst-out (FIFO) manner. In this case, input unit clauses are preferred to more recently derived unit clauses. UP : Two queues are maintained: Q 1 and Q 2.WhenUP starts the search for an inconsistent subformula, Q 1 contains all the unit clauses of φ (more recently derived unit clauses are at the end of Q 1 ), and Q 2 is empty. The unit clauses Fig. 5 Conflict graphs

10 Constraints (2010) 15: derived during the search are stored in Q 2, and the clauses in Q 1 are not propagated until Q 2 is empty. UP S : All unit clauses are stored in a stack and are propagated in a first-in-last-out (FILO) manner. In this case, the most recently derived unit clause is selected. Observe that UP Q propagates all the existing unit clauses in φ before propagating any newly derived unit clause. On the other hand, UP S and UP propagate all the unit clauses derived by an existing unit clause before propagating another unit clause. Let S be the set of unit clauses of an instance φ, andlett be the set of newly derived unit clauses after propagating a unit clause of S. On the one hand, UP and UP S propagate the unit clauses of T and all subsequently derived unit clauses before propagating another unit clause of S. If an empty clause is derived before propagating another unit clause of S, this avoids consuming other unit clauses of S in the derivation of an empty clause. On the other hand, UP Q propagates other unit clauses of S before propagating unit clauses of T. Ifanemptyclauseis derived, several unit clauses of S may be involved in the derivation of this empty clause. This analysis suggests that UP and UP S are of better quality than UP Q because they may consume fewer unit clauses of S when detecting an inconsistent subformula. As a result, unit propagation might be able to detect further inconsistent subformulas if there are more unit clauses in S. This might provide an explanation of the superiority of UP and UP S over UP Q in a considerable number of experiments of our empirical investigation. Nevertheless, as the following examples shows, UP and UP S, depending on the ordering of unit clauses in S, may consume more input unit clauses than UP Q in the detection of an inconsistent subformula. Example 4 Let φ ={x 1, x 1 x 2, x 2 x 3, x 3 x 4, x 4 x 5, x 5 x 6, x 6, x 6 x 7, x 6 x 8, x 7 x 8 },andlets ={x 1, x 6 }. Assume that x 1 is propagated first. The propagation of UP and UP S is as follows : x 1 x 2 x 3 x 4 x 5 x 6, detecting the inconsistent subformula {x 1, x 1 x 2, x 2 x 3, x 3 x 4, x 4 x 5, x 5 x 6, x 6 },which contains two unit clauses. However, UP Q propagates unit clauses as follows: at the beginning, Q={x 1, x 6 }. When x 1 is propagated, Q={x 6, x 2 }. When x 6 is propagated, Q={x 2, x 5, x 7, x 8 }.Whenx 2 is propagated, Q={x 5, x 7, x 8, x 3 }.Whenx 5 is propagated, Q={x 7, x 8, x 3, x 4 }.Whenx 7 is propagated, Q={x 8, x 3, x 4, x 8 }.When x 8 is propagated, an empty clause is derived, detecting the inconsistent subformula {x 6, x 6 x 7, x 6 x 8, x 7 x 8 }, which contains exactly one unit clause. In the previous example, if UP and UP S propagate x 6 before x 1,theyalsofindan inconsistent subformula containing exactly one unit clause. Actually, UP and UP S may find an inconsistent subformula not containing more unit clauses than UP Q if an optimal ordering is used to propagate the clauses of S. To see this, suppose that UP Q finds an inconsistent subformula containing unit clauses u 1, u 2,..., u k. Then, it is enough that UP and UP S begin by propagating u 1,andthenu 2 after propagating all the unit clauses implied by u 1, and so on. They will possibly find an inconsistent subformula containing even fewer unit clauses because of the newly derived unit clauses. Our experimental results show that, for the tested benchmarks, UP S and UP find more inconsistent subformulas than UP Q because they consume, in general, fewer input unit clauses. Now, we compare UP S and UP. The following example suggests that UP S tends to consume more clauses (not necessarily unit clauses) than UP when detecting

11 466 Constraints (2010) 15: inconsistent subformulas; i.e., UP S tends to find larger inconsistent subformulas than UP. Example 5 Let φ ={x 1, x 1 x 2, x 1 x 3, x 1 x 4, x 2 x 3, x 4 x 5, x 5 x 6, x 6 x 7, x 7 x 8, x 7 x 9, x 8 x 9 }. UP S : Initially, S ={x 1 }.Whenx 1 is propagated, unit clauses x 2, x 3,andx 4 are added to S (S ={x 4, x 3, x 2 }). When x 4 is propagated, unit clause x 5 is added to S (S ={x 5, x 3, x 2 }). When x 5 is propagated, unit clause x 6 is added to S (S = {x 6, x 3, x 2 }). When x 6 is propagated, unit clause x 7 is added to S (S ={x 7, x 3, x 2 }). When x 7 is propagated, unit clauses x 8 and x 9 are added to S (S ={x 9, x 8, x 3, x 2 }). When x 9 is propagated, unit clause x 8 is added to S. (S ={x 8, x 8, x 3, x 2 }). When x 8 is propagated, the empty clause is derived. The inconsistent subformula detected by UP S is {x 1, x 1 x 4, x 4 x 5, x 5 x 6, x 6 x 7, x 7 x 8, x 7 x 9, x 8 x 9 }, which contains 8 clauses. UP : Initially, Q 1 ={x 1 } and Q 2 is empty. When x 1 is propagated, unit clauses x 2, x 3,andx 4 are added to Q 2 (Q 2 ={x 2, x 3, x 4 }). When x 2 is propagated, unit clause x 3 is added to Q 2 (Q 2 ={x 3, x 4, x 3 }). When x 3 is propagated, the empty clause is derived. The inconsistent subformula detected by UP is {x 1, x 1 x 2, x 1 x 3, x 2 x 3 }, which contains 4 clauses. In an implication graph, the level in the graph of a node l corresponding to an input unit clause is 0, and the level in the graph of a node l, that has been added using the clause c = l l 1 l k because the nodes l 1,...,l k are already in the graph, is the maximum of the levels of l 1,...,l k plus 1. We believe that UP S tends to find larger inconsistent subformulas than UP because UP S always uses the node with the maximum level to add new nodes into the implication graph, so that the derived empty clause has a high level. However, UP always uses the node with the minimum level to add new nodes, so that the derived empty clause has a low level. The fact that an empty clause with high level has more chances of involving more clauses than an empty clause with low level could explain the behaviour observed in the previous example. Observe that the use of two queues in UP is to avoid propagating the input unit clauses (level 0) before other unit clauses. We would like to emphasize that UP Q, UP S,andUP are just heuristics. In Partial MaxSAT and Weighted Partial MaxSAT, the hard clauses in the inconsistent subsformulas detected by UP Q, UP S,andUP may produce different performance profiles for the following reasons: It is generally easier to find inconsistent subformulas in Partial MaxSAT and Weighted Partial MaxSAT because only the soft clauses of the detected inconsistent subformula are removed every time a contradiction is found. The hard clauses involved in any contradiction are used to detect further inconsistencies. Since there is always the same number of hard clauses for detecting inconsistencies, consuming one or two more soft unit clauses for deriving a contradiction may have a more limited impact in the performance profile. In particular, its impact could be more limited in instances where all the soft clauses are unit. When comparing the size of inconsistent subformulas containing hard clauses, we should take into account the number of soft clauses. It could be more useful detecting a larger inconsistent subformula containing very few soft clauses than

12 Constraints (2010) 15: detecting a smaller inconsistent subformula containing a larger number of soft clauses. 4 Augmenting UP with failed literal detection One limitation of UP is that it needs at least one unit clause for deriving a contradiction. One way to overcome this drawback is to augment UP with failed literal detection. The advantage is that the time complexity of determining whether a literal is failed is linear in the size of the formula [6], and it allows to detect inconsistent subformulas such as {x 1 x 2, x 1 x 2, x 1 x 2, x 1 x 2 }, which are beyond the reach of unit propagation. We augment UP with failed literal detection as follows: Given a MaxSAT instanceφ, we first computeup(φ).let S be the sum of the inconsistent subformulas detected by UP. Then, UP is applied to both (φ \ S) {x} and (φ \ S) { x} for a variable x occurring in φ \ S. IfUPfinds a contradiction in (φ \ S) {x} (i.e., x is a failed literal in φ) and another contradiction in (φ \ S) { x} (i.e., x is a failed literal in φ), then the sum of the two inconsistent subformulas detected by UP, once we have removed the unit clauses x and x, is an inconsistent subformula of φ (say, ϕ). This process is repeated for another variable x, but now the contradictions must be found in ((φ \ S) \ ϕ) {x } and ((φ \ S) \ ϕ) { x }. The correctness of the lower bound UP augmented with failed literal detection follows from the following proposition, whose result has been claimed before in [6]. Proposition 2 Let φ be a MaxSAT instance, let x be a variable of φ such that x and x are failed literals in φ, letϕ 1 be the inconsistent subformula detected by UP in φ {x}, and let ϕ 2 be the inconsistent subformula detected by UP in φ { x}. Then, (ϕ 1 ϕ 2 ) \ {x, x} is an inconsistent subformula of φ. Proof We can produce a proof of x by applying resolution to ϕ 1 \{x}, and a proof of x by applying resolution to ϕ 2 \{ x}. The proof of x (x) is obtained by removing, in the unit refutation that can be produced from the clauses in ϕ 1 {x} (ϕ 2 { x}), the resolution steps resolving the unit clause x ( x) and a clause of ϕ 1 (ϕ 2 ) containing the literal x (x). This way, the last resolution step derives the unit clause x (x) instead of the empty clause. Observe that now we apply resolution but not necessarily unit resolution. If we put the two proofs together and resolve x and x, we get a refutation from (ϕ 1 ϕ 2 ) \{x, x}. In the following, UP augmented with failed literal detection is denoted by UP FL, and the corresponding extensions of UP Q, UP,andUP S are denoted by UP Q FL, UP FL,andUPS FL. Observe that, while UP only identifies unit refutations, UP FL identifies non-unit refutations too. Since the number of variables occurring in many MaxSAT instances can be relatively large, the detection of failed literals in UP FL is applied to a reduced number of variables in practice [14]: Let Var(φ) be the set of propositional variables occurring in φ such that (i) they do not occur in unit clauses; and (ii) they have at least two positive occurrences and two negatives occurrences in binary clauses. UP FL is just applied to the variables in Var(φ).

13 468 Constraints (2010) 15: Failed literal detection is not applied to the variables occurring in unit clauses because they do not lead to a contradiction if UP FL is applied to φ (remind that UPis applied before searching for failed literals). The selection of variables with at least two positive occurrences and two negatives occurrences in binary clauses was determined empirically. These variables produce at least two new unit clauses when they are set to a truth value. In contrast toup, the following proposition shows that the inconsistent subformulas detected by UP FL are not necessarily minimally unsatisfiable. This result opens the door to devise simplifications techniques for reducing the size of the detected inconsistent subformula. Proposition 3 The inconsistent subformula detected by U P FL minimally unsatisf iable. are not necessarily Proof We provide a counterexample. Let φ be the MaxSAT instance {x 1 x 2, x 1 x 2, x 2 x 3, x 2 x 3, x 2 x 3, x 2 x 3 }.IfweapplyUP FL using x 1 as branching variable, then the inconsistent subformula detected by UP FL is the whole φ. Note that if we remove the clause x 1 x 2 and the clause x 1 x 2 from φ, the resulting formula is unsatisfiable. So, the inconsistent subformula detected by UP FL is not minimally unsatisfiable. 5 Guiding the application of inference to cycle structures In the previous sections, we have defined a number of methods for computing underestimations that improve the quality of LBs. As we have already explained, in solvers like MaxSatz and MiniMaxSat, the underestimation component guides the application of inference rules. In these solvers, inference is triggered if both a conflict is found, and a certain structure is identified. In contrast, there are solvers like Max- DPLL [9] in which the LB relies on applying inference rules without being guided by a powerful underestimation component. We say that a formula contains a cycle structure if it contains a subformula of the form l 1 l 2, l 1 l 3, l 2 l 3. Max-DPLL, MaxSatz and MiniMaxSat apply MaxSAT resolution to cycle structures but under certain conditions which are reviewed below (c.f. Section 5.2). Our objective now is to define a clever heuristic that guides the application of MaxSAT resolution to cycle structures when the underestimation component is UP FL, and show that our heuristic yields an improved LB. This section is organized as follows. Section 5.1 presents the MaxSAT resolution rule because the inference discussed in this paper relies on applying refinements of this rule. Section 5.2 surveys the application of MaxSAT resolution to cycle structures in contemporary solvers. Section 5.3 motivates the benefits of applying MaxSAT resolution to cycle structures contained in an inconsistent subformula detected by UP FL, and proves that the detected inconsistent subformulas are smaller than the inconsistent subformulas detected without applying MaxSAT resolution, and additional clauses, that may improve the quality of the LB, are generated. Section 5.4 defines a clever heuristic for guiding the application of MaxSAT resolution when the underestimation component is UP FL.

14 Constraints (2010) 15: Fig. 6 The MaxSAT resolution rule 5.1 The MaxSAT resolution rule The resolution rule applied in SAT (i.e., l D, l D D D ) preserves satisfiability but not the number of unsatisfied clauses, and therefore cannot be applied to MaxSAT instances. Independently and in parallel, Bonet et al. [3, 4], and Heras and Larrosa [7] defined a MaxSAT resolution rule that preserves, for every possible truth assignment, the number of unsatisfied clauses (i.e., it is sound). Figure 6 shows the MaxSAT resolution rule. Applying the rule amounts to replacing the premises with the conclusion. If the conclusion is added to the premises as in SAT, the number of clauses which are falsified by an assignment might increase. The tautologies concluded by the rule are removed, and the repeated literals in a clause are collapsed into one. Moreover, Bonet et al. [3, 4] proved the completeness of MaxSAT resolution: By saturating successively w.r.t. all the variables, one derives as many empty clauses as the minimum number of unsatisfied clauses in the MaxSAT input instance. Saturating w.r.t. a variable amounts to applying the MaxSAT resolution rule to clauses containing that variable until every possible application of the inference rule only introduces clauses containing that variable (since tautologies are eliminated). Once a MaxSAT instance is saturated w.r.t. a variable, all the clauses containing that variable are not considered to saturate w.r.t. another variable. We refer to [4] for further technical details and for the weighted version of the rule. In the next section we present sound, but not complete, inference rules which are refinements of the MaxSAT resolution rule. They have the advantages that they can be applied very efficiently. 5.2 Cycle structures in inference rules MaxSatz applies MaxSAT resolution when UPdetects a conflict, and the generated subformula captures certain structures. More precisely, MaxSatz incorporates the following rules (also called Rules (1), (2), (3), and (4) in this paper): l 1, l 1 l 2, l 2 =, l 1 l 2 (1) l 1, l 1 l 2, l 2 l 3,..., l k l k+1, l k+1 =, l 1 l 2, l 2 l 3,..., l k l k+1 (2)

15 470 Constraints (2010) 15: l 1, l 1 l 2, l 1 l 3, l 2 l 3 =, l 1 l 2 l 3, l 1 l 2 l 3 (3) l 1, l 1 l 2, l 2 l 3,..., l k l k+1, l k+1 l k+2, l k+1 l k+3, l k+2 l =, l 1 l 2, l 2 l 3,..., l k l k+1, k+3 l k+1 l k+2 l k+3, l (4) k+1 l k+2 l k+3 Observe that MaxSatz, via Rules (3) and(4), applies MaxSAT resolution to cycle structures only if they are contained in an inconsistent subformula generated by UP. In contrast, Max-DPLL applies MaxSAT resolution to all the cycle structures occurring in a MaxSAT instance, and does not combine its application with the underestimation component. Max-DPLL implements, among others, the cycle resolution rule: l 1 l 2, l 1 l 3, l 2 l 3 = l1, l 1 l 2 l 3, l 1 l 2 l 3 (5) MiniMaxSat incorporates LB UP, and applies MaxSAT resolution to the inconsistent subformula detected by UPif the largest resolvent in the refutation has arity less than 4; otherwise, it just increments the underestimation. Therefore, MiniMaxSat applies MaxSAT resolution to cycles structures contained in certain inconsistent subformulas. 5.3 Cycle structures in UP FL The next example illustrates the usefulness of applying MaxSAT resolution to cycle structures occurring in MaxSAT instances not containing unit clauses, and using a solver that incorporates UP FL and the inference rules of MaxSatz (Rules (1), (2), (3), and (4)). Example 6 Assume that a MaxSAT instance φ contains x 1 x 2, x 2 x 3, x 2 x 4, x 3 x 4, x 1 x 5, x 5 x 6, x 1 x 7, x 6 x 7 x 8 x 2, x 8 x 3, x 8 x 4, x 8 x 9, x 8 x 10, x 8 x 11, x 9 x 10 x 11 Rules (3) and(4) cannot be applied because φ does not contain unit clauses. Failed literal detection on the variable x 1 finds the inconsistent subformula in the first line. After removing this subformula, it cannot detect further inconsistent subformulas. The underestimation is only incremented by 1. However, if MaxSAT resolution is applied to x 2 x 3, x 2 x 4, x 3 x 4, these clauses are replaced with x 2, x 2 x 3 x 4, x 2 x 3 x 4, and then the underestimation component detects 2 inconsistent subformulas instead of 1. The first with failed literal detection on the variable x 1 : x 1 x 2, x 2, x 1 x 5, x 5 x 6, x 1 x 7, x 6 x 7 and the second with failed literal detection on the variable x 8 : x 8 x 2, x 8 x 3, x 8 x 4, x 8 x 9, x 8 x 10, x 8 x 11, x 9 x 10 x 11, x 2 x 3 x 4 Example 6, together with the analysis of Rules (3)and(4), shows that, in this case, applying MaxSAT resolution to cycle structures contained in an inconsistent subformula improves the quality of LBs. We now generalize this observation by proving

16 Constraints (2010) 15: that if we apply MaxSAT resolution to cycle structures contained in an inconsistent subformula, apart from incrementing the underestimation by 1, this transformation generates two new ternary clauses that can be used to derive additional inconsistent subformulas, allowing to obtain better LBs. To prove that results, we first need to prove the following lemmas: Lemma 1 Let l be a failed literal in a MaxSAT instance φ, andlets l be the set of clauses detected by U P FL when proving that l is failed. If S l contains the cycle structure l 1 l 2, l 1 l 3, l 2 l 3,thenl 1 was set to true during the application of unit propagation. Proof Except for the empty clause, every clause in S l becomes unit when it is used for propagation, meaning that every clause in S l is satisfied by at most one literal in the unit propagation. If l 1 was set to false in the propagation, then at least one of the three clauses l 1 l 2, l 1 l 3, l 2 l 3 would be satisfied by two literals and could not belong to S l. So, l 1 was set to true during unit propagation. Lemma 2 If l is a failed literal in a MaxSAT instance, and S l contains the cycle structure l 1 l 2, l 1 l 3, l 2 l 3,thenl 1 was assigned a truth value before l 2 and l 3 during the application of unit propagation. Proof Except for the empty clause, every clause in S l was unit when it was satisfied in the unit propagation. We assume that l 2 was assigned a truth value before l 1 and show that this is impossible. If l 2 was assigned true, clause l 1 l 2 would be satisfied without being unit; if l 2 was assigned false, then l 3 would be assigned true before l 2 was assigned false, since otherwise clause l 2 l 3 would be satisfied before being unit. But in the latter case, clause l 1 l 3 would be satisfied without being unit. Lemma2alsomeansthatifS l contains a cycle structure, then the cycle structure must be formed by the last three binary clauses in the implication graph detecting S l, which makes the identification of the cycle structure in S l fast and easy. Proposition 4 Let l be a failed literal in a MaxSAT instance φ, andlets l be the set of clauses detected by U P FL.IfS l contains the cycle structure l 1 l 2, l 1 l 3, l 2 l 3, and S l is S l after applying MaxSAT resolution to the cycle structure, then S l \{l 1 l 2 l 3, l 1 l 2 l 3 } is inconsistent. Proof By Lemmas 1 and 2, l 1 was assigned true in UP(φ {l}) independently of the three clauses l 1 l 2, l 1 l 3, l 2 l 3, which are replaced with { l 1, l 1 l 2 l 3, l 1 l 2 l 3 } in S l. Therefore, unit propagation in S l \{l 1 l 2 l 3, l 1 l 2 l 3 } derives an empty clause from the unit clause l 1. Proposition 4 means that, if both l and l are failed literals, and S l contains the cycle structure l 1 l 2, l1 l 3, l2 l 3, we can apply MaxSAT resolution in φ (replacing these three binary clauses with one unit clause l 1 and two ternary clauses l 1 l 2 l 3 and l 1 l 2 l 3 ) to obtain S l, and then transform the inconsistent

17 472 Constraints (2010) 15: subformula (S l S l ) \{l, l} into a smaller inconsistent subformula (S l S l ) \{l, l, l 1 l 2 l 3, l 1 l 2 l 3 } of φ. So, apart from incrementing the underestimation by 1, this transformation generates two ternary clauses from (S l S l ) \{l, l} that can be used to derive additional independent inconsistent subformulas, allowing to obtain better LBs. In Example 6, (S x1 S x1 ) \{ x 1, x 1 } is equal to {x 1 x 2, x 2 x 3, x 2 x 4, x 3 x 4, x 1 x 5, x 5 x 6, x 1 x 7, x 6 x 7 } which is transformed by applying MaxSAT resolution to the cycle structure x 2 x 3, x 2 x 4, x 3 x 4 into a smaller inconsistent subformula {x 1 x 2, x 2, x 1 x 5, x 5 x 6, x 1 x 7, x 6 x 7 } So, after incrementing the underestimation by 1, we have two additional ternary clauses (x 2 x 3 x 4 and x 2 x 3 x 4 ) generated from the cycle structure which can be used to detect further inconsistent subformulas. The benefit of applying MaxSAT resolution to a cycle structure not contained in an inconsistent subformula is not so clear. The next example, as well as the results reported in the empirical part, provide evidence that this application may lead to a worse LB. Example 7 Assume that a MaxSAT instance φ contains x 1 x 2, x 1 x 3, x 2 x 3, x 2 x 6, x 3 x 6, x 1 x 7, x 1 x 8, x 7 x 9, x 8 x 9, x 4, x 4 x 2, x 3 x 5, x 5 Without activating the inference component, UP FL subformula detects an inconsistent {x 2 x 3, x 4, x 4 x 2, x 3 x 5, x 5 } Then, after removing this subformula from φ, failed literal detection on x 1 detects the second inconsistent subformula {x 1 x 2, x 1 x 3, x 2 x 6, x 3 x 6, x 1 x 7, x 1 x 8, x 7 x 9, x 8 x 9 } Note that the first three clauses of φ form a cycle structure but do not belong to a same inconsistent subformula detected by UP FL. If MaxSAT resolution is applied to the cycle structure, φ becomes x 1, x 1 x 2 x 3, x 1 x 2 x 3, x 2 x 6, x 3 x 6, x 1 x 7, x 1 x 8, x 7 x 9, x 8 x 9, x 4, x 4 x 2, x 3 x 5, x 5. Once UP FL detects the inconsistent subformula {x 1, x 1 x 2 x 3, x 4, x 4 x 2, x 3 x 5, x 5 }

18 Constraints (2010) 15: φ becomes (after removing the inconsistent subformula) {x 1 x 2 x 3, x 2 x 6, x 3 x 6, x 1 x 7, x 1 x 8, x 7 x 9, x 8 x 9 } and is consistent. So, only one inconsistent subformula is detected when MaxSAT resolution is applied to the cycle structure, making the LB worse. 5.4 Heuristics for applying MaxSAT resolution in cycle structures The previous analysis suggests that it is better to apply MaxSAT resolution to cycle structures contained in an inconsistent subformula detected by UP FL than applying it exhaustively. In practice, when we identify a cycle structure at a node of the search tree, we distinguish three cases: 1. The cycle structure is contained in an inconsistent subformula. 2. The cycle structure is not contained in an inconsistent subformula at the current node, but probably belongs to an inconsistent subformula in the subtree below the current node. 3. The cycle structure is not contained in an inconsistent subformula at the current node, and will not probably belong to an inconsistent subformula in the subtree below the current node. We define a heuristic that applies MaxSAT resolution in the first two cases. As we will see, the benefit of applying MaxSAT resolution in the second case is twofold: two ternary clauses are generated in advance, and the probable inconsistent subformula containing the cycle structure will be easier and faster to detect in the subtree with the application of MaxSAT resolution. This heuristic is implemented in Algorithm 1, where occ2(l) is the number of occurrences of literal l in the binary clauses of φ. Between the two literals of a variable x that have reasonable probability to be failed (since their satisfaction results in at least two new unit clauses), Algorithm 1 considers first the literal l with more occurrences in binary clauses. Note that l has a smaller probability of being failed than l since its satisfaction produces fewer new unit clauses than the satisfaction of l. If l is a failed literal and S l contains a cycle structure, the cycle structure is replaced to obtain S l before detecting l. If l also is a failed literal, the inconsistent subformula (S l S l ) \{l, l} is transformed into a smaller inconsistent subformula to generate two ternary clauses. If l is not a failed literal in the current node, failed literal detection does not detect an inconsistent subformula containing the cycle structure of S l, but S l is now smaller thanks to MaxSAT resolution because it becomes now S l \{l 1 l 2 l 3, l 1 l 2 l 3 } by Proposition 4, and it will be easier to re-detect in the subtree. Note that l has reasonable probability to be a failed literal in the subtree, i.e., the cycle structure in the original S l has reasonable probability to be contained in an inconsistent subformula in the subtree. As soon as l fails in the subtree, Algorithm 1 will detect the smaller inconsistent subformula (S l S l ) \{l, l} with smaller cost (remind that S l is now smaller).

19 474 Constraints (2010) 15: On the contrary, if l is not a failed literal, Algorithm 1 does not detect an inconsistent subformula, l is not detected and no inference is applied to S l even if l is a failed literal, avoiding the application of MaxSAT resolution to a cycle structure not contained in an inconsistent subformula. 6 Empirical investigation and analysis The first block of experiments was devoted to compare the different variants of UP and UP FL on the random Max-2SAT and Max-Cut instances considered in [13], as 1000 Max-2SAT variables 1e+07 Max-2SAT variables time (log scale) UP Q UP S 0.1 UP S FL UP Q FL UP * 0.01 UP * FL number of clauses branches (log scale) 1e UP S UP S FL UP Q 1000 UP * UP Q FL 100 UP * FL number of clauses Fig. 7 Max-2SAT: mean time (left plot) and mean number of branches (right plot)