A New Approach to Proving Upper Bounds for MAX-2-SAT
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1 A New Approach to Proving Upper Bounds for MAX-2-SAT Arist Kojevnikov Alexander S. Kulikov Abstract In this paper we present a new approach to proving upper bounds for the maximum 2-satisfiability problem (MAX-2-SAT). We present a new 2 K/5.5 -time algorithm for MAX-2-SAT, where K is the number of clauses in an input formula. We also obtain a 2 N/6 bound, where N is the number of variables in an input formula, for a particular case of MAX-2-SAT, where each variable appears in at most three 2-clauses. This immediately implies a 2 N/6 bound, where N is the number of vertices in an input graph, for the independent set problem on 3-regular graphs. The key point of our improvement is a combined complexity measure for estimating the running time of an algorithm. By using a new complexity measure we able to provide a much simpler proof of new upper bounds for MAX-2-SAT than proofs of previously known bounds. 1 Introduction 1.1 The MAX-SAT and MAX-2-SAT Problems. The maximum satisfiability problem (MAX- SAT) is: Given a formula in conjunctive normal form (CNF), find a maximal possible number of simultaneously satisfiable clauses of this formula. It is one of the most important optimization problems (MAX-SAT is a strengthening of the well-known satisfiability problem). Many practical and theoretical problems can be formulated in terms of MAX-SAT. For example, the maximum cut, vertex cover and maximum independent set problems can be reduced to MAX-2-SAT [14]. Also, MAX-SAT has applications in artificial intelligence and database systems [1, 6]. 1.2 Upper bounds. The decision version of MAX- SAT (where one is asked whether there exists an assignment to variables of a given formula satisfying at arist@logic.pdmi.ras.ru. St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, , St. Petersburg, Russia. Research supported in part by Russian Science Support Foundation, RFBR grant , and INTAS grant kulikov@logic.pdmi.ras.ru. St. Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, , St. Petersburg, Russia. Research supported in part by grant NSh of the President of Russian Federation for Leading Scientific School Support and INTAS grant Table 1: Known upper bounds for MAX-SAT and MAX-2-SAT MAX-SAT MAX-2-SAT N 2 N 2 N/1.261 [16] K 2 K/2.465 [3] 2 K/5.217 [9] L 2 L/6.892 [2] 2 L/ [7] least a given number of clauses) is NP-complete even in case, where each clause of an input formula contains at most two literals (this particular case of the MAX-SAT problem is called MAX-2-SAT). Moreover, it is known [15] that the decision version of MAX-2- SAT is NP-complete, if each variable appears in an input formula at most three times. In Table 1 we give the best known upper bounds for MAX-SAT and MAX-2- SAT w.r.t. the three standard formula complexity measures, namely the number of variables N, the number of clauses K, and the length L of an input formula. For simplicity, throughout all the paper we give exponential upper bounds without polynomial factors. Almost all algorithms from Table 1 (as well as our new algorithm) are based on a splitting method and thus use polynomial space. The only exception is the algorithm by Williams [16], which is based on different ideas (finding triangles in an auxiliar graph) and uses exponential space. The William s bound is the first improving the trivial 2 N bound. Recently, by proving a 2 N/6 bound for the pathwidth of cubic graphs Fomin et al. [5] obtained an exponential-space 2 N/6 -time algorithm for the independent set problem (and several other NP-hard graph problems). By using this result Kneis et al. [8] designed an exponential-space 2 K/ time algorithm for MAX- 2-SAT. 1.3 Our Result. In this paper we present a new 2 K/5.5 upper bound for MAX-2-SAT. The key point of our improvement is a combined complexity measure for estimating the running time of an algorithm. Let us explain informally what advantages one can get by using non-classical complexity measures (all formal definitions and proofs are given in the following sections).
2 Let (i, j)-literal denote a literal that occurs in a formula in i clauses positively and in j clauses negatively. Note that an (i, j)-literal provides an (i, j)-splitting w.r.t. the number of clauses K. Now let (i, j)-literal denote a literal occurring in i 2-clauses positively and in j 2-clauses negatively. Then an (i, j)-literal immediately gives an (i + j, i + j)-splitting w.r.t. the number of 2-clauses K 2. Moreover, if one proves an upper bound c K2 for MAX- 2-SAT, then a bound c K also holds, as K 2 K for any formula in 2-CNF. So, estimating the complexity of an algorithm not in terms of K, but in terms of K 2 is more convenient. Gramm et al. [7] use this idea to design a 2 K/5 -time algorithm for MAX-2-SAT. We also use a non-standard measure to prove a 2 K/5.5 upper bound for MAX-2-SAT. Our measure is obtained from K 2 by eliminating the dependency on variables that occur in at most two 2-clauses (such variables can be eliminated from a formula by simplification rules) and reducing the dependency on variables that occur in either three or four 2-clauses. It is known [14] that the maximum independent set problem can be easily reduced to MAX-2-SAT. For graph with n vertices and K edges, the reduction produces a 2-CNF formula having n variables and K 2-clauses. The maximum independent set problem is known to be NP-complete even for 3-regular graphs. Our 2 N/6 bound for the particular case of MAX-2- SAT where each variable appears in at most three 2- clauses in an input formula implies a 2 N/6 bound for the independent set problem on 3-regular graphs. 1.4 Automated Proofs of Upper Bounds. The algorithm of this paper is based on a splitting method. Currently best known upper bounds for many NP-hard problems are proven by using exactly this method. Usually the analysis of a splitting algorithm contains a big number of cases. This implies that many modern papers describing splitting algorithms are quite difficult to read and verify. For this reason, we also used a computer for analyzing the cases corresponding to the proof of the new bound. More precisely, we asked a computer to prove that the MAX-2-SAT problem can be solved in 2 N/6 time for formulas containing only variables occurring in at most three 2-clauses (the main case analysis of this paper is devoted to exactly this case). The automated proof of this fact is available at kulikov/autoproofs. The used framework for automated proofs of upper bounds on the running time of splitting algorithms is described in detail in [4]. This program uses the so-called general simplification rule [10], which generalizes such simplification rules as pure literal, dominating unit clause, almost dominating unit clause, rare variable. Note that the automated proof contains a list of bottleneck cases, which can be useful for further improving upper bounds for MAX-2-SAT. 2 General Setting 2.1 Basic Definitions. Let V be a set of Boolean variables. By v we denote the negation of a variable v V. A literal is either a Boolean variable or its negation. For a literal l, by var(l) we denote the variable corresponding to l. A clause is a disjunction of several literals l 1, l 2,..., l n, such that var(l i ) var(l j ) for i j. The length of a clause is the number of its literals. A k-clause is a clause consisting of exactly k literals. A formula in conjunctive normal form (CNF) is a conjunction of clauses. The length of a formula is the number of literals in it, that is, the sum of the lengths of all its clauses. A formula in k-cnf is a formula in CNF containing only clauses of length at most k. Following the notation of [7], in this paper we allow a formula to contain also a special true clause T, which is satisfied by any assignment. Informally speaking, the number of such clauses is the number of already satisfied clauses. By OptV al(f ) we denote the maximal number of simultaneously satisfiable clauses of F. We say that a literal l occurs in a clause, if this clause contains l. However, when we say that a variable x occurs in a clause, we mean that this clause contains either x or x. By #(F, l) we denote the number of occurrences of a literal l in a formula F, by # k (F, l) we mean the number of k-clauses of F containing l. By weight of a variable we mean the number of 2-clauses containing this variable (thus, if x is a variable of weight k, then # 2 (F, x) + # 2 (F, x) = k). By V (F ) we denote the set of all variables of F. Neighbors of a variable x are all variables occurring in 2-clauses with x. For a set of variables V 0 V (F ), by N(F, V 0 ) we denote the set of all neighbors of variables from V 0 except for the variables from V 0. By G(F ) we denote the following undirected multi-graph: V (F ) is the set of vertices, two vertices are connected by an edge if their corresponding variables occur in a 2-clause in F (note that this definition allows multiple edges between two vertices). We usually omit F when it is clear from the context. For a formula F and a literal l, by F [l] we denote a formula obtained from F by the following way: replace each clause of F containing l by T, eliminate all occurrences of l from other clauses, eliminate all empty clauses from F. By F [l 1 = l 2 ] we mean the following formula: replace all occurrences of l 1 by l 2 and all occurrences of l 1 by l 2, replace all clauses that contain a literal together with its negation by T, for each clause containing a pair of equal literals remove one of these
3 literals from this clause. 2.2 Complexity Measures. Usually to prove an upper bound on the running time of a splitting algorithm one fixes a formula complexity measure and shows that the algorithm always splits the formula into several formulas each having smaller complexity than the initial formula. The natural complexity measures are the number of variables N, the number of clauses K and the length L of a formula. A lot of bounds w.r.t. these measures for the SAT and MAX-SAT problems and their restricted versions are known. Sometimes however it is possible to use a different complexity measure to prove an upper bound w.r.t. one of the standard complexity measures. Such a trick is used, e.g., by Gramm et al. [7] to prove an upper bound 2 K/5 for MAX-2-SAT and by Kullmann [11] to prove an upper bound N for 3-SAT. In this paper we also use a non-standard complexity measure to prove a better upper bound for MAX-2-SAT w.r.t. K, its construction is given below. Let F denote a 2-CNF formula F. By N i (F ) we denote the number of variables of F occurring in exactly i 2-clauses in F. It is easy to see that the following equality holds: K 2 (F ) = N(F ) i=1 i N i (F ) 2 where K 2 (F ) is the number of 2-clauses in F. Our new complexity measure is obtained by a slight change of coefficients before the first four N i : N(F ) γ(f ) = N 3 (F ) N 4 (F ) + i=5, i N i (F ) 2 Let us explain (informally) how this change of coefficients helps to improve an upper bound for MAX-2-SAT. Note that splitting on a variable of weight w in general case gives a (w, w)-splitting w.r.t. K 2 (as in both branches all 2-clauses containing this variable are eliminated or become 1-clauses). Now consider a variable x of weight 5. If all neighbors of x are different and have weight at most 5, then splitting on x is a (5.5, 5.5)- splitting w.r.t. γ, as in both branches x is eliminated (γ is reduced by 2.5) and the weight of all its neighbors is decreased (γ is reduced by 0.6 5). Our analysis in the following sections shows that we can guarantee such a splitting in all other cases. 2.3 Estimating the Running Time. For estimating the running time of an algorithm we use the wellknown technique of recurrent inequalities [12]. For a formula F, a literal l and a complexity measure µ, we. say that a splitting F [l], F [ l] is an (i, j)-splitting w.r.t. µ, if µ(f ) µ(f [l]) i and µ(f ) µ(f [ l]) j. A splitting number of an (i, j)-splitting is the unique positive root of the equation x i + x j = 1, we denote it by τ(i, j). It is known that if an algorithm always splits with a splitting number not exceeding some constant c, then its running time on a formula F is bounded by c µ(f ). Our algorithm always performs (4, 10)- and (5.5, 5.5)-splitting w.r.t. γ. This implies that it works in time 2 γ/5.5 (as = τ(4, 10) < τ(5.5, 5.5) = 2 1/5.5 = ). 3 Simplification Rules We use the procedure Simp for simplifying a formula, see Fig. 1. It is easy to see that the running time of this procedure is polynomial in the length of an input formula. Its correctness as well as several additional properties are given in Lemma 3.1. We call a formula simplified if the procedure Simp does not change it. Lemma 3.1. Let F be a formula in 2-CNF and let F = Simp(F ). Then the following hold: 1. OptV al(f ) = OptV al(f ); 2. F does not contain variables of weight at most 2; 3. for any variable x, # 2 (F, x) # 2 (F, x); 4. γ(f ) γ(f ); 5. let F be a simplified formula, a, x be neighbors in F, and a has weight 3; then at least in one of the formulas F [x] and F [ x] the simplification rules assign a Boolean value to a. Proof. 1. Let F be a formula and l be a literal of F such that either #(F, l) = 0 or # 1 (F, l) #(F, l). Note that for any assignment A for all variables of the formula F, such that l A, an assignment A\{ l} {l} satisfies not less clauses of F than A. This shows that the pure literal and dominating unit clause rules preserve OptV al(f ). The correctness of the frequently meeting variable rule follows from the correctness of the pure literal and dominating unit clause rules. The almost common clauses rule is correct as any assignment satisfies at least one of the clauses C and D, and an assignment satisfies both of them iff it satisfies C\{l}. 2. F does not contain variables of weight at most 2, as all such variables can be eliminated by the frequently meeting variables rule.
4 Procedure Simp Input: A formula F in 2-CNF. Output: A simplified formula F in 2-CNF. Method. 1. Almost common clauses. If F = F 0 {C, D}, where C\{l} = D\{ l} for a literal l, then return Simp(F 0 {C\{x}, T }). 2. Pure literal. If #(F, l) = 0 for a literal l F, then return Simp(F [l]). 3. Dominating unit clause. If # 1 (F, l) #(F, l) for a literal l F, then return Simp(F [l]). 4. Frequently meeting variables. Let F contains two variables x 1 and x 2, such that x 1 occurs in F in at most one 2-clause without x 2. Note that after assigning x 2 a Boolean value, x 1 appears in at most one 2-clause, so either the pure literal or dominating unit clause rule assigns a Boolean value to x 1 (before that the almost common clauses rule possibly removes several unit clauses with x 1 ). Let α, β be Boolean values that the simplification rules assign to x 1 in F [x 2 ] and F [ x 2 ], respectively. Depending on the values of α and β the frequently meeting variables rule simplifies the formula F as follows: if α = 0, β = 0, then return Simp(F [ x 1 ]) if α = 0, β = 1, then return Simp(F [x 1 = x 2 ]) if α = 1, β = 0, then return Simp(F [x 1 = x 2 ]) if α = 1, β = 1, then return Simp(F [x 1 ]) 5. Return F. Figure 1: A procedure for simplifying a formula 3. It is easy to see that assigning a Boolean value to a variable as well as replacing two clauses by their common part do not increase the weight of variables. Let us show that the frequently meeting variables rule also does not increase the weight of variables. The only variable whose weight can possibly be increased is x 2. However, F [x 1 = x 2 ] (as well as F [x 1 = x 2 ]) does not contain 2-clauses of F consisting of x 1 and x 2 (at least one clause) and contains at most one new 2-clause (as x 1 appears without x 2 at most once). 4. Follows immediately from 3 and the definition of γ. 5. To prove this we need to consider several cases of other occurrences of the variable a. There are no other possible cases as F is simplified. (a) (ax)(a?)(ā?) after assigning x = 0 the dominating unit clause rule assigns a = 1; (b) (ax)(ā?)(ā?) after assigning x = 1 the pure literal rule assigns a = 0; (c) (ax)(a?)(a?)(ā) after assigning x = 0 the almost common clauses rule eliminates (a) and (ā), and then the pure literal rule assigns a = 1; (d) (ax)(a?)(a?)(ā)(ā) after assigning x = 1 the dominating until clause rule assigns a = 0; (e) (ax)(a?)(ā?)(ā) after assigning x = 1 the dominating unit clause rule assigns a = 0; (f) (ax)(ā?)(ā?)(a) after assigning x = 0 the dominating unit clause rule assigns a = 1. 4 Algorithm Our new algorithm is presented in Fig. 2. It is a typical splitting algorithm, that is, it first simplifies an input formula, then splits the simplified formula into two simpler formulas and recursively calls for them, and finally returns the answer according to the answers returned by both recursive calls. An upper bound on the running time of the algorithm is proven in Theorem 4.1. Theorem 4.1. Given a formula in 2-CNF, the algorithm MaxSatAlg returns the maximal possible number of simultaneously satisfiable clauses of this formula in time 2 K/5.5, where K is the number of clauses in the formula. Proof. The correctness of MaxSatAlg follows immediately from the correctness of the procedure Simp. To prove the stated upper bound we show that MaxSatAlg always splits with a splitting number not exceeding
5 Algorithm MaxSatAlg Input: A formula F in 2-CNF. Output: OptV al(f ). Method. 1. F = Simp(F ) 2. if F contains only (T ) clauses, then return L(F ) 3. if F = F 1 F 2, where F 1 and F 2 do not share variables, then return MaxSatAlg(F 1 ) + MaxSatAlg(F 2 ) 4. choose the variable x such that τ(γ(f ) γ(simp(f [x])), γ(f ) γ(simp(f [ x]))) is minimal and return max(m axsatalg(f [x]), M axsatalg(f [ x])) Figure 2: A 2 K/5.5 -Algorithm for MAX-2-SAT 2 1/5.5 w.r.t. γ. This is sufficient as γ(f ) K(F ) for any 2-CNF formula F. Consider a 2-CNF formula F, such that F is simplified and does not contain closed subformulas. Note that F does not contain variables of weight at most two. Below we consider several cases depending on the maximal weight of the variables of F. In each case we show that F contains a variable such that splitting on it is at least (5.5, 5.5)-splitting w.r.t. γ. Suppose that F contains a variable x of weight w 6. Observe that all variables of F (except for x) appear in at least two 2-clauses without x (otherwise, the frequently meeting variables rule is applicable). Also, reducing the weight of a variable of weight at least 3 reduces γ at least by 1/2 (as the difference between any two coefficients of N i in the definition of γ is at least 1/2). Thus, after assigning x a Boolean value, γ is reduced at least by w/2 (as x is eliminated) plus w/2 (as the weight of the neighbors of x is reduced). This means that splitting on x is at least (6, 6)-splitting w.r.t. γ. Now suppose that F contains only variables of weight at most 5. Let x be a variable of weight exactly 5. For 1 i j 5, let k ij denote the number of neighbors of x, that have weight j and occur i times with x. Since F is simplified, k ij = 0 for all i, j such that either j 2 (as the simplification rules eliminate variables of weight at most 2) or j i 1 (as the frequently meeting variables rule eliminates such variables). Thus, only the following k ij -s may have positive values: k 13, k 14, k 15, k 24, k 25, k 35. Since x occurs in five 2-clauses, the following equality holds: k 13 + k 14 + k k k k 35 = 5. Note that if a variable has weight j and occurs i times with x, after assigning x a Boolean value it occurs in (j i) 2-clauses. Let F be a formula obtained from F by assigning a Boolean value to x. Then, γ(f ) γ(f ) = (k (k 14 + k 24 ) + 2.5(k 15 + k 25 + k )) (k 14 + k k 15 ) 0.6(k 13 + k 14 + k k k k 35 ) = = 5.5. This shows that splitting on x is at least (5.5, 5.5)- splitting w.r.t. γ (as the procedure Simp does not increase γ). The case when F contains a variable of weight 4 (and no variables of higher weight) is similar. The condition that a variable occurs in four 2-clauses looks as follows: k 13 + k k 24 = 4, the difference of complexities is estimated as follows: γ(f ) γ(f ) = (k (k 14 + k )) (k 14 ) 0.9(k 13 + k k 24 ) = = 5.5. The only remaining case is when F contains only variables of weight 3. This case is handled in Theorem 4.2. Theorem 4.2. If a simplified formula F contains only variables of weight 3, then F contains a variable, such that splitting on it is at least (6, 6)-splitting w.r.t. γ. Proof. It is easy to see that γ(f ) = N(F ), so in the following we show that it is possible to find a (6, 6)- splitting w.r.t. the number of variables. Note that any formula consisting of variables of weight 3 only contains even number of variables. Also, if a formula is simplified, then it contains only variables of weight 3. Thus, each splitting in F is a (2k, 2l)-splitting for some integers k and l. Moreover, splitting on any variable of F immediately gives us a (4, 4)-splitting, since each variable has exactly three neighbors (as the frequently meeting variables rule is not applicable) and the simplification rules eliminate variable that occurs in less than three 2-clauses. In the following we show that F contains a variable x, such that splitting on it is either (4, 10)- or (6, 6)-splitting w.r.t. N. First, suppose that F contains three variables, such that any two of them appear in a 2-clause (such variables form a triangle in G(F )). Let a, b, c be the variables of this triangle. Observe that N({a, b, c}) contains at least two variables (otherwise, a, b, c and its neighbor form
6 least in one of the branches a, b, y, c, z are assigned Boolean values, all its neighbors are further eliminated). Figure 3: Main cases a closed subformula). At least on of these neighbors appears with only one variable from {a, b, c}, call it d and suppose w.l.o.g. that it occurs with a (see Fig. 3(a)). We state that splitting on d is a (6, 6)- splitting. After assigning d a Boolean value, the variable a appears only in 2-clauses with b and c and possibly in some unit clauses. The simplification rules either assign a Boolean value to a or replace both these 2-clauses by a clause consisting of variables b and c. In the former case both b and c are also eliminated by simplification rules. In the latter the frequently meeting variables rule eliminates one of the variables b and c. Thus, after assigning d a Boolean variable at least four variables are eliminated (three neighbors of d and at least one of b and c). This shows that splitting on d is a (6, 6)- splitting (as any splitting in F is a (2k, 2l)-splitting). Now consider the case when G(F ) does not contain triangles. Suppose that F contains a variable x with neighbors a, b, c, such that in both F [x] and F [ x] the simplification rules assign a Boolean value to at least one of the variables a, b, c. It is easy to see that splitting on x is a (6, 6)-splitting. Consider one of the branches F [x] and F [ x]. W.l.o.g. suppose that that a is assigned a Boolean value in this branch. Clearly, all the neighbors of a are eliminated. Since N({a}) {b, c} = (as G(F ) does not contain triangles), at least 4 variables are eliminated in this branch. Thus, in the following we suppose that after splitting on any variable of F the simplification rules assign Boolean values to all its neighbors in one branch (recall that due to Lemma 3.1 every neighbor is assigned a Boolean value in at least one branch). Consider a variable x with N({x}) = {a, b, c}. If N({a, b, c}) 5, then splitting on x is a (4, 10)-splitting. So, assume that N({a, b, c}) < 5. This means that either there is a variable y x, such that N(y) = {a, b, c} (see Fig. 3(b)), or there are two variables y, z x, such that N(y) {a, b, c} = 2 and N(z) {a, b, c} = 2 (see Fig. 3(c)). In the former case splitting on b is a (6, 6)- splitting (after eliminating x and y by the simplification rules, one of the variables a and c is also eliminated). In the latter case splitting on x is a (4, 10)-splitting (at 5 Conclusions and Further Directions We have presented the algorithm for MAX-2-SAT with the running time 2 K/5.5, where K is the number of clauses of an input formula. The key point of our improvement is a new complexity measure for estimating the running time of the algorithm. It would be interesting to design new measures to improve upper bounds for other NP-hard problems. It would also be interesting to apply the ideas from [13] to prove better upper bounds for MAX-2- SAT. Let us give several informal ideas showing how an 2 K/5 bound can be proven for MAX-2-SAT. Consider a splitting algorithm which uses the simplification rules of this paper, but splits on randomly chosen variables. Let us estimate for a variable x of weight w the probability that this algorithm will not split on x, that is, that x will be eliminated by the simplification rules. We know that the simplification rules eliminate variables of weight at most 2. Thus, in assumption that all neighbors of x are different, the required probability is at least 3/(w + 1), as it is exactly the probability that the algorithm will select x after all its neighbors except for at most two of them. Thus, the expected number of variables on which this algorithm splits is at most N(F ) i 2 i=1 i+1 N i(f ). It is easy to see that this sum does not exceed K 2 /5, i/10 for any positive integer i. Of course, this is only an informal proof, which needs to be done more accurate (for example, one has to consider cases when several neighbors of a variable are equal). We believe that this proof can be made precise. After doing this it is possible to estimate the probability that neighbors of a variable are eliminated not during splitting, but by as i 2 i+1 the simplification rules. This would allow to prove a stronger bound on the probability that the algorithm will not split on a variable. But such an estimation seems to be a more complicated work. Acknowledgments We would like to thank our supervisor Edward A. Hirsch for help in writing this paper. Also, we would like to thank the anonymous referees for helpful comments. References [1] P. Asirelli, M. De Santis, and M. Martelli. Integrity constraints in logic databases. Journal of Logic Programming, 3: , [2] N. Bansal and V. Raman. Upper bounds for MaxSat: Further improved. In Proceedings of ISAAC 99, pages , 1999.
7 [3] J. Chen and I. Kanj. Improved exact algorithms for MAX-SAT. In Proceedings of the 5th LATIN, volume 2286 of LNCS, pages , [4] S. S. Fedin and A. S. Kulikov. Automated proofs of upper bounds on the running time of splitting algorithms. Zapiski nauchnykh seminarov POMI, 316: , [5] F. V. Fomin and K. Høie. Pathwidth of cubic graphs and exact algorithms. Information Processing Letters, To appear. [6] H. Gallaire, J. Minker, and J.-M. Nicolas. Logic and databases: A deductive approach. ACM Computing Surveys, 16(2): , June [7] J. Gramm, E. A. Hirsch, R. Niedermeier, and P. Rossmanith. Worst-case upper bounds for MAX-2-SAT with an application to MAX-CUT. Discrete Applied Mathematics, 130(2): , [8] J. Kneis, D. MD. Mölle, S. Richter, and P. Rossmanith. Pathwidth of cubic graphs and exact algorithms. In Proceedings of the 31st International Workshop on Graph-Theoretic Concepts in Computer Science, To appear. [9] J. Kneis and P. Rossmanith. A new satisfiability algorithm with applications to Max-Cut. Technical Report AIB , Department of Computer Science, RWTH Aachen University, April [10] A. S. Kulikov. Automated generation of simplification rules for SAT and MAXSAT. In Proceedings of the Eighth International Conference on Theory and Applications of Satisfiability Testing (SAT 2005), volume 3569 of LNCS, pages Springer Verlag, [11] O. Kullmann. New methods for 3-SAT decision and worst-case analysis. Theoretical Computer Science, 223:1 72, [12] O. Kullmann and H. Luckhardt. Algorithms for SAT/TAUT decision based on various measures. Preprint, [13] R. Paturi, P. Pudlák, M. E. Saks, and F. Zane. An improved exponential-time algorithm for k-sat. In Proceedings of the 39th Annual Symposium on Foundations of Computer Science. (FOCS-98), pages , Palo Alto, CA, November [14] S. Poljak and Z. Tuza. Maximum cuts and largest bipartite subgraphs. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 20: , [15] V. Raman, B. Ravikumar, and S. Srinivasa Rao. A simplified NP-complete MAXSAT problem. Information Processing Letters, 65(1):1 6, [16] R. Williams. A new algorithm for optimal constraint satisfaction and its implications. In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP), volume 3142 of LNCS, pages Springer Verlag, 2004.
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