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35
Generating satisfiable problem instances
 In AAAI/IAAI
, 2000
"... A major difficulty in evaluating incomplete local search style algorithms for constraint satisfaction problems is the need for a source of hard problem instances that are guaranteed to be satisfiable. A standard approach to evaluate incomplete search methods has been to use a general problem generat ..."
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Cited by 84 (9 self)
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A major difficulty in evaluating incomplete local search style algorithms for constraint satisfaction problems is the need for a source of hard problem instances that are guaranteed to be satisfiable. A standard approach to evaluate incomplete search methods has been to use a general problem generator and a complete search method to filter out the unsatisfiable instances. Unfortunately, this approach cannot be used to create problem instances that are beyond the reach of complete search methods. So far, it has proven to be surprisingly difficult to develop a direct generator for satisfiable instances only. In this paper, we propose a generator that only outputs satisfiable problem instances. We also show how one can finely control the hardness of the satisfiable instances by establishing a connection between problem hardness and a new kind of phase transition phenomenon in the space of problem instances. Finally, we use our problem distribution to show the easyhardeasy pattern in search complexity for local search procedures, analogous to the previously reported pattern for complete search methods.
UnitWalk: A new SAT solver that uses local search guided by unit clause elimination
, 2002
"... In this paper we present a new randomized algorithm for SAT, i.e., the satisfiability problem for Boolean formulas in conjunctive normal form. Despite its simplicity, this algorithm performs well on many common benchmarks ranging from graph coloring problems to microprocessor verification. ..."
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Cited by 65 (1 self)
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In this paper we present a new randomized algorithm for SAT, i.e., the satisfiability problem for Boolean formulas in conjunctive normal form. Despite its simplicity, this algorithm performs well on many common benchmarks ranging from graph coloring problems to microprocessor verification.
Adding New Clauses for Faster Local Search
, 1996
"... A primary concern when using local search methods for CNF satisfiability is how to get rid of local minimas. Among many other heuristics, Weighting by Morris (1993) and Selman and Kautz (1993) works overwhelmingly better than others (Cha and Iwama 1995). Weighting increases the weight of each clause ..."
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Cited by 44 (2 self)
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A primary concern when using local search methods for CNF satisfiability is how to get rid of local minimas. Among many other heuristics, Weighting by Morris (1993) and Selman and Kautz (1993) works overwhelmingly better than others (Cha and Iwama 1995). Weighting increases the weight of each clause which is unsatisfied at a local minima. This paper introduces a more sophisticated weighting strategy, i.e., adding new clauses (ANC) that are unsatisfied at the local minima. As those new clauses, we choose resolvents of the clauses unsatisfied at the local minima and randomly selected neighboring clauses. The idea is that ANC is to make the slope of search space more smooth than the simple weighting. Experimental data show that ANC is faster than simple weighting: (i) When the number of variables is 200 or more, ANC is roughly four to ten times as fast as weighting in terms of the number of search steps. (ii) It might be more important that the divergence of computation time for each try ...
Backbone Fragility and the Local Search Cost Peak
 Journal of Artificial Intelligence Research
, 2000
"... The local search algorithm WSat is one of the most successful algorithms for solving the satisfiability (SAT) problem. It is notably e#ective at solving hard Random 3SAT instances near the socalled `satisfiability threshold', but still shows a peak in search cost near the threshold and lar ..."
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Cited by 43 (3 self)
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The local search algorithm WSat is one of the most successful algorithms for solving the satisfiability (SAT) problem. It is notably e#ective at solving hard Random 3SAT instances near the socalled `satisfiability threshold', but still shows a peak in search cost near the threshold and large variations in cost over di#erent instances. We make a number of significant contributions to the analysis of WSat on highcost random instances, using the recentlyintroduced concept of the backbone of a SAT instance. The backbone is the set of literals which are entailed by an instance. We find that the number of solutions predicts the cost well for smallbackbone instances but is much less relevant for the largebackbone instances which appear near the threshold and dominate in the overconstrained region. We show a very strong correlation between search cost and the Hamming distance to the nearest solution early in WSat's search. This pattern leads us to introduce a measure of the ba...
Performance tests of local search algorithms using new types of random cnf formulas
 14th International Joint Conference on Arti cial Intelligence
, 1995
"... New types of testinstance generators have been developed for generating random CNF (Conjunctive Normal Form) formulas with controlled attributes. In this paper, we use these generators to test the performance of localsearchbased SAT algorithms. For this purpose, the generator which produces formul ..."
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Cited by 38 (4 self)
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New types of testinstance generators have been developed for generating random CNF (Conjunctive Normal Form) formulas with controlled attributes. In this paper, we use these generators to test the performance of localsearchbased SAT algorithms. For this purpose, the generator which produces formulas having exactly one satisfying truth assignment is especially desirable. It is shown that (i) among several different strategies of local search, the weighting strategy is overwhelmingly faster than the others and that (ii) local search works significantly better for instances of larger clause/variable ratio, which allows us to come up with a new strategy for making local search even faster. 1
On Computing Minimum Unsatisfiable Cores
, 2003
"... Certifying the correctness of a SAT solver is straightforward for satisfiable instances of SAT. Given a ..."
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Cited by 35 (3 self)
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Certifying the correctness of a SAT solver is straightforward for satisfiable instances of SAT. Given a
A New Look at the EasyHardEasy Pattern of Combinatorial Search Difficulty
 Journal of Artificial Intelligence Research
, 1997
"... The easyhardeasy pattern in the difficulty of combinatorial search problems as constraints are added has been explained as due to a competition between the decrease in number of solutions and increased pruning. We test the generality of this explanation by examining one of its predictions: if the ..."
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Cited by 15 (2 self)
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The easyhardeasy pattern in the difficulty of combinatorial search problems as constraints are added has been explained as due to a competition between the decrease in number of solutions and increased pruning. We test the generality of this explanation by examining one of its predictions: if the number of solutions is held fixed by the choice of problems, then increased pruning should lead to a monotonic decrease in search cost. Instead, we find the easyhardeasy pattern in median search cost even when the number of solutions is held constant, for some search methods. This generalizes previous observations of this pattern and shows that the existing theory does not explain the full range of the peak in search cost. In these cases the pattern appears to be due to changes in the size of the minimal unsolvable subproblems, rather than changing numbers of solutions. 1. Introduction Recently, many authors have shown that the solution cost for various kinds of combinatorial search probl...
Hiding satisfying assignments: two are better than one
 In Proceedings of AAAI’04
, 2004
"... The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random kSAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment ..."
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Cited by 13 (2 self)
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The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random kSAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment A. Unfortunately, instances generated in this manner tend to be relatively easy and can be solved efficiently by practical heuristics. Roughly speaking, as the formula’s density increases, for a number of different algorithms, A acts as a stronger and stronger attractor. Motivated by recent results on the geometry of the space of satisfying truth assignments of random kSAT and NAEkSAT formulas, we introduce a simple twist on this basic model, which appears to dramatically increase its hardness. Namely, in addition to forbidding the clauses violated by the hidden assignment A, we also forbid the clauses violated by its complement, so that both A and A are satisfying. It appears that under this “symmetrization ” the effects of the two attractors largely cancel out, making it much harder for algorithms to find any truth assignment. We give theoretical and experimental evidence supporting this assertion. 1
From spin glasses to hard satisfiable formulas
 In Proceedings of SAT’04
, 2004
"... Abstract. We introduce a highly structured family of hard satisfiable 3SAT formulas corresponding to an ordered spinglass model from statistical physics. This model has provably “glassy ” behavior; that is, it has many local optima with large energy barriers between them, so that local search algo ..."
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Cited by 13 (0 self)
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Abstract. We introduce a highly structured family of hard satisfiable 3SAT formulas corresponding to an ordered spinglass model from statistical physics. This model has provably “glassy ” behavior; that is, it has many local optima with large energy barriers between them, so that local search algorithms get stuck and have difficulty finding the true ground state, i.e., the unique satisfying assignment. We test the hardness of our formulas with two DavisPutnam solvers, Satz and zChaff, the recently introduced Survey Propagation (SP), and two local search algorithms, WalkSAT and RecordtoRecord Travel (RRT). We compare our formulas to random 3XORSAT formulas and to two other generators of hard satisfiable instances, the minimum disagreement parity formulas of Crawford et al., and Hirsch’s hgen2. For the complete solvers the running time of our formulas grows exponentially in √ n, and exceeds that of random 3XORSAT formulas for small problem sizes. SP is unable to solve our formulas with as few as 25 variables. For WalkSAT, our formulas appear to be harder than any other known generator of satisfiable instances. Finally, our formulas can be solved efficiently by RRT but only if the parameter d is tuned to the height of the barriers between local minima, and we use this parameter to measure the barrier heights in random 3XORSAT formulas as well. 1
Generating hard satisfiable formulas by hiding solutions deceptively
 In AAAI
, 2005
"... To test incomplete search algorithms for constraint satisfaction problems such as 3SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this m ..."
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Cited by 13 (3 self)
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To test incomplete search algorithms for constraint satisfaction problems such as 3SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this method tends to produce easy problems, since the majority of literals point toward the “hidden ” assignment A. Last year, (Achlioptas, Jia, & Moore 2004) proposed a problem generator that cancels this effect by hiding both A and its complement A. While the resulting formulas appear to be just as hard for DPLL algorithms as random 3SAT formulas with no hidden assignment, they can be solved byWalkSAT in only polynomial time. Here we propose a new method to cancel the attraction to A, by choosing a clause with t> 0 literals satisfied by A with probability proportional to q t for some q < 1. By varying q, we can generate formulas whose variables have no bias, i.e., which are equally likely to be true or false; we can even cause the formula to “deceptively ” point away from A. We present theoretical and experimental results suggesting that these formulas are exponentially hard both for DPLL algorithms and for incomplete algorithms such asWalkSAT.