Tradeoffs in the Complexity of Backdoor Detection

Dilkina, Bistra; Gomes, Carla P.; Sabharwal, Ashish

doi:10.1007/978-3-540-74970-7_20

Cited by 27 publications

(25 citation statements)

References 20 publications

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“…which strong backdoors have been studied in depth are 2-SAT, Horn-SAT, and RenamableHorn-SAT [cf. 2,11,12]. Most of such syntactic classes satisfy a natural property, namely, they are closed under clause removal.…”

Section: Backdoor Sets For Clause Learning Sat Solversmentioning

confidence: 99%

Backdoors in the Context of Learning

Dilkina

Gomes

Sabharwal

2009

Lecture Notes in Computer Science

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Abstract. The concept of backdoor variables has been introduced as a structural property of combinatorial problems that provides insight into the surprising ability of modern satisfiability (SAT) solvers to tackle extremely large instances. This concept is, however, oblivious to "learning" during search-a key feature of successful combinatorial reasoning engines for SAT, mixed integer programming (MIP), etc. We extend the notion of backdoors to the context of learning during search. We prove that the smallest backdoors for SAT that take into account clause learning and order-sensitivity of branching can be exponentially smaller than "traditional" backdoors. We also study the effect of learning empirically.

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Section: Backdoor Sets For Clause Learning Sat Solversmentioning

confidence: 99%

Backdoors in the Context of Learning

Dilkina

Gomes

Sabharwal

2009

Lecture Notes in Computer Science

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“…Instance domains Williams et al [13] DPLL structured Interian [6] 2SAT, Horn random 3SAT Dilkina et al [2] DPLL, Horn, RHorn graph coloring, planning, game theory, automotive configuration Paris et al [9] RHorn random 3SAT, SAT competition Kottler et al [8] 2SAT, Horn, RHorn SAT competition, automotive configuration Samer & Szeider [11] Horn, RHorn automotive configuration Ruan et al [10] DPLL quasigroup completion, graph coloring Kilby et al [7] DPLL random 3SAT Gregory et al [4] DPLL planning, graph coloring, quasigroup Dilkina et al [3] DPLL planning, circuits…”

Section: Sub-solversmentioning

confidence: 99%

Finding Small Backdoors in SAT Instances

Beek

2011

Advances in Artificial Intelligence

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Abstract. Although propositional satisfiability (SAT) is NP-complete, state-of-the-art SAT solvers are able to solve large, practical instances. The concept of backdoors has been introduced to capture structural properties of instances. A backdoor is a set of variables that, if assigned correctly, leads to a polynomial-time solvable sub-problem. In this paper, we address the problem of finding all small backdoors, which is essential for studying value and variable ordering mistakes. We discuss our definition of sub-solvers and propose algorithms for finding backdoors. We experimentally compare our proposed algorithms to previous algorithms on structured and real-world instances. Our proposed algorithms improve over previous algorithms for finding backdoors in two ways. First, our algorithms often find smaller backdoors. Second, our algorithms often find a much larger number of backdoors.

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“…The sub-solver applied by CPLEX at each search node of the branch-andbound routine uses a dual simplex LP algorithm in conjunction with a variety of cuts. In our previous study [3] of backdoors in Satisfiability problems, we investigated the sub-solver routine used in Satz [9] which applied probing to each search node. Similarly here, we set CPLEX to use strong branching, adding a lot of additional inference at each node.…”

Section: Experimental Evaluationmentioning

confidence: 99%

Backdoors to Combinatorial Optimization: Feasibility and Optimality

Dilkina

Gomes

Malitsky

et al. 2009

Lecture Notes in Computer Science

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Abstract. There has been considerable interest in the identification of structural properties of combinatorial problems that lead, directly or indirectly, to the development of efficient algorithms for solving them. One such concept is that of a backdoor set-a set of variables such that once they are instantiated, the remaining problem simplifies to a tractable form. While backdoor sets were originally defined to capture structure in decision problems with discrete variables, here we introduce a notion of backdoors that captures structure in optimization problems, which often have both discrete and continuous variables. We show that finding a feasible solution and proving optimality are characterized by backdoors of different kinds and size. Surprisingly, in certain mixed integer programming problems, proving optimality involves a smaller backdoor set than finding the optimal solution. We also show extensive results on the number of backdoors of various sizes in optimization problems. Overall, this work demonstrates that backdoors, appropriately generalized, are also effective in capturing problem structure in optimization problems.

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Tradeoffs in the Complexity of Backdoor Detection

Cited by 27 publications

References 20 publications

Backdoors in the Context of Learning

Backdoors in the Context of Learning

Finding Small Backdoors in SAT Instances

Backdoors to Combinatorial Optimization: Feasibility and Optimality

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