## Guiding SAT Diagnosis with Tree Decompositions (2004)

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Venue: | in Sixth International Conference on Theory and Applications of Satisfiability Testing (SAT |

Citations: | 21 - 0 self |

### BibTeX

@INPROCEEDINGS{Bjesse04guidingsat,

author = {Per Bjesse and James Kukula and Robert Damiano and Ted Stanion and Yunshan Zhu},

title = {Guiding SAT Diagnosis with Tree Decompositions},

booktitle = {in Sixth International Conference on Theory and Applications of Satisfiability Testing (SAT},

year = {2004},

pages = {315--329},

publisher = {Springer}

}

### Years of Citing Articles

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### Abstract

A tree decomposition of a hypergraph is a construction that captures the graph's topological structure. Every tree decomposition has an associated tree width, which can be viewed as a measure of how treelike the original hypergraph is. Tree decomposition has proven to be a very useful theoretical vehicle for generating polynomial algorithms for subclasses of problems whose general solution is NP-complete. As a rule, this is done by designing the algorithms so that their runtime is bounded by some polynomial times a function of the tree width of a tree decomposition of the original problem. Problem instances that have bounded tree width can thus be solved by the resulting algorithms in polynomial time.

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Citation Context ...ons by one or more orders of magnitude. 2 Preliminaries In the remainder of this paper, we will focus on augmenting GRASP-like [16] implementations of the Davis-Putnam-Loveland-Logemann (DPLL) method =-=[11]-=- with tree decomposition guidance. We refer readers unfamiliar with decision, deduction, and diagnosis components of such algorithms, including conflict graphs and backjumping, to [23]. 3 Tree Decompo... |

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Citation Context ...re orders of magnitude. 1 Introduction Tree decomposition [6] is a graph theoretic concept which abstractly captures topological structure in a variety of problems [4] such as constraint satisfaction =-=[12]-=-, Gaussian elimination [21], database query processing [9], and image computation [13]. The topological structure of a Conjunctive Normal Form (CNF) formula can be represented as a hypergraph, where t... |

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Citation Context ...width, where our tree-sat method decreases the number of necessary decisions by one or more orders of magnitude. 2 Preliminaries In the remainder of this paper, we will focus on augmenting GRASP-like =-=[16]-=- implementations of the Davis-Putnam-Loveland-Logemann (DPLL) method [11] with tree decomposition guidance. We refer readers unfamiliar with decision, deduction, and diagnosis components of such algor... |

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Citation Context ... so our method has the potential of being effective even on some classes of problems that have an intractable cutwidth. Finding a minimal width tree decomposition for an arbitrary graph is NPcomplete =-=[5]-=-, so an efficient general algorithm is unlikely to exist. For fixed k, checking if a graph has treewidth k and constructing a width k tree decomposition if it does can be done in time linear in the si... |

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Citation Context ...to (1) modify the conflict clause generation; and (2) control the selection of decision variables so that we only use compatible candidates. We solve the first problem by modifying the standard 1-UIP =-=[22]-=- conflict clause generation slightly so that the conflict clauses that it returns are forced to contain variables exclusively from the last node a decision variable was selected from (the choice node)... |

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Citation Context ...ion. We also present experimental results demonstrating that the method we propose can decrease the number of necessary decisions by one or more orders of magnitude. 1 Introduction Tree decomposition =-=[6]-=- is a graph theoretic concept which abstractly captures topological structure in a variety of problems [4] such as constraint satisfaction [12], Gaussian elimination [21], database query processing [9... |

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Citation Context ... of necessary decisions by one or more orders of magnitude. 1 Introduction Tree decomposition [6] is a graph theoretic concept which abstractly captures topological structure in a variety of problems =-=[4]-=- such as constraint satisfaction [12], Gaussian elimination [21], database query processing [9], and image computation [13]. The topological structure of a Conjunctive Normal Form (CNF) formula can be... |

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Citation Context ...t general algorithm is unlikely to exist. For fixed k, checking if a graph has treewidth k and constructing a width k tree decomposition if it does can be done in time linear in the size of the graph =-=[7]-=-. Unfortunately, current algorithms grow in cost very rapidly with k, and are only practical for (-2 4 7) (-0 1 2) ( 0 1-3) ( 1 2 8) ( 0-1 6) ( 2 3-8) (1-2 4) ( 2-3 5) (-2 4 7) (-0 1 2) ( 0 1-3) ( 0-1... |

131 | The Quest for Efficient Boolean Satisfiability Solvers
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Citation Context ...n (DPLL) method [11] with tree decomposition guidance. We refer readers unfamiliar with decision, deduction, and diagnosis components of such algorithms, including conflict graphs and backjumping, to =-=[23]-=-. 3 Tree Decomposition Given a hypergraph G = (V; E), where V is a set of vertices and E a set of hyperedges with e ` V for each e 2 E, a tree decomposition of G is a triple (N; F; ) where 1. N is a s... |

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Citation Context ...6] is a graph theoretic concept which abstractly captures topological structure in a variety of problems [4] such as constraint satisfaction [12], Gaussian elimination [21], database query processing =-=[9]-=-, and image computation [13]. The topological structure of a Conjunctive Normal Form (CNF) formula can be represented as a hypergraph, where the vertices of the hypergraph correspond to the variables ... |

90 | Efficient Approximation for Triangulation of Minimum Treewidth
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Citation Context ... ( 0-1 6) (1-2 4) A:(1 2 3) B:(2 3) Fig. 4. Constructing a tree decomposition very small k, roughly ks4 [19]. Much more efficient algorithms have been developed for approximating k with bounded error =-=[2]-=-, but even these appear to be too costly for industrial problems. Due to the limitations of the direct approaches to computing a tree decomposition, we have taken a different approach. We rely on the ... |

57 | Approximating treewidth, pathwidth, frontsize, and shortest elimination tree
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Citation Context ...ions of the direct approaches to computing a tree decomposition, we have taken a different approach. We rely on the facts that tree decompositions can be derived from orderings of hypergraph vertices =-=[8]-=----CNF variables in our case---and that a plethora of robust and scalable CNF variable ordering methods is available. Given an ordering of variables, we use the following algorithm to compute a decomp... |

35 | Compiling knowledge into decomposable negation normal form
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Citation Context ...s of the CNF and the hyperedges correspond to the clauses. Given a small treewidth tree decomposition for a hypergraph of a CNF formula, a variety of methods are known for deciding its satisfiability =-=[12, 10, 3]-=-. In this paper we report on our effort to learn whether satisfiability solving guided by tree decomposition can be made to work in practice. To do this, we attempt to find tree decompositions of smal... |

27 | Faster SAT and smaller BDDs via common function structure
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(Show Context)
Citation Context ...riable for which j S e3v ej is smallest, being the size of the tree node to be constructed to eliminate the variable. The second heuristic we explored is built using the MINCE linear placement engine =-=[1]-=-. The objects placed by MINCE are the hyperedges of the graph, corresponding to the CNF clauses. MINCE then generates a linear order of the clauses, attempting to reduce the cutwidth, the largest numb... |

26 | A structure-based variable ordering heuristic for SAT
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(Show Context)
Citation Context ...o be gained: In a parallel development to our original conference paper in SAT 2003, Huang and Darwiche have introduced a variable ordering heuristic that is a continuation of Darwiche's work on DNNF =-=[14]-=-. They guide decision variable selection in a DPLL solver using DTrees---the subclass of tree decompositions that correspond to full binary trees. Huang and Darwiche's variable selection heuristic use... |

25 | Partition-based Decision Heuristics for Image Computation using SAT and BDDs
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- 2001
(Show Context)
Citation Context ...cept which abstractly captures topological structure in a variety of problems [4] such as constraint satisfaction [12], Gaussian elimination [21], database query processing [9], and image computation =-=[13]-=-. The topological structure of a Conjunctive Normal Form (CNF) formula can be represented as a hypergraph, where the vertices of the hypergraph correspond to the variables of the CNF and the hyperedge... |

18 | Why is ATPG easy
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Citation Context ...we have described will not provide a useful bound on the sizes of conflict clauses or their number. But we expect that many practical problems will have small width. For example, it has been observed =-=[18]-=- that digital circuits tend to have small cutwidth. Small cutwidth implies small treewidth, so we expect our method to be effective on a large fraction of digital circuits. Moreover, the reverse is no... |

14 |
editors. The second DIMACS implementation challenge
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Citation Context ...lems are a mixture of industrial and public benchmarks: The Dubois problems are a series of random benchmarks generated by the gensathard program that is included in the DIMACS benchmark distribution =-=[15]-=-. The remaining problems are inhouse-generated equivalence checking problems. In particular, the addm x y examples are equivalence checks between different ways of implementing the addition of x y-bit... |

13 | Solving Satisfiability using Decomposition and the Most Constrained Subproblem
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(Show Context)
Citation Context ...s of the CNF and the hyperedges correspond to the clauses. Given a small treewidth tree decomposition for a hypergraph of a CNF formula, a variety of methods are known for deciding its satisfiability =-=[12, 10, 3]-=-. In this paper we report on our effort to learn whether satisfiability solving guided by tree decomposition can be made to work in practice. To do this, we attempt to find tree decompositions of smal... |

9 |
and S.Malik, â€śChaff: Engineering an Efficient SAT-solver
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(Show Context)
Citation Context ...der will have small width. 6 Implementing a Tree-based DPLL Solver Let us consider the practicalities of integrating the tree decomposition approach to satisfiability solving into a modern Chaff-like =-=[17]-=- DPLL engine. In order to make use of a tree decomposition, we need to (1) modify the conflict clause generation; and (2) control the selection of decision variables so that we only use compatible can... |

8 |
Algorithmic aspects of vertex elimination directed graphs
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(Show Context)
Citation Context ...ntroduction Tree decomposition [6] is a graph theoretic concept which abstractly captures topological structure in a variety of problems [4] such as constraint satisfaction [12], Gaussian elimination =-=[21]-=-, database query processing [9], and image computation [13]. The topological structure of a Conjunctive Normal Form (CNF) formula can be represented as a hypergraph, where the vertices of the hypergra... |

1 |
Tree Decomposition: A Feasibility Study
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(Show Context)
Citation Context ...4) D:(2) (-2 4 7) (-0 1 2) ( 0 1-3) (1-2 4) A:(1 2 3) B:(2 3) C:(0 1) (-2 4 7) (-0 1 2) ( 0 1-3) ( 0-1 6) (1-2 4) A:(1 2 3) B:(2 3) Fig. 4. Constructing a tree decomposition very small k, roughly ks4 =-=[19]-=-. Much more efficient algorithms have been developed for approximating k with bounded error [2], but even these appear to be too costly for industrial problems. Due to the limitations of the direct ap... |