Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable
Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical D P -complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+ k clauses can be recognized in time n O(k) . We imp...
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| Vydané v: | Journal of computer and system sciences Ročník 69; číslo 4; s. 656 - 674 |
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01.12.2004
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| Abstract | Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical
D
P
-complete problem. It was shown recently that minimal unsatisfiable formulas with
n variables and
n+
k clauses can be recognized in time
n
O(k)
. We improve this result and present an algorithm with time complexity
O(2
kn
4)
; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999).
Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses
F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by
k, then we can decide in time
O(2
kn
3)
whether
F is satisfiable;
k is called the maximum deficiency of
F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails. |
|---|---|
| AbstractList | Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical
D
P
-complete problem. It was shown recently that minimal unsatisfiable formulas with
n variables and
n+
k clauses can be recognized in time
n
O(k)
. We improve this result and present an algorithm with time complexity
O(2
kn
4)
; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999).
Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses
F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by
k, then we can decide in time
O(2
kn
3)
whether
F is satisfiable;
k is called the maximum deficiency of
F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails. |
| Author | Szeider, Stefan |
| Author_xml | – sequence: 1 givenname: Stefan surname: Szeider fullname: Szeider, Stefan email: stefan.szeider@durham.ac.uk organization: Department of Computer Science, University of Durham, Durham, England, UK |
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| Cites_doi | 10.1016/0166-218X(84)90081-7 10.1016/0022-0000(88)90042-6 10.1016/S0166-218X(00)00221-3 10.2178/bsl/1203350879 10.1145/368273.368557 10.1016/0097-3165(86)90060-9 10.1016/S0304-3975(01)00337-1 10.1016/0166-218X(85)90050-2 10.1137/0608024 10.1109/SFCS.2002.1181983 10.1016/S0004-3702(02)00182-0 10.1016/S0022-0000(03)00079-5 10.1016/S0166-218X(00)00245-6 10.1109/CCC.2000.856741 10.1145/321033.321034 10.1016/S0166-218X(99)00184-5 10.1007/978-3-662-53622-3_7 10.1016/S0166-218X(01)00358-4 10.1007/BF01215352 10.1016/0095-8956(86)90030-4 10.1023/A:1018924526592 10.1016/S0166-218X(02)00406-7 10.1016/S0166-218X(99)00038-4 |
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| Keywords | Tree-width Fixed-parameter complexity SAT problem Minimal unsatisfiability Bipartite matching Expansion D P -complete problem |
| Language | English |
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| References | Papadimitriou (BIB23) 1994 Aharoni, Linial (BIB1) 1986; 43 Downey, Fellows (BIB10) 1999 Heusch, Porschen, Speckenmeyer (BIB16) 2003; 67 Seymour, Thomas (BIB26) 1994; 14 R. Diestel, Graph Theory, Graduate Texts in Mathematics, Vol. 173, 2nd Edition, Springer, New York, 2000. Kleine Büning (BIB18) 2000; 107 Zhang, Malik (BIB31) 2002; Vol. 2404 Monien, Speckenmeyer (BIB22) 1985; 10 Kullmann (BIB20) 2003; 130 Urquhart (BIB30) 1995; 1 Fleischner, Kullmann, Szeider (BIB12) 2002; 289 Arnborg, Corneil, Proskurowski (BIB3) 1987; 8 Davis, Putnam (BIB7) 1960; 7 Lovász, Plummer (BIB21) 1986; Vol. 29 Davydov, Davydova, Kleine Büning (BIB8) 1998; 23 Szeider (BIB28) 2004; Vol. 2919 Davis, Logemann, Loveland (BIB6) 1962; 5 Franco, Van Gelder (BIB14) 2003; 125 S. Szeider, Generalizations of matched CNF formulas, Ann. Math. Artif. Intell., Special issue with selected papers from the Fifth International Symposium on the Theory and Applications of Satisfiability Testing (SAT’02), to appear. M. Alekhnovich, A.A. Razborov, Satisfiability, branch-width and Tseitin tautologies, in: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’02), 2002, pp. 593–603. O. Kullmann, An application of matroid theory to the SAT problem, in: Fifteenth Annual IEEE Conference on Computational Complexity, 2000, pp. 116–124. Courcelle, Makowsky, Rotics (BIB4) 2001; 108 Franco, Goldsmith, Schlipf, Speckenmeyer, Swaminathan (BIB13) 1999; 96/97 Papadimitriou, Wolfe (BIB24) 1988; 37 Robertson, Seymour (BIB25) 1986; 41 Tovey (BIB29) 1984; 8 Fellows (BIB11) 2003; Vol. 2880 Courcelle, Olariu (BIB5) 2000; 101 Kleine Büning (BIB17) 1999; Vol. 1584 Gottlob, Scarcello, Sideri (BIB15) 2002; 138 Papadimitriou (10.1016/j.jcss.2004.04.009_BIB23) 1994 Kleine Büning (10.1016/j.jcss.2004.04.009_BIB17) 1999; Vol. 1584 Gottlob (10.1016/j.jcss.2004.04.009_BIB15) 2002; 138 Szeider (10.1016/j.jcss.2004.04.009_BIB28) 2004; Vol. 2919 Downey (10.1016/j.jcss.2004.04.009_BIB10) 1999 Franco (10.1016/j.jcss.2004.04.009_BIB14) 2003; 125 Papadimitriou (10.1016/j.jcss.2004.04.009_BIB24) 1988; 37 Seymour (10.1016/j.jcss.2004.04.009_BIB26) 1994; 14 Franco (10.1016/j.jcss.2004.04.009_BIB13) 1999; 96/97 Fellows (10.1016/j.jcss.2004.04.009_BIB11) 2003; Vol. 2880 Courcelle (10.1016/j.jcss.2004.04.009_BIB4) 2001; 108 10.1016/j.jcss.2004.04.009_BIB2 Aharoni (10.1016/j.jcss.2004.04.009_BIB1) 1986; 43 Zhang (10.1016/j.jcss.2004.04.009_BIB31) 2002; Vol. 2404 10.1016/j.jcss.2004.04.009_BIB9 10.1016/j.jcss.2004.04.009_BIB19 Tovey (10.1016/j.jcss.2004.04.009_BIB29) 1984; 8 Robertson (10.1016/j.jcss.2004.04.009_BIB25) 1986; 41 Davis (10.1016/j.jcss.2004.04.009_BIB6) 1962; 5 Davydov (10.1016/j.jcss.2004.04.009_BIB8) 1998; 23 Kleine Büning (10.1016/j.jcss.2004.04.009_BIB18) 2000; 107 Heusch (10.1016/j.jcss.2004.04.009_BIB16) 2003; 67 Monien (10.1016/j.jcss.2004.04.009_BIB22) 1985; 10 Kullmann (10.1016/j.jcss.2004.04.009_BIB20) 2003; 130 Fleischner (10.1016/j.jcss.2004.04.009_BIB12) 2002; 289 Lovász (10.1016/j.jcss.2004.04.009_BIB21) 1986; Vol. 29 Urquhart (10.1016/j.jcss.2004.04.009_BIB30) 1995; 1 Arnborg (10.1016/j.jcss.2004.04.009_BIB3) 1987; 8 Davis (10.1016/j.jcss.2004.04.009_BIB7) 1960; 7 10.1016/j.jcss.2004.04.009_BIB27 Courcelle (10.1016/j.jcss.2004.04.009_BIB5) 2000; 101 |
| References_xml | – volume: 37 start-page: 2 year: 1988 end-page: 13 ident: BIB24 article-title: The complexity of facets resolved publication-title: J. Comput. Syst. Sci – volume: 14 start-page: 217 year: 1994 end-page: 241 ident: BIB26 article-title: Call routing and the ratcatcher publication-title: Combinatorica – reference: R. Diestel, Graph Theory, Graduate Texts in Mathematics, Vol. 173, 2nd Edition, Springer, New York, 2000. – volume: 108 start-page: 23 year: 2001 end-page: 52 ident: BIB4 article-title: On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic publication-title: Discrete Appl. Math – volume: 107 start-page: 83 year: 2000 end-page: 98 ident: BIB18 article-title: On subclasses of minimal unsatisfiable formulas publication-title: Discrete Appl. Math – volume: Vol. 2919 start-page: 188 year: 2004 end-page: 202 ident: BIB28 article-title: On fixed-parameter tractable parameterizations of SAT publication-title: Theory and Applications of Satisfiability, Sixth International Conference, SAT 2003, Selected and Revised Papers – reference: S. Szeider, Generalizations of matched CNF formulas, Ann. Math. Artif. Intell., Special issue with selected papers from the Fifth International Symposium on the Theory and Applications of Satisfiability Testing (SAT’02), to appear. – volume: 125 start-page: 177 year: 2003 end-page: 214 ident: BIB14 article-title: A perspective on certain polynomial time solvable classes of satisfiability publication-title: Discrete Appl. Math – volume: 8 start-page: 85 year: 1984 end-page: 89 ident: BIB29 article-title: A simplified NP-complete satisfiability problem publication-title: Discrete Appl. Math – volume: 23 start-page: 229 year: 1998 end-page: 245 ident: BIB8 article-title: An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF publication-title: Ann. Math. Artif. Intell – volume: 1 start-page: 425 year: 1995 end-page: 467 ident: BIB30 article-title: The complexity of propositional proofs publication-title: Bull. Symbol. Logic – volume: 43 start-page: 196 year: 1986 end-page: 204 ident: BIB1 article-title: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas publication-title: J. Combin. Theory Ser. A – volume: 96/97 start-page: 89 year: 1999 end-page: 106 ident: BIB13 article-title: An algorithm for the class of pure implicational formulas publication-title: Discrete Appl. Math – volume: 138 start-page: 55 year: 2002 end-page: 86 ident: BIB15 article-title: Fixed-parameter complexity in AI and nonmonotonic reasoning publication-title: Artif. Intell – volume: 289 start-page: 503 year: 2002 end-page: 516 ident: BIB12 article-title: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference publication-title: Theoret. Comput. Sci – volume: 41 start-page: 92 year: 1986 end-page: 114 ident: BIB25 article-title: Graph minors. V. Excluding a planar graph publication-title: J. Combin. Theory Ser. B – volume: Vol. 1584 start-page: 171 year: 1999 end-page: 178 ident: BIB17 article-title: An upper bound for minimal resolution refutations publication-title: CSL’98 – volume: Vol. 2404 start-page: 17 year: 2002 end-page: 36 ident: BIB31 article-title: The quest for efficient boolean satisfiability solvers publication-title: Computer Aided Verification: 14th International Conference (CAV 2002) – volume: 67 start-page: 772 year: 2003 end-page: 788 ident: BIB16 article-title: Improving a fixed parameter tractability time bound for the shadow problem publication-title: J. Comput. Syst. Sci – volume: 5 start-page: 394 year: 1962 end-page: 397 ident: BIB6 article-title: A machine program for theorem-proving publication-title: Comm. ACM – year: 1994 ident: BIB23 publication-title: Computational Complexity – year: 1999 ident: BIB10 publication-title: Parameterized Complexity – volume: Vol. 29 year: 1986 ident: BIB21 article-title: Matching Theory publication-title: Annals of Discrete Mathematics – reference: O. Kullmann, An application of matroid theory to the SAT problem, in: Fifteenth Annual IEEE Conference on Computational Complexity, 2000, pp. 116–124. – reference: M. Alekhnovich, A.A. Razborov, Satisfiability, branch-width and Tseitin tautologies, in: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science (FOCS’02), 2002, pp. 593–603. – volume: 130 start-page: 209 year: 2003 end-page: 249 ident: BIB20 article-title: Lean clause-sets publication-title: Discrete Appl. Math – volume: Vol. 2880 start-page: 1 year: 2003 end-page: 12 ident: BIB11 article-title: Blow-ups, win/win's, and crown rules: some new directions in FPT publication-title: Graph-Theoretic Concepts in Computer Science (WG 2003) – volume: 10 start-page: 287 year: 1985 end-page: 295 ident: BIB22 article-title: Solving satisfiability in less than 2 publication-title: Discrete Appl. Math – volume: 8 start-page: 277 year: 1987 end-page: 284 ident: BIB3 article-title: Complexity of finding embeddings in a publication-title: SIAM J. Algebraic Discrete Meth – volume: 101 start-page: 77 year: 2000 end-page: 114 ident: BIB5 article-title: Upper bounds to the clique width of graphs publication-title: Discrete Appl. Math – volume: 7 start-page: 201 year: 1960 end-page: 215 ident: BIB7 article-title: A computing procedure for quantification theory publication-title: J. ACM – volume: 8 start-page: 85 issue: 1 year: 1984 ident: 10.1016/j.jcss.2004.04.009_BIB29 article-title: A simplified NP-complete satisfiability problem publication-title: Discrete Appl. Math doi: 10.1016/0166-218X(84)90081-7 – volume: 37 start-page: 2 issue: 1 year: 1988 ident: 10.1016/j.jcss.2004.04.009_BIB24 article-title: The complexity of facets resolved publication-title: J. Comput. Syst. Sci doi: 10.1016/0022-0000(88)90042-6 – volume: 108 start-page: 23 issue: 1–2 year: 2001 ident: 10.1016/j.jcss.2004.04.009_BIB4 article-title: On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic publication-title: Discrete Appl. Math doi: 10.1016/S0166-218X(00)00221-3 – volume: 1 start-page: 425 issue: 4 year: 1995 ident: 10.1016/j.jcss.2004.04.009_BIB30 article-title: The complexity of propositional proofs publication-title: Bull. Symbol. Logic doi: 10.2178/bsl/1203350879 – volume: 5 start-page: 394 year: 1962 ident: 10.1016/j.jcss.2004.04.009_BIB6 article-title: A machine program for theorem-proving publication-title: Comm. ACM doi: 10.1145/368273.368557 – volume: Vol. 2880 start-page: 1 year: 2003 ident: 10.1016/j.jcss.2004.04.009_BIB11 article-title: Blow-ups, win/win's, and crown rules: some new directions in FPT – volume: 43 start-page: 196 year: 1986 ident: 10.1016/j.jcss.2004.04.009_BIB1 article-title: Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas publication-title: J. Combin. Theory Ser. A doi: 10.1016/0097-3165(86)90060-9 – year: 1999 ident: 10.1016/j.jcss.2004.04.009_BIB10 – ident: 10.1016/j.jcss.2004.04.009_BIB27 – volume: 289 start-page: 503 issue: 1 year: 2002 ident: 10.1016/j.jcss.2004.04.009_BIB12 article-title: Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference publication-title: Theoret. Comput. Sci doi: 10.1016/S0304-3975(01)00337-1 – volume: Vol. 2919 start-page: 188 year: 2004 ident: 10.1016/j.jcss.2004.04.009_BIB28 article-title: On fixed-parameter tractable parameterizations of SAT – volume: 10 start-page: 287 year: 1985 ident: 10.1016/j.jcss.2004.04.009_BIB22 article-title: Solving satisfiability in less than 2n steps publication-title: Discrete Appl. Math doi: 10.1016/0166-218X(85)90050-2 – volume: Vol. 2404 start-page: 17 year: 2002 ident: 10.1016/j.jcss.2004.04.009_BIB31 article-title: The quest for efficient boolean satisfiability solvers – volume: 8 start-page: 277 issue: 2 year: 1987 ident: 10.1016/j.jcss.2004.04.009_BIB3 article-title: Complexity of finding embeddings in a k-tree publication-title: SIAM J. Algebraic Discrete Meth doi: 10.1137/0608024 – year: 1994 ident: 10.1016/j.jcss.2004.04.009_BIB23 – volume: Vol. 1584 start-page: 171 year: 1999 ident: 10.1016/j.jcss.2004.04.009_BIB17 article-title: An upper bound for minimal resolution refutations – ident: 10.1016/j.jcss.2004.04.009_BIB2 doi: 10.1109/SFCS.2002.1181983 – volume: 138 start-page: 55 issue: 1–2 year: 2002 ident: 10.1016/j.jcss.2004.04.009_BIB15 article-title: Fixed-parameter complexity in AI and nonmonotonic reasoning publication-title: Artif. Intell doi: 10.1016/S0004-3702(02)00182-0 – volume: 67 start-page: 772 issue: 4 year: 2003 ident: 10.1016/j.jcss.2004.04.009_BIB16 article-title: Improving a fixed parameter tractability time bound for the shadow problem publication-title: J. Comput. Syst. Sci doi: 10.1016/S0022-0000(03)00079-5 – volume: 107 start-page: 83 issue: 1–3 year: 2000 ident: 10.1016/j.jcss.2004.04.009_BIB18 article-title: On subclasses of minimal unsatisfiable formulas publication-title: Discrete Appl. Math doi: 10.1016/S0166-218X(00)00245-6 – ident: 10.1016/j.jcss.2004.04.009_BIB19 doi: 10.1109/CCC.2000.856741 – volume: 7 start-page: 201 issue: 3 year: 1960 ident: 10.1016/j.jcss.2004.04.009_BIB7 article-title: A computing procedure for quantification theory publication-title: J. ACM doi: 10.1145/321033.321034 – volume: Vol. 29 year: 1986 ident: 10.1016/j.jcss.2004.04.009_BIB21 article-title: Matching Theory – volume: 101 start-page: 77 issue: 1–3 year: 2000 ident: 10.1016/j.jcss.2004.04.009_BIB5 article-title: Upper bounds to the clique width of graphs publication-title: Discrete Appl. Math doi: 10.1016/S0166-218X(99)00184-5 – ident: 10.1016/j.jcss.2004.04.009_BIB9 doi: 10.1007/978-3-662-53622-3_7 – volume: 125 start-page: 177 year: 2003 ident: 10.1016/j.jcss.2004.04.009_BIB14 article-title: A perspective on certain polynomial time solvable classes of satisfiability publication-title: Discrete Appl. Math doi: 10.1016/S0166-218X(01)00358-4 – volume: 14 start-page: 217 issue: 2 year: 1994 ident: 10.1016/j.jcss.2004.04.009_BIB26 article-title: Call routing and the ratcatcher publication-title: Combinatorica doi: 10.1007/BF01215352 – volume: 41 start-page: 92 issue: 1 year: 1986 ident: 10.1016/j.jcss.2004.04.009_BIB25 article-title: Graph minors. V. Excluding a planar graph publication-title: J. Combin. Theory Ser. B doi: 10.1016/0095-8956(86)90030-4 – volume: 23 start-page: 229 year: 1998 ident: 10.1016/j.jcss.2004.04.009_BIB8 article-title: An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF publication-title: Ann. Math. Artif. Intell doi: 10.1023/A:1018924526592 – volume: 130 start-page: 209 issue: 2 year: 2003 ident: 10.1016/j.jcss.2004.04.009_BIB20 article-title: Lean clause-sets publication-title: Discrete Appl. Math doi: 10.1016/S0166-218X(02)00406-7 – volume: 96/97 start-page: 89 year: 1999 ident: 10.1016/j.jcss.2004.04.009_BIB13 article-title: An algorithm for the class of pure implicational formulas publication-title: Discrete Appl. Math doi: 10.1016/S0166-218X(99)00038-4 |
| SSID | ssj0011563 |
| Score | 1.8690916 |
| Snippet | Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical
D
P
-complete... |
| SourceID | crossref elsevier |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 656 |
| SubjectTerms | Bipartite matching DP-complete problem Expansion Fixed-parameter complexity Minimal unsatisfiability SAT problem Tree-width |
| Title | Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable |
| URI | https://dx.doi.org/10.1016/j.jcss.2004.04.009 |
| Volume | 69 |
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