Solving Max SAT and #SAT on structured CNF formulas

Transcription

1 Solving Max SAT and #SAT on structured CNF formulas Sigve Hortemo Sæther, Jan Arne Telle, Martin Vatshelle University of Bergen July 14, 2014 Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

2 Outline Equivalence of assignments Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

3 Outline Equivalence of assignments CNF formulas with few equivalence classes Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

4 Outline Equivalence of assignments CNF formulas with few equivalence classes Algorithm for MAX SAT Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

5 Outline Equivalence of assignments CNF formulas with few equivalence classes Algorithm for MAX SAT Polynomial time solvable cases Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

6 Equivalence of assignments Equivalence of assignments to CNF formulas When are two truth assignments to a CNF formula equivalent? Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

7 Equivalence of assignments Equivalence of assignments to CNF formulas When are two truth assignments to a CNF formula equivalent? When they satisfy the same set of clauses. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

8 Equivalence of assignments Equivalence of assignments to CNF formulas When are two truth assignments to a CNF formula equivalent? When they satisfy the same set of clauses. The number of equivalence classes is called the ps-value. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

9 Equivalence of assignments Equivalence of assignments to CNF formulas When are two truth assignments to a CNF formula equivalent? When they satisfy the same set of clauses. The number of equivalence classes is called the ps-value. We need two standard definitions before we give a sufficient condition for low ps-value. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

10 CNF formulas with few equivalence classes Incidence graph (a b c) ( a c d) ( b d) Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

11 CNF formulas with few equivalence classes Incidence graph (a b c) ( a c d) ( b d) a b c d C 1 C 2 C 3 Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

12 CNF formulas with few equivalence classes Incidence graph (a b c) ( a c d) ( b d) a b c d C 1 C 2 C 3 Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

13 CNF formulas with few equivalence classes Incidence graph (a b c) ( a c d) ( b d) a b c d C 1 C 2 C 3 For a CNF formula F we denote the incedence graph by I(F ). Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

14 CNF formulas with few equivalence classes Induced matching Given a bipartite graph M is an induced matching if: a b c d e f g h Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

15 CNF formulas with few equivalence classes Induced matching Given a bipartite graph M is an induced matching if: a b c d e f g h 1 M is a matching. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

16 CNF formulas with few equivalence classes Induced matching Given a bipartite graph M is an induced matching if: a b c d e f g h 1 M is a matching. 2 No other edge is adjacent to 2 edges in M. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

17 CNF formulas with few equivalence classes CNF formulas with low ps-value Lemma Let F be a CNF formula and k be the maximum size of an induced matching in I(F). The ps-value of F is at most cla(f) k + 1 Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

18 CNF formulas with few equivalence classes CNF formulas with low ps-value Lemma Let F be a CNF formula and k be the maximum size of an induced matching in I(F). The ps-value of F is at most cla(f) k + 1 Sketch of proof 1 Let U be the set of unsatisfied clauses, and L the variables appearing in some clause of U, then there is a unique assignment for L. 2 There exist U U of size at most k such that L is uniquely defined by U. 3 There is at most cla(f) k + 1 choices for U Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

19 CNF formulas with few equivalence classes CNF formulas with low ps-value Any CNF formula F where I(F) has a maximum induced matching of size 1 will be on the form: Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

20 CNF formulas with few equivalence classes CNF formulas with low ps-value Any CNF formula F where I(F) has a maximum induced matching of size 1 will be on the form: (a b) Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

21 CNF formulas with few equivalence classes CNF formulas with low ps-value Any CNF formula F where I(F) has a maximum induced matching of size 1 will be on the form: (a b) ( a b c) Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

22 CNF formulas with few equivalence classes CNF formulas with low ps-value Any CNF formula F where I(F) has a maximum induced matching of size 1 will be on the form: (a b) ( a b c) (a b c d) Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

23 CNF formulas with few equivalence classes CNF formulas with low ps-value Any CNF formula F where I(F) has a maximum induced matching of size 1 will be on the form: (a b) ( a b c) (a b c d)... Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

24 Algorithm for MAX SAT DP algorithm strategy 1 Make an ordering of clauses and variables. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

25 Algorithm for MAX SAT DP algorithm strategy 1 Make an ordering of clauses and variables. Let the i first elements of the ordering define a cut S i. 2 Store one solution for each equivalence class of the cut S i. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

26 Algorithm for MAX SAT DP algorithm strategy 1 Make an ordering of clauses and variables. Let the i first elements of the ordering define a cut S i. 2 Store one solution for each equivalence class of the cut S i. 3 Use solutions for S i to find solutions of S i+1. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

27 Algorithm for MAX SAT DP algorithm strategy 1 Make an ordering of clauses and variables. Let the i first elements of the ordering define a cut S i. 2 Store one solution for each equivalence class of the cut S i. 3 Use solutions for S i to find solutions of S i+1. Note: the linear ordering can be replaced by a tree-like decomposition. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

28 Algorithm for MAX SAT Equivalence over a cut Let F be a CNF formula, X var(f ) and C cla(f ) defines a cut. X C C X Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

29 Algorithm for MAX SAT Equivalence over a cut Let F be a CNF formula, X var(f ) and C cla(f ) defines a cut. X C C X Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

30 Algorithm for MAX SAT Equivalence over a cut Let F be a CNF formula, X var(f ) and C cla(f ) defines a cut. X C C X Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

31 Algorithm for MAX SAT Equivalence over a cut Let F be a CNF formula, X var(f ) and C cla(f ) defines a cut. X C C X We define ps-value of a cut as the ps-value of the formula where each variable is removed from all clauses on the same side as the variable. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

32 Algorithm for MAX SAT PS-width 1 The ps-width of a decomposition (ordering) is the max ps-value over all cuts defined by the decomposition. 2 The ps-width of a CNF formula is the min ps-width over all decompositions. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

33 Algorithm for MAX SAT Runningtime for MAX SAT Theorem Given a formula F over n variables and m clauses and of total size s, and a decomposition of F of ps-width k, we solve #SAT, and weighted MAXSAT in time O(k 3 s(m + n)). If the decomposition is a linear order the runningtime can be improved by a factor k. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

34 Polynomially solvable cases Formulas of linear PS-width We say a formula has an interval order if: Each variable and clause can be assigned an interval of the real line. Such that a variable x is in a clause c if and only if the interval of x intersects the interval of c. Incidence graphs of such formulas are called interval bigraphs. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

35 Polynomially solvable cases Formulas of linear PS-width We say a formula has an interval order if: Each variable and clause can be assigned an interval of the real line. Such that a variable x is in a clause c if and only if the interval of x intersects the interval of c. Incidence graphs of such formulas are called interval bigraphs. Many other classes of bipartite intersection graphs correspond to formulas which have polynomial ps-width. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

36 Polynomially solvable cases Relation to graph decomposition A tree-decomposition of I(F) of tree-width O(log(n)) can be turned into a decomposition of polynomial ps-width. A clique-decomposition of I(F) of constant clique-width can be turned into a decomposition of polynomial ps-width. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

37 Polynomially solvable cases Relation to graph decomposition A tree-decomposition of I(F) of tree-width O(log(n)) can be turned into a decomposition of polynomial ps-width. A clique-decomposition of I(F) of constant clique-width can be turned into a decomposition of polynomial ps-width. An active community study a wide range of width parameters such as: tree-width, branch-width, rank-width, boolean-width, clique-width, cut-width, MM-width, MIM-width... Bounding any of these parameters would prove polynomial ps-width. Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

38 Future research Polynomially solvable cases 1 Can we approximate ps-width? 2 Can we recognize graphs of MIM-width 1? 3 Does real world SAT instances have low ps-width? Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15

39 Polynomially solvable cases THANK YOU Sæther, Telle, Vatshelle (UiB) Max SAT on structured formulas July 14, / 15