An algorithm for Exact Satisfiability analysed with the number of clauses as parameter

Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2 n n O(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of these problems. Our algorithms are based on the principle of inclusion-exclusion and the zeta transform.In effect we get exact algorithms in 2 n n O(1) time for several well-studied partition problems including Domatic Number, Chromatic Number, Maximum k-Cut, Bin Packing, List Colouring, and the Chromatic Polynomial. We also have applications to Bayesian learning with decision graphs and to model-based data clustering.If only polynomial space is available, our algorithms run in time 3 n n O(1) if membership in F can be decided in polynomial time. We solve Chromatic Number in O(2.2461 n ) time and Domatic Number in O(2.8718 n ) time.Finally, we present a family of polynomial space approximation algorithms that find a number between χ(G) and (1 + )χ(G) in time O(1.2209 n + 2.2461 e − n ).

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“…Acknowledgments. The first two authors are indebted to Bolette A. Madsen [36] for making them think about these problems. M.K.…”

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confidence: 99%

Set Partitioning via Inclusion-Exclusion

2009

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“…The calculation of Nishio(S) may be expressed as an exact satisfiability problem with 3n clauses, one for each row, column, and block of the puzzle. Known methods for exact satisfiability of instances with c clauses take time 2 c c O (1) [5,20], and can therefore be used to compute Nishio(S) in time 8 n n O (1) . Alternatively it can be expressed as a 3dimensional matching problem, and solved in randomized expected time 2 n n O(1) [4].…”

Section: Nishiomentioning

confidence: 99%

Solving Single-Digit Sudoku Subproblems

Eppstein

2012

Lecture Notes in Computer Science

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“…However, many authors seem to have missed this paper as they published algorithms with slightly worse upper bounds on the running time [7,8]. The currently fastest algorithm for this problem is due to Byskov et al and (1) )-time and polynomial-space algorithm by Madsen [18]. These results were improved by Björklund and Husfeldt who gave an O(2 m m O (1) )-time and polynomial-space algorithm [3].…”

Section: "For Every Polynomial-time Algorithm You Have There Is An Ementioning

confidence: 99%

Partition Into Triangles on Bounded Degree Graphs

2012

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We consider the PARTITION INTO TRIANGLES problem on bounded degree graphs. We show that this problem is polynomial-time solvable on graphs of maximum degree three by giving a linear-time algorithm. We also show that this problem becomes N P-complete on graphs of maximum degree four. Moreover, we show that there is no subexponential-time algorithm for this problem on graphs of maximum degree four unless the Exponential-Time Hypothesis fails. However, the PARTITION INTO TRIANGLES problem on graphs of maximum degree at most four is in many cases practically solvable as we give an algorithm for this problem that runs in O(1.02220 n ) time and linear space.

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An algorithm for Exact Satisfiability analysed with the number of clauses as parameter

Cited by 11 publications

References 10 publications

Set Partitioning via Inclusion-Exclusion

Set Partitioning via Inclusion-Exclusion

Solving Single-Digit Sudoku Subproblems

Partition Into Triangles on Bounded Degree Graphs

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