Algorithmica

, Volume 68, Issue 3, pp 739–757

# Fixed-Parameter Tractability of Satisfying Beyond the Number of Variables

• Robert Crowston
• Gregory Gutin
• Mark Jones
• Venkatesh Raman
• Saket Saurabh
• Anders Yeo
Article

## Abstract

We consider a CNF formula F as a multiset of clauses: F={c 1,…,c m }. The set of variables of F will be denoted by V(F). Let B F denote the bipartite graph with partite sets V(F) and F and with an edge between vV(F) and cF if vc or $$\bar{v} \in c$$. The matching number ν(F) of F is the size of a maximum matching in B F . In our main result, we prove that the following parameterization of MaxSat (denoted by (ν(F)+k)-SAT) is fixed-parameter tractable: Given a formula F, decide whether we can satisfy at least ν(F)+k clauses in F, where k is the parameter.

A formula F is called variable-matched if ν(F)=|V(F)|. Let δ(F)=|F|−|V(F)| and δ (F)=max F′⊆F δ(F′). Our main result implies fixed-parameter tractability of MaxSat parameterized by δ(F) for variable-matched formulas F; this complements related results of Kullmann (IEEE Conference on Computational Complexity, pp. 116–124, 2000) and Szeider (J. Comput. Syst. Sci. 69(4):656–674, 2004) for MaxSat parameterized by δ (F).

To obtain our main result, we reduce (ν(F)+k)-SAT into the following parameterization of the Hitting Set problem (denoted by (mk)-Hitting Set): given a collection $$\mathcal{C}$$ of m subsets of a ground set U of n elements, decide whether there is XU such that CX≠∅ for each $$C\in \mathcal{C}$$ and |X|≤mk, where k is the parameter. Gutin, Jones and Yeo (Theor. Comput. Sci. 412(41):5744–5751, 2011) proved that (mk)-Hitting Set is fixed-parameter tractable by obtaining an exponential kernel for the problem. We obtain two algorithms for (mk)-Hitting Set: a deterministic algorithm of runtime $$O((2e)^{2k+O(\log^{2} k)} (m+n)^{O(1)})$$ and a randomized algorithm of expected runtime $$O(8^{k+O(\sqrt{k})} (m+n)^{O(1)})$$. Our deterministic algorithm improves an algorithm that follows from the kernelization result of Gutin, Jones and Yeo (Theor. Comput. Sci. 412(41):5744–5751, 2011).

## Keywords

Bipartite Graph Special Instance Deterministic Algorithm Maximum Match Truth Assignment
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Notes

### Acknowledgements

This research was partially supported by an International Joint grant of the Royal Society.

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## Authors and Affiliations

• Robert Crowston
• 1
• Gregory Gutin
• 1
• Mark Jones
• 1
• Venkatesh Raman
• 2
• Saket Saurabh
• 2
• Anders Yeo
• 3
1. 1.Royal HollowayUniversity of LondonEgham, SurreyUK
2. 2.The Institute of Mathematical SciencesChennaiIndia
3. 3.University of JohannesburgJohannesburgSouth Africa