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

• Robert Crowston
• Gregory Gutin
• Mark Jones
• Venkatesh Raman
• Saket Saurabh
• Anders Yeo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7317)

## 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 an edge between v ∈ V(F) and c ∈ F if v ∈ c 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 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 (2000) and Szeider (2004) for MaxSat parameterized by δ *(F).

To prove our main result, we obtain an O((2e)2k k O(logk) (m + n) O(1))-time algorithm for the following parameterization of the Hitting Set problem: given a collection $$\cal C$$ of m subsets of a ground set U of n elements, decide whether there is X ⊆ U such that C ∩ X ≠ ∅ for each $$C\in \cal C$$ and |X| ≤ m − k, where k is the parameter. This improves an algorithm that follows from a kernelization result of Gutin, Jones and Yeo (2011).

## Keywords

Polynomial Time Bipartite Graph Polynomial Kernel 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.

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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 Holloway, University of LondonEghamUK
2. 2.The Institute of Mathematical SciencesChennaiIndia
3. 3.University of JohannesburgSouth Africa