# Simple Approximation Algorithms for MAXNAESP and Hypergraph *2-colarability*

## Abstract

Hypergraph *2-colorability*, also known as set splitting, is a widely studied problem in graph theory. In this paper we study the maximization version of the same. We recast the problem as a special type of satisfiability problem and give approximation algorithms for it. Our results are valid for hypergraph *2-colorability*, set splitting and MAX-CUT (which is a special case of hypergraph *2-colorability*) because the reductions are approximation preserving. Here we study the MAXNAESP problem, the optimal solution to which is a truth assignment of the literals that maximizes the number of clauses satisfied. As a main result of the paper, we show that any locally optimal solution (a solution is locally optimal if its value cannot be increased by complementing assignments to literals and pairs of literals) is guaranteed a performance ratio of \(
\frac{1}
{2} + \in
\)
. This is an improvement over the ratio of \(
\frac{1}
{2}
\)
attributed to another local improvement heuristic for MAX-CUT [6]. In fact we provide a bound of \(
\frac{k}
{{k + 1}}
\)
for this problem, where *k* ≥ 3 is the minimum number of literals in a clause. Such locally optimal algorithms appear to subsume typical greedy algorithms that have been suggested for problems in the general domain of satisfiability. It should be noted that the NAESP problem where each clause has exactly two literals, is equivalent to MAX-CUT. However, obtaining good approximation ratios using semi-definite programming techniques [3] appears difficult. Also, the randomized rounding algorithm as well as the simple randomized algorithm both [4] yield a bound of \(
\frac{1}
{2}
\)
for the MAXNAESP problem. In contrast to this, the algorithm proposed in this paper obtains a bound of \(
\frac{1}
{2} + \in
\)
for this problem.

## Keywords

Approximation Algorithms*Hypergraph*2-colorability Set Splitting MAXNAESP MAX-CUT

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