Analysis of the two-for-one swap heuristic for approximating the maximum independent set in a k-polymatroid

Abstract

Let $f:2^N\to Z^{+}$ be a polymatroid (an integer-valued non-decreasing submodular set function with $f(\varnothing )=0$). A k-polymatroid satisfies that $f\left(e\right)\le k$ for all $e\in N$. We call $S\subseteq N$ independent if $f\left(S\right)={\sum }_{e\in S}f\left(e\right)$ and $f\left(e\right)>0$ for all $e\in S$. Such a set was also called a matching. Finding a maximum-size independent set in a 2-polymatroid has been studied and polynomial-time algorithms are known for linear polymatroids. For $k\ge 3$, the problem is NP-hard, and a $\left(\right(2/k)-ϵ)$-approximation is known and is obtained by swapping as long as possible a subset of up to ${(1/ϵ)}^{{\log }_{k-1}(2k+1)}$ elements from the current solution by a set with one more element.

Here we give a simple analysis of the more particular two-for-one repeated swapping heuristic, obtaining a tight (weaker) $(2/(k+1\left)\right)$-approximation.