An Improved Approximation for Maximum $k$-Dependent Set on Bipartite Graphs

TL;DR

This paper introduces an improved approximation algorithm for the Maximum $k$-dependent Set problem on bipartite graphs, achieving a better ratio than previous methods and applicable to K"{o}nig-Egerváry graphs.

Contribution

It presents a new approximation algorithm with a tighter ratio for bipartite graphs, extending its applicability to K"{o}nig-Egerváry graphs.

Findings

Achieves a $(1+rac{k}{k+2})$-approximation ratio.

Runs in $O(k m oot{n} )$ time for graphs with $n$ vertices and $m$ edges.

Improves upon the previous ratio of $1+rac{k}{k+1}$.

Abstract

We present a $(1+\frac{k}{k+2})$-approximation algorithm for the Maximum $k$-dependent Set problem on bipartite graphs for any $k\ge1$. For a graph with $n$ vertices and $m$ edges, the algorithm runs in $O(k m \sqrt{n})$ time and improves upon the previously best-known approximation ratio of $1+\frac{k}{k+1}$ established by Kumar et al. [Theoretical Computer Science, 526: 90--96 (2014)]. Our proof also indicates that the algorithm retains its approximation ratio when applied to the (more general) class of K\"{o}nig-Egerv\'{a}ry graphs.