Covering vertices by sequential stars

2026-05-23

AI Summary

This study addresses the vertex-disjoint star cover problem in graphs, where each star must contain a number of leaves within the interval $[k, \ell]$ (with $1 \leq k < \ell$ and $\ell$ possibly infinite), aiming to maximize the number of covered vertices. The authors propose a novel algorithm based on local search and amortized analysis, enhanced by augmented configurations that connect distant neighborhoods to enable effective local improvements. Their main contributions include proving that the problem is APX-hard for $k=2$, thereby resolving an open complexity question, and establishing optimal or first-known approximation guarantees in four key settings: a $(k+1)/2$-approximation for $k \geq 3$ and $\ell = \infty$, a $4/3$-approximation for $k=2$ and $\ell = \infty$, a $1 + \ell/(\ell+1)$-approximation for $k=2 < \ell$, and the first approximation algorithm for the general case $3 \leq k < \ell$.

Abstract

We study the problem of covering the maximum number of vertices in a graph by a collection of vertex-disjoint stars, each with a number of satellites in a given interval $[k, \ell]$, where $1 \le k < \ell$ and $\ell$ can be infinity. This is referred to as sequential {\sc $[k, \ell]$-Star Packing} problem. It is solvable in polynomial time when $k = 1$, but becomes strongly NP-hard when $k \ge 2$. In this paper, we propose either the first or an improved approximation algorithm for the following four sequential settings: 1) a $\frac {k+1}2$-approximation algorithm when $k \ge 3$ and $\ell = \infty$, improving the previous best ratio of $\frac {(k+1)^2}{2k+1}$; 2) a $\frac 43$-approximation algorithm when $k = 2$ and $\ell = \infty$, improving the previous best ratio of $\frac 32$; 3) the first $(1 + \frac \ell{\ell+1})$-approximation algorithm when $2 = k < \ell$; and 4) the first $(1 + \max\left\{\frac {k-1}2, \frac {(k+1) \ell}{3 (\ell+1)}\right\})$-approximation algorithm when $3 \le k < \ell$. Besides the main algorithmic techniques being local search coupled with amortized analysis, we observe augmenting configurations to bridge two distant neighborhoods for a local improvement operation. Additionally, the problem has been shown APX-hard when $k \ge 3$; we prove its APX-hardness for the last remaining case where $k = 2$.

Problem

• star packing
• vertex cover
• approximation algorithm
• NP-hard
• graph theory

Innovation

• approximation algorithm
• star packing
• local search
• amortized analysis
• APX-hardness