Computing Coverage Kernels Under Restricted Settings

TL;DR

This paper studies the computational complexity of the Minimum Coverage Kernel problem for sets of boxes, showing NP-hardness persists under restrictions and providing approximation algorithms for practical solutions.

Contribution

It demonstrates NP-hardness of the problem in restricted cases and introduces two polynomial-time approximation algorithms.

Findings

NP-hardness persists even in restricted instances

Two polynomial-time approximation algorithms are proposed

Provides insights into the problem's complexity and approximability

Abstract

We consider the Minimum Coverage Kernel problem: given a set $B$ of $d$-dimensional boxes, find a subset of $B$ of minimum size covering the same region as $B$. This problem is $\mathsf{NP}$-hard, but as for many $\mathsf{NP}$-hard problems on graphs, the problem becomes solvable in polynomial time under restrictions on the graph induced by $B$. We consider various classes of graphs, show that Minimum Coverage Kernel remains $\mathsf{NP}$-hard even for severely restricted instances, and provide two polynomial time approximation algorithms for this problem.