An Improved Time-Efficient Approximate Kernelization for Connected Treedepth Deletion Set

TL;DR

This paper introduces a polynomial-sized approximate kernelization scheme for the CONNECTED -TREEDEPTH DELETION problem, enabling efficient approximate solutions for a complex graph modification problem.

Contribution

The authors develop a (1+)-approximate kernel with size polynomial in the parameter, advancing kernelization techniques for connected treedepth deletion.

Findings

Existence of a polynomial-sized approximate kernelization scheme for the problem.

The kernel size depends exponentially on  and , providing a tunable approximation.

The approach generalizes previous kernelization limitations for similar problems.

Abstract

We study the CONNECTED \eta-TREEDEPTH DELETION problem where the input instance is an undireted graph G = (V, E) and an integer k. The objective is to decide if G has a set S \subseteq V(G) of at most k vertices such that G - S has treedepth at most \eta and G[S] is connected. As this problem naturally generalizes the well-known CONNECTED VERTEX COVER, when parameterized by solution size k, the CONNECTED \eta-TREEDEPTH DELETION does not admit polynomial kernel unless NP \subseteq coNP/poly. This motivates us to design an approximate kernel of polynomial size for this problem. In this paper, we show that for every 0 < \epsilon <= 1, CONNECTED \eta-TREEDEPTH DELETION SET admits a (1+\epsilon)-approximate kernel with O(k^{2^{\eta + 1/\epsilon}}) vertices, i.e. a polynomial-sized approximate kernelization scheme (PSAKS).