Search-Space Reduction Via Essential Vertices Revisited: Vertex Multicut and Cograph Deletion

TL;DR

This work develops a LP-based framework linking the integrality gap of v-avoiding relaxations to the existence of $c$-essential detectors for vertex-hitting-set problems, enabling systematic search-space reduction in FPT algorithms. The authors obtain concrete upper bounds, including $3$-Essential detection for Vertex Multicut, $5$-Essential detection for Directed Vertex Multicut, and $3.5$-Essential detection for Cograph Deletion, with corresponding fixed-parameter time bounds that depend on the number of non-$c$-essential vertices. They further compare these to the standard LP relaxations, proving a tight $4$-gap for the standard Cograph Deletion LP via probabilistic constructions, and offer hardness results under the UGC showing that the detected thresholds (e.g., $2- ext{varepsilon}$ for DFVS and $1.5- ext{varepsilon}$ for VC) are likely optimal. The results illuminate when preprocessing can dramatically shrink the search space and point to future work on stronger relaxations and broader problem classes.

Abstract

For an optimization problem $Π$ on graphs whose solutions are vertex sets, a vertex $v$ is called $c$-essential for $Π$ if all solutions of size at most $c \cdot OPT$ contain $v$. Recent work showed that polynomial-time algorithms to detect $c$-essential vertices can be used to reduce the search space of fixed-parameter tractable algorithms solving such problems parameterized by the size $k$ of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for $3$-Essential detection for Vertex Multicut, which translates into an algorithm that finds a minimum multicut of an undirected $n$-vertex graph $G$ in time $2^{O(\ell^3)} \cdot n^{O(1)}$, where $\ell$ is the number of vertices in an optimal solution that are not $3$-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of $c$, the detection task becomes NP-hard assuming the Unique Games Conjecture. For example, we show that ($2-\varepsilon$)-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects $2$-essential vertices is best-possible.

Theorems & Definitions (21)

• Theorem 1: schrijver-combopt
• Theorem 2
• Lemma 3
• Theorem 4
• Corollary 5
• Lemma 6
• Theorem 7
• Lemma 8
• Theorem 9
• Corollary 10