Search-Space Reduction via Essential Vertices

TL;DR

This paper introduces a preprocessing technique that identifies essential vertices in graphs to reduce problem size and improve fixed-parameter tractable algorithms for vertex-subset problems.

Contribution

It defines c-essential vertices and demonstrates polynomial-time algorithms to find solution subsets containing these vertices for classic problems.

Findings

Polynomial-time algorithms for identifying essential vertices.

Reduced search space for FPT algorithms.

Improved fixed-parameter algorithms based on essential vertices.

Abstract

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.