Directed Acyclic Subgraph Problem Parameterized above the Poljak-Turzik Bound

TL;DR

This paper proves that finding a large acyclic subgraph in a directed graph, exceeding a known bound by a parameter k, is fixed-parameter tractable and provides algorithms for solving and reducing the problem efficiently.

Contribution

It establishes the fixed-parameter tractability of the directed acyclic subgraph problem above the Poljak-Turzik bound and offers polynomial kernelization algorithms.

Findings

The problem is fixed-parameter tractable with a runtime of (12k)!n^{O(1)}.

Existence of a polynomial-time algorithm for kernelization to size O(k^2).

Provides algorithms for both solving and reducing the problem efficiently.

Abstract

An oriented graph is a directed graph without directed 2-cycles. Poljak and Turz\'{i}k (1986) proved that every connected oriented graph $G$ on $n$ vertices and $m$ arcs contains an acyclic subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$ arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does $G$ have an acyclic subgraph with least $\frac{m}{2}+\frac{n-1}{4}+k$ arcs, where $k$ is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runtime $(12k)!n^{O(1)}$. Thus, the problem is fixed-parameter tractable. We also prove that there is a polynomial time algorithm that either establishes that the input instance of the problem is a Yes-instance or reduces the input instance to an equivalent one of size $O(k^2)$.