A Faster Algorithm for the Maximum Even Factor Problem

TL;DR

This paper introduces a new, faster algorithm with $O(n^3 \, \log n)$ complexity for finding maximum even factors in odd-cycle symmetric digraphs, improving upon the previous $O(n^4)$ method.

Contribution

The paper presents a novel sparse recovery technique and an improved algorithm for maximum even factor problem in a specific class of digraphs.

Findings

Achieved $O(n^3 \log n)$ running time for the problem.

Developed a new sparse recovery technique.

Enhanced the efficiency of maximum even factor computation.

Abstract

Given a digraph $G = (VG,AG)$, an \emph{even factor} $M \subseteq AG$ is a subset of arcs that decomposes into a collection of node-disjoint paths and even cycles. Even factors in digraphs were introduced by Geleen and Cunningham and generalize path matchings in undirected graphs. Finding an even factor of maximum cardinality in a general digraph is known to be NP-hard but for the class of \emph{odd-cycle symmetric} digraphs the problem is polynomially solvable. So far, the only combinatorial algorithm known for this task is due to Pap; it has the running time of $O(n^4)$ (hereinafter $n$ stands for the number of nodes in $G$). In this paper we present a novel \emph{sparse recovery} technique and devise an $O(n^3 \log n)$-time algorithm for finding a maximum cardinality even factor in an odd-cycle symmetric digraph.