The Shortest Even Cycle Problem is Tractable

TL;DR

This paper presents a polynomial-time algorithm for finding the shortest even cycle in a directed graph, solving a long-standing open problem using algebraic fingerprinting, randomized polynomial identity testing, and novel ring constructions.

Contribution

The authors introduce a new algebraic approach utilizing specialized rings and polynomial identity testing to efficiently detect shortest even cycles in directed graphs, a problem previously unsolved in polynomial time.

Findings

Developed a randomized polynomial-time algorithm for shortest even cycle detection.

Designed finite rings supporting efficient permanent computations and characteristic 2 emulation.

Extended algebraic techniques to graphs with bounded genus for faster algorithms.

Abstract

Given a directed graph, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997) gave an efficiently testable structural characterisation of even-cycle-free directed graphs. Methodologically, our algorithm relies on algebraic fingerprinting and randomized polynomial identity testing over a finite field, and uses a generating polynomial implicit in Vazirani and Yannakakis ( Discrete Appl. Math. 1989) that enumerates weighted cycle covers as a difference of a permanent and a determinant polynomial. The need to work with the permanent is where our main technical contribution occurs. We design a family of finite commutative rings of characteristic 4 that simultaneously (i) give a nondegenerate representation for the generating polynomial identity via the permanent and the determinant, (ii) support efficient permanent computations, and (iii) enable emulation of finite-field arithmetic in characteristic 2. Here our work is foreshadowed by that of Bj\"orklund and Husfeldt (SIAM J. Comput. 2019), who used a considerably less efficient ring design to obtain a polynomial-time algorithm for the shortest two disjoint paths problem. Building on work of Gilbert and Tarjan (Numer. Math. 1978) as well as Alon and Yuster (J. ACM 2013), we also show how ideas from the nested dissection technique for solving linear equation systems leads to faster algorithm designs when we have control on the separator structure of the input graph; for example, this happens when the input has bounded genus.