The Shortest Even Cycle Problem Is Tractable

Given a directed graph as input, 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), 150 (1999), pp. 929–975) and, independently, McCuaig (Electron. J. Combin., 11 (2004), R7900) (announced jointly at STOC 1997) gave an efficiently testable structural characterization of even-cycle-free directed graphs. Methodologically, our algorithm relies on the standard framework of algebraic fingerprinting and randomized polynomial identity testing over a finite field and, in fact, relies on a generating polynomial implicit in a paper of Vazirani and Yannakakis (Discrete Appl. Math., 25 (1989), pp. 179–190) that enumerates weighted cycle covers by the parity of their number of cycles as a difference of a permanent and a determinant polynomial. The need to work with the permanent—known to be #P-hard apart from a very restricted choice of coefficient rings (L. G. Valiant, Theoret. Comput. Sci., 8 (1979), pp. 189–201)—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 by extension of Valiant’s techniques, and (iii) enable emulation of finite-field arithmetic in characteristic 2. Here our work is foreshadowed by that of Björklund and Husfeldt (SIAM J. Comput., 48 (2019), pp. 1698–1710) who used a considerably less efficient commutative ring design—in particular, one lacking finite-field emulation—to obtain a polynomial-time algorithm for the shortest two disjoint paths problem in undirected graphs. Building on work of Gilbert and Tarjan (Numer. Math., 50 (1986), pp. 377–404) as well as Alon and Yuster (J. ACM, 42 (2013), pp. 844–856), we also show how ideas from the nested dissection technique for solving linear equation systems—introduced by George (SIAM J. Numer. Anal., 10 (1973), pp. 345–363) for symmetric positive definite real matrices—leads to faster algorithm designs in our present finite-ring randomized context when we have control of the separator structure of the input graph; for example, this happens when the input has bounded genus.