Finding a Shortest Even Hole in Polynomial Time
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An even (respectively, odd) hole in a graph is an induced cycle with even (respectively, odd) length that is at least four. Bienstock [DM 1991 and 1992] proved that detecting an even (respectively, odd) hole containing a given vertex is NP-complete. Conforti, Chornuéjols, Kappor, and Vušković [FOCS 1997] gave the first known polynomial-time algorithm to determine whether a graph contains even holes. Chudnovsky, Kawarabayashi, and Seymour [JGT 2005] estimated that Conforti et al.'s algorithm runs in O(n^40) time on an n-vertex graph and reduced the required time to O(n^31). Subsequently, da Silva and Vušković [JCTB 2013], Chang and Lu [JCTB 2017], and Lai, Lu, and Thorup [STOC 2020] improved the time to O(n^19), O(n^11), and O(n^9), respectively. The tractability of determining whether a graph contains odd holes has been open for decades until the algorithm of Chudnovsky, Scott, Seymour, and Spirkl [JACM 2020] that runs in O(n^9) time, which Lai et al. also reduced to O(n^8). By extending Chudnovsky et al.'s techniques for detecting odd holes, Chudnovsky, Scott, and Seymour [Combinatorica 2020 to appear] (respectively, [arXiv 2020]) ensured the tractability of finding a long (respectively, shortest) odd hole. They also ensured the NP-hardness of finding a longest odd hole, whose reduction also works for finding a longest even hole. Recently, Cook and Seymour ensured the tractability of finding a long even hole. An intriguing missing piece is the tractability of finding a shortest even hole, left open for at least 15 years by, e.g., Chudnovsky et al. [JGT 2005] and Johnson [TALG 2005]. We resolve this long-standing open problem by giving the first known polynomial-time algorithm, running in O(n^31) time, for finding a shortest even hole in an n-vertex graph that contains even holes.
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