Computer Science > Computational Complexity
[Submitted on 23 Nov 2022 (v1), last revised 19 Jun 2023 (this version, v3)]
Title:Odd Paths, Cycles and $T$-joins: Connections and Algorithms
View PDFAbstract:Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization" (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem and some multiflow problems. We clarify the interconnections between some of these problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in undirected graphs without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lovász (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''). On the other hand, we provide an efficient algorithm to the closely related and well-studied Minimum-weight Odd $T$-Join problem for non-negative weights: our algorithm runs in FPT time parameterized by $c$, where $c$ is the number of connected components in some efficiently computed minimum-weight $T$-join. If negative weights are also allowed, then finding a minimum-weight odd $\{s,t\}$-join is equivalent to the Minimum-weight Odd $T$-Join problem for arbitrary weights, whose complexity is still only conjectured to be polynomial-time solvable. The analogous problems for digraphs are also considered.
Submission history
From: Ildikó Schlotter [view email][v1] Wed, 23 Nov 2022 11:11:04 UTC (74 KB)
[v2] Fri, 9 Jun 2023 12:06:43 UTC (85 KB)
[v3] Mon, 19 Jun 2023 19:57:16 UTC (85 KB)
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