Problem
Research questions and friction points this paper is trying to address.
Exact Matching
bipartite graph
perfect matching
red-blue edge coloring
deterministic polynomial-time
Bipartite Exact Matching in P
🤖 AI Summary
This work resolves a forty-year-old open problem concerning the deterministic polynomial-time decidability of whether a bipartite graph contains a perfect matching with exactly $t$ red edges. By employing tight cut decompositions to reduce the problem to brace components, and leveraging McCuaig’s inductive characterization of braces together with a matching-induced Two-extra Hall theorem and a $q$-circuit lemma to handle redundant edge configurations, the authors establish the non-vanishing affine slice conjecture for bipartite braces. The study presents the first deterministic $O(n^6)$ algorithm for the Exact Matching problem and provides a complete formal verification of the result in Lean 4.
📝 Abstract
The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly t red edges. Introduced by Papadimitriou and Yannakakis in 1982, the problem has resisted deterministic polynomial-time algorithms for over four decades, despite admitting a randomized solution via the Schwartz-Zippel lemma since 1987. We prove the Affine-Slice Nonvanishing Conjecture (ASNC) for all bipartite braces and give a deterministic O(n^6) algorithm for Exact Matching on all bipartite graphs. The algorithm follows via the tight-cut decomposition, which reduces the decision problem to brace blocks.
The proof proceeds by structural induction on McCuaig's brace decomposition. We establish the McCuaig exceptional families, the replacement determinant algebra, and the narrow-extension cases (KA, J3 to D1). For the superfluous-edge step, we introduce two closure tools: a matching-induced Two-extra Hall theorem that resolves the rank-(m-2) branch via projective-collapse contradiction, and a distinguished-state q-circuit lemma that eliminates the rank-(m-1) branch entirely by showing that any minimal dependent set containing the superfluous state forces rank m-2. The entire proof has been formally verified in the Lean 4 proof assistant.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Exact Matching
bipartite graphs
deterministic algorithm
tight-cut decomposition
formal verification
🔎 Similar Papers
No similar papers found.
