Math-1877 · mathematics · Combinatorics

Exact matching in red-blue bipartite graphs

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Let $G=(U\sqcup W,E)$ be a finite bipartite graph with $|U|=|W|=n$. Each edge is labelled red or blue, and an integer $k$ with $0\le k\le n$ is specified. A perfect matching is a set $M\subseteq E$ in which each vertex belongs to exactly one edge. The exact matching decision problem asks whether \[ |M\cap E_{\rm red}|=k \] for some perfect matching $M$.

Find a deterministic polynomial-time algorithm for this decision problem, and for constructing a matching when it exists, or a useful combinatorial characterization leading to such an algorithm. The source already records randomized polynomial algorithms, so the unresolved algorithmic target must distinguish deterministic computation from those methods.

The prescribed number of red edges is exact. Merely finding the smallest and largest attainable red counts does not justify that every intermediate integer is attainable. The graph is not assumed complete, and neither colour class need itself be a matching. There are exactly two colours; allowing an arbitrary number of colours with individual exact quotas would give a different problem. A feasibility certificate on a yes instance consists simply of the matching, while the main challenge is to recognize both yes and no instances efficiently and deterministically.

Short English statement

Can exact red-edge counts in bipartite perfect matchings be decided and realized in deterministic polynomial time?

Sources

S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3000034 (AMR-029-0034), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
S2 Egres Open - List of open problems, Open-problem page: Exact matching in red-blue bipartite graphs. Original source indexed by AMR-029-0034. https://oldlemon.cs.elte.hu/egres/open/List_of_open_problems.
S3 EGRES Open, Exact matching in red-blue bipartite graphs. Individual source page and explanatory remarks. https://lemon.cs.elte.hu/egres/open/Exact_matching_in_red-blue_bipartite_graphs.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

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Record details

Subjects: Combinatorics
Collection: Top Open Problems
Mathematics rank: 1877
UnsolvedMath: Catalogue entry