Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, November 28, 2023, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Adrian Taubner

Exact Matching: Algorithms and Related Problems

The Exact Matching Problem is a graph problem arising in computer science. Given a graph with each edge colored as red or blue, the question is whether there exists a perfect matching with exactly k red edges for a given integer k. While the problem was shown to be solvable in randomized polynomial time in the 1980s, a polynomial time deterministic algorithm for the Exact Matching has so far only been found for restricted classes such as complete and complete bipartite graphs. In this work, we provide additional results and ideas for finding a polynomial time deterministic algorithm for the Exact Matching Problem.

Arguably our most interesting result is using outcomes of the Edmond Blossom Weighted Algorithm to show a deterministic polynomial time algorithm for finding a Second Minimum Matching. We demonstrate how this could be extended to an algorithm finding a k-Minimum Matching and solving the Minimum + C Matching Problem in exponential time, with k, C serving as exponents.

Last but not least, we provided a deterministic polynomial time modification that transforms a Correct Parity Matching into a Second Minimum Matching in bipartite graphs.

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