Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, September 26, 2023, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: Nicolas El Maalouly

Top-k Perfect Matching: Motivation and LP Relaxation

In the Top-k Perfect Matching (TkPM) we are given a weighted graph and an integer k and the goal is to output a perfect matching that maximizes the top-k weight function (the top-k weight function is the sum of the k highest weights in a given set of edges). Such an objective function has been studied for problems like k-Clustering and load balancing but not yet for matching problems. The main motivation to study this problem however, is that it relates to the Exact Matching problem. Here we are given a graph with edges colored red and blue, and an integer k and the goal is to output a perfect matching with exactly k red edges. After introducing the problem in 1982, Papadimitriou and Yannakakis conjectured it to be NP-hard. Soon after however, Mulmuley et al. proved in 1987 that it can be solved in randomized polynomial time, which makes it unlikely to be NP-hard. Determining whether Exact Matching is in P remains an open problem and very little progress has been made towards that goal. The hope is that by studying Top-k perfect matching we can gain more insight into Exact Matching. The main focus of this talk will be the LP relaxation of the problem and its properties.

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