Problem
Research questions and friction points this paper is trying to address.
weighted k-matroid intersection
integrality gap
LP relaxation
matchoid
Integrality-Gap Bounds for Weighted Matchoids and Matroid Intersection
🤖 AI Summary
研究解决了加权k-拟阵交问题的整数间隙界问题，通过改进LP松弛方法将上界从k提升到k-1+1/k，并推广至p-匹配拟阵。
📝 Abstract
The weighted $k$-matroid intersection problem asks for a maximum-weight set that is independent in each of $k$ matroids on a common ground set. The natural LP relaxation optimizes over the intersection of the $k$ matroid independent set polytopes. It is conjectured that this LP has integrality gap at most $k-1$. The conjecture is known for $k\le3$, but for $k\ge4$ the best general upper bound was $k$. We improve this bound to $k-1+1/k$. More generally, we prove that the natural LP of a $p$-matchoid has integrality gap at most $p-1+1/p$, with a deterministic LP-relative algorithm attaining the same factor. The matchoid extension resolves the $p$-matchoid part of a conjecture of Lee, Sviridenko, and Vondrák; projective planes give explicit tight instances whenever one of order $p-1$ exists.
Innovation
Methods, ideas, or system contributions that make the work stand out.
weighted k-matroid intersection
integrality gap
linear programming (LP)
p-matchoid
deterministic LP-relative algorithm
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