LP integrality gap for weighted k-matroid intersection
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What is the integrality gap of the natural linear program for weighted intersection of k matroids? The general conjectured factor is k−1; a weaker target is k−1+1/k.
Why it matters and what counts as progress
Why it matters: A smaller LP gap gives a stronger approximation guarantee relative to a computable relaxation.
What counts as progress: A deterministic LP-relative factor below k for general k≥4, a proof reaching k−1, or a larger lower-bound example; the known k=3 factor is already two.
How the problem developed
- 2013 · Conjecture [1]
Jon Lee, Maxim Sviridenko and Jan Vondrák Conjectured an integrality gap of k−1 for the natural LP of weighted k-matroid intersection.
- 18 Sept 2026 · Improved upper bound [2]
Yu Cong and Yajie Zhao, with GPT-5.6 Sol assisting the proof and algorithm development Improved the general weighted k-matroid-intersection LP upper bound from k to k−1+1/k; the paper also resolves the related p-matchoid conjecture component.
AI contribution
Integrality-Gap Bounds for Weighted Matchoids and Matroid Intersection
For weighted k-matroid intersection with k≥2, the natural LP has integrality gap at most k−1+1/k. A deterministic LP-relative algorithm achieves that factor; the paper separately extends the bound to p-matchoids.
Sources
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Publication dates and editorial changes.
AI-assisted proof improves weighted matroid-intersection gap