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 Vondr\'ak; projective planes give explicit tight instances whenever one of order $p-1$ exists.
A Las Vegas algorithm using $\tilde O(n\sqrt r)$ expected independence queries and a deterministic algorithm using $\tilde O(nr^{2/3})$ queries are given.
For positive integers $n,r,t$, let $\delta(n,r,t)$ denote the maximum possible minimum degree of a balanced $r$-partite graph with parts of size $n$ and chromatic number at most $t$. Lo, Treglown and Zhao established a general upper bound for this parameter and used it, together with explicit constructions, to determin...
We study prophet inequalities under intersections of $q$ partition matroids, where an online algorithm irrevocably selects elements with independent nonnegative values drawn from known distributions and revealed in an adversarial order. We prove an $\Omega(q/\log q)$ lower bound on the competitive ratio. Together with...
This work proves the existence of a $(q, 1/(1+q))-balanced contention resolution scheme for the integral knapsack polytope for every $q \in [0, 1]$, which may be of independent interest.
Alternative parameterizations of the k-core problem are investigated to identify conditions under which it can be placed into sub-polynomial complexity classes, and the boundary of parallel tractability for the k-core problem is explored.
Y. S. To, Cristina G. Fernandes· 0 citations
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