The matroid secretary problem asks an online algorithm to select a high-weight independent set from elements arriving in uniformly random order, with immediate and irrevocable decisions. Singla (2026) recently gave a $4$-competitive algorithm for arbitrary matroids using only the number of elements and independence queries on already-arrived elements. Following his approach, we obtain an improved competitive ratio of $2+\sqrt3\approx3.7321$ in the same information model. Our algorithm accepts every element of a fixed canonical optimum with probability at least $2-\sqrt3$ and uses $O(n^2)$ independence queries. The algorithm modifies Singla's reversible reference process by retaining a randomly chosen part of the sample as a reserve whose membership in the reference greedy solution is not frozen. Balancing the remaining sample and post-sample elements preserves reversibility and allows an exact calculation of the probability that an exchange partner blocks a target element. The resulting guarantee has a direct analytic proof.
We consider the weighted matroid intersection problem in the independence-oracle model. A sequence of works by Huang--Kakimura--Kamiyama [SODA'16 \&Math. Program'19], Chekuri--Quanrud [SODA'16], Quanrud [ICALP'24], and Dudeja--Grilnberger [IPCO'26] has developed efficient $(1-\varepsilon)$-approximation algorithms for...
The matroid secretary problem (MSP) is one of the cleanest, and most captivating open problems in online algorithms. The famous MSP conjecture stipulates that there exists a constant-competitive algorithm, yet to date the best known algorithms are $O(\log \log (\text{rank}))$ competitive. It is widely believed that all...
Paul Dütting, R. P. Leme, Martin Pál et al.· 5 citations
It is proved that, for arbitrary $p and arbitrary initial orderings $\sigma$ and arbitrary initial orderings $\sigma$, after polynomially many queries in the number of items, the expected cost of a query is at most $\mathsf{OPT} + O(1)$.
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.
Karger's randomized contraction algorithm finds a minimum-weight cocircuit of a matroid whenever the cogirth-density ratio is bounded. We prove that the same hypothesis yields a deterministic algorithm with the same exponent. If every contraction minor of rank at least $r_0$ of a matroid $M$ has cogirth-density ratio a...
It is demonstrated how to compute the largest value of $\alpha$ for which a committee still fails the axiom using this mincut formulation, and a direct mincut formulation of the problem on a bipartite graph is given.
Drew Springham· 0 citations
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