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 at most $c$, then a minimum-weight cocircuit of $M$ is computable deterministically in $m^{O(r_0)} n^{O(c)}$ time when the contraction minors of bounded rank have at most $m$ parallel classes, by an algorithm that knows neither $r_0$ nor $c$. As a consequence, we give a deterministic algorithm computing the cogirth of rank-$p$ perturbed graphic matroids in $2^{O(p^2)} n^{O(1)}$ time, fixed-parameter tractable in $p$, settling the cogirth side of a question of Geelen and Kapadia (2018). The extensions of the contraction method carry over deterministically: enumerating all near-minimum 1-cocycles, computing a minimum-weight $k$-cocycle, and computing the Pareto frontier under several positive criteria.
We consider the determinant maximization problem under partition constraints: Given an $n\times n$ PSD matrix A and a partition matroid $M$ on $[n]$, find a base $S$ of $M$ that maximizes $\det(A_{S,S})$. We give an $e^{O(k)}$-approximation algorithm to find such a set $S$, where $k$ is the rank of $M$. This improves u...
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.
Albertson's conjecture asserts that every finite simple graph $G$ with $\chi(G) \ge r$ satisfies $\operatorname{cr}(G) \ge \operatorname{cr}(K_r)$. Building on Cranston's verification for $r \le 24$ and his reduction of $r \in \{25,26\}$ to three residual orders, we eliminate those residual cases and then prove the cas...
We prove that there is an absolute positive constant $c$ such that every bipartite graph with $N$ vertices in each part and at least $N^2/2$ edges contains a complete bipartite graph $K_{t,t}$ whenever $N\ge c\, 2^{t}$. This improves the classical K\H{o}v\'ari-S\'os-Tur\'an bound requiring $N$ of order $t\,2^t $. As a...
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.
The soundness uses a result of Haeupler, Saha, and Srinivasan building on the proof of Moser and Tardos, to upper-bound the probability that a fixed relatively large subset is an independent set after the Moser-Tardos algorithm terminates.
Édouard Bonnet· 0 citations
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