arXiv subjects

Theophile Thiery

Publications and source records attributed to Theophile Thiery.

6 recordsLinked to original sources

arXiv2026

Submodular Maximization over Many Matroids via Ordered Local Search

Given a monotone submodular function, we consider the problem of finding a maximum-valued set in the intersection of $k$ matroids. Our main result is a polynomial time local search based algorithm achieving a $\frac{k}{2} + o(k)$ approximation guarantee. This asymptotically matches the best-known guarantee of $\frac{k}{2} + \epsilon$ in the unweighted setting by Lee, Sviridenko, and Vondr\'ak (2009). Prior to this work, the state-of-the-art was a $\frac{\ln(4)k}{1+\ln(2)} + o(k)$-approximation algorithm obtained by Feldman and Ward (2026). Our approach extends to Matroid $k$-Parity yielding the same approximation guarantee. In contrast to the weight bucketing approach underlying the recent advances of Singer and Thiery (2025) and Feldman and Ward (2026), our algorithm processes elements greedily in decreasing order of marginal value and searches for sufficiently profitable swaps, whose gain exceeds a parameter $\alpha$ given as a function of $k$. We further combine this idea with the weight bucketing approach to obtain improved guarantees for weighted $k$-Set Packing. Our second main result is a $\frac{\ln(4)k}{3} + o(k)$-approximation algorithm for weighted $k$-Set Packing, improving on the state of the art $\frac{k}{2.00561} + O(1)$-approximation by Neuwohner (2023).

cs.DS↗

arXiv2024

Better Approximation for Weighted $k$-Matroid Intersection

We consider the problem of finding an independent set of maximum weight simultaneously contained in $k$ matroids over a common ground set. This $k$-matroid intersection problem appears naturally in many contexts, for example in generalizing graph and hypergraph matching problems. In this paper, we provide a $(k+1)/(2 \ln 2)$-approximation algorithm for the weighted $k$-matroid intersection problem. This is the first improvement over the longstanding $(k-1)$-guarantee of Lee, Sviridenko and Vondr\'ak (2009). Along the way, we also give the first improvement over greedy for the more general weighted matroid $k$-parity problem. Our key innovation lies in a randomized reduction in which we solve almost unweighted instances iteratively. This perspective allows us to use insights from the unweighted problem for which Lee, Sviridenko, and Vondr\'ak have designed a $k/2$-approximation algorithm. We analyze this procedure by constructing refined matroid exchanges and leveraging randomness to avoid bad local minima.

cs.DS↗

arXiv2024

Asymptotically Optimal Hardness for $k$-Set Packing and $k$-Matroid Intersection

For any $\varepsilon > 0$, we prove that $k$-Dimensional Matching is hard to approximate within a factor of $k/(12 + \varepsilon)$ for large $k$ unless $\textsf{NP} \subseteq \textsf{BPP}$. Listed in Karp's 21 $\textsf{NP}$-complete problems, $k$-Dimensional Matching is a benchmark computational complexity problem which we find as a special case of many constrained optimization problems over independence systems including: $k$-Set Packing, $k$-Matroid Intersection, and Matroid $k$-Parity. For all the aforementioned problems, the best known lower bound was a $\Omega(k /\log(k))$-hardness by Hazan, Safra, and Schwartz. In contrast, state-of-the-art algorithms achieved an approximation of $O(k)$. Our result narrows down this gap to a constant and thus provides a rationale for the observed algorithmic difficulties. The crux of our result hinges on a novel approximation preserving gadget from $R$-degree bounded $k$-CSPs over alphabet size $R$ to $kR$-Dimensional Matching. Along the way, we prove that $R$-degree bounded $k$-CSPs over alphabet size $R$ are hard to approximate within a factor $\Omega_k(R)$ using known randomised sparsification methods for CSPs.

cs.CC↗

arXiv2023

Two-Sided Weak Submodularity for Matroid Constrained Optimization and Regression

We study the following problem: Given a variable of interest, we would like to find a best linear predictor for it by choosing a subset of $k$ relevant variables obeying a matroid constraint. This problem is a natural generalization of subset selection problems where it is necessary to spread observations amongst multiple different classes. We derive new, strengthened guarantees for this problem by improving the analysis of the residual random greedy algorithm and by developing a novel distorted local-search algorithm. To quantify our approximation guarantees, we refine the definition of weak submodularity by Das and Kempe and introduce the notion of an upper submodularity ratio, which we connect to the minimum $k$-sparse eigenvalue of the covariance matrix. More generally, we look at the problem of maximizing a set function $f$ with lower and upper submodularity ratio $γ$ and $β$ under a matroid constraint. For this problem, our algorithms have asymptotic approximation guarantee $1/2$ and $1-e^{-1}$ as the function is closer to being submodular. As a second application, we show that the Bayesian A-optimal design objective falls into our framework, leading to new guarantees for this problem as well.

cs.DS↗

arXiv2023

An Improved Approximation for Maximum Weighted $k$-Set Packing

We consider the weighted $k$-set packing problem, in which we are given a collection of weighted sets, each with at most $k$ elements and must return a collection of pairwise disjoint sets with maximum total weight. For $k = 3$, this problem generalizes the classical 3-dimensional matching problem listed as one of the Karp's original 21 NP-complete problems. We give an algorithm attaining an approximation factor of $1.786$ for weighted 3-set packing, improving on the recent best result of $2-\frac{1}{63,700,992}$ due to Neuwohner. Our algorithm is based on the local search procedure of Berman that attempts to improve the sum of squared weights rather than the problem's objective. When using exchanges of size at most $k$, this algorithm attains an approximation factor of $\frac{k+1}{2}$. Using exchanges of size $k^2(k-1) + k$, we provide a relatively simple analysis to obtain an approximation factor of 1.811 when $k = 3$. We then show that the tools we develop can be adapted to larger exchanges of size $2k^2(k-1) + k$ to give an approximation factor of 1.786. Although our primary focus is on the case $k = 3$, our approach in fact gives slightly stronger improvements on the factor $\frac{k+1}{2}$ for all $k > 3$. As in previous works, our guarantees hold also for the more general problem of finding a maximum weight independent set in a $(k+1)$-claw free graph.

cs.DS↗

arXiv2021

Improved Multi-Pass Streaming Algorithms for Submodular Maximization with Matroid Constraints

We give improved multi-pass streaming algorithms for the problem of maximizing a monotone or arbitrary non-negative submodular function subject to a general $p$-matchoid constraint in the model in which elements of the ground set arrive one at a time in a stream. The family of constraints we consider generalizes both the intersection of $p$ arbitrary matroid constraints and $p$-uniform hypergraph matching. For monotone submodular functions, our algorithm attains a guarantee of $p+1+\varepsilon$ using $O(p/\varepsilon)$-passes and requires storing only $O(k)$ elements, where $k$ is the maximum size of feasible solution. This immediately gives an $O(1/\varepsilon)$-pass $(2+\varepsilon)$-approximation algorithms for monotone submodular maximization in a matroid and $(3+\varepsilon)$-approximation for monotone submodular matching. Our algorithm is oblivious to the choice $\varepsilon$ and can be stopped after any number of passes, delivering the appropriate guarantee. We extend our techniques to obtain the first multi-pass streaming algorithm for general, non-negative submodular functions subject to a $p$-matchoid constraint with a number of passes independent of the size of the ground set and $k$. We show that a randomized $O(p/\varepsilon)$-pass algorithm storing $O(p^3k\log(k)/\varepsilon^3)$ elements gives a $(p+1+\barγ+O(\varepsilon))$-approximation, where $\bar{gamma}$ is the guarantee of the best-known offline algorithm for the same problem.

cs.DS↗