Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.00920

A Faster Auction Algorithm for Weighted Matroid Intersection

Tatsuya Terao
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 this problem. We present a simple deterministic auction algorithm that, given two matroids on a common ground set of size $n$, computes a $(1-\varepsilon)$-approximate maximum-weight common independent set using $O(n \varepsilon^{-2} \log^2(n))$ independence-oracle queries. This is the first deterministic $(1-\varepsilon)$-approximation algorithm for the weighted matroid intersection problem whose query complexity is nearly linear in $n$ and polynomial in $1/\varepsilon$. Our algorithm builds on the auction algorithm for unweighted matroid intersection by Huang--Kobayashi ['26], together with the analysis of the auction algorithm for weighted bipartite matching by Liu--Ke--Khuller [APPROX'23].

Explore related subjects

cs.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tatsuya Terao. 2026-10-01. A Faster Auction Algorithm for Weighted Matroid Intersection. https://arxiv.org/abs/2610.00920

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Instance-Optimality of Bidirectional PageRank Estimation

We study the problem of estimating a vertex's PageRank within a constant relative error, with constant probability. We prove that an adaptive variant of the simple classic bidirectional algorithm is instance-optimal up to a polylogarithmic factor for all directed graphs of order $n$ whose maximum in- and out-degrees are at most a constant fraction of $n$. In other words, there is no correct algorithm that can be faster than our algorithm on any such graph by more than a polylogarithmic factor. We further extend the instance-optimality to all graphs in which at most a polylogarithmic number of vertices have unbounded degrees. This covers all sparse graphs with $\tilde{O}(n)$ edges. In addition, we provide a counterexample showing that the bidirectional algorithm is not instance-optimal for graphs whose degrees are mostly equal to $n$. We also consider weighted graphs and multigraphs. We show that the bidirectional algorithm is instance-optimal on \emph{all} multigraphs, but for weighted simple graphs, we have almost the same limitations as for unweighted simple graphs.

cs.DS↗
arXiv2026

Solving Hypergraph Laplacian Systems in Almost-Linear Time

For a connected weighted hypergraph, we give a randomized almost-linear-time solver for the Poisson problem for the cut-based hypergraph Laplacian in the natural input size $P=\sum_{e\in E}|e|$, the sum of hyperedge sizes. For every fixed constant $C>0$, our randomized algorithm runs in $P^{1+o(1)}$ time and, with high probability over its internal randomness, returns a primal point and a dual certificate, with additive optimality gap at most $\exp(-\log^C P)$. A key step is to rewrite the Fenchel dual as a convex-flow problem on an auxiliary $O(P)$-arc graph, yielding a near-optimal dual flow. The main difficulty is primal recovery, because this flow does not by itself determine a primal potential. Our main new ingredient is a recovery theorem showing that, for primal recovery, the detailed routing of the dual flow inside each hyperedge gadget can be discarded: one nonnegative scalar per hyperedge is enough. After the necessary finite-precision rounding, these scalars define a linear-cost min-cost-flow instance on the auxiliary graph, and solving it exactly recovers a primal potential. Finally, a ground-vertex reduction from regularized objectives to the Poisson solver gives randomized almost-linear-time resolvent/proximal primitives for the same cut-based hypergraph Laplacian.

cs.DS↗
arXiv2026

Efficiently Listing Projected Trees, and Equivalence of Listing and Enumeration

The subgraph isomorphism problem and its generalizations, such as conjunctive queries where some nodes are projected, are among the most fundamental problems in graph algorithms and database theory. In this paper, we study the listing and enumeration variants of these problems and present two main results. The first result is an algorithm for enumerating projected trees with preprocessing time $\widetilde{O}(n^{17.42})$ and delay $\mathrm{polylog}(n)$. Prior to this work, for trees on $k$ nodes all algorithms in the literature required preprocessing time $n^{Ω(k)}$ or delay $n^{Ω(1)}$ or assumed $ω=2$. Our result generalizes to arbitrary projected hypergraphs, achieving enumeration in preprocessing time $\widetilde{O}(m^{17.42 \, \mathrm{subw}(H)})$ and polylogarithmic delay, where $\mathrm{subw}(H)$ is the submodular width of the pattern hypergraph $H$. We heavily rely on fast (rectangular and output-sensitive) matrix multiplication, which we complement by fine-grained lower bounds indicating that any algorithm beating preprocessing time $n^{Ω(k)}$ with polylogarithmic delay must rely on fast matrix multiplication. The second result is a generic enumeration-to-listing reduction, establishing that listing and enumeration are equivalent under natural assumptions. For (colored) subgraph isomorphism, our reduction transforms any listing algorithm running in time $O(f(n,m) + t \cdot g(n,m))$ into an enumeration algorithm with preprocessing time $O\left( (f(n,m)+g(n,m)+n+m) \log^2 n \right)$ and delay $O(g(n,m))$. We utilize this reduction to prove our first main result, and we expect that our generic reduction will find many future applications.

cs.DS↗