Breaking the quadratic barrier for matroid intersection

Abstract

The matroid intersection problem is a fundamental problem that has been extensively studied for half a century. In the classic version of this problem, we are given two matroids M 1 = (V, I 1 ) and M 2 = (V, I 2 ) on a comment ground set V of n elements, and then we have to find the largest common independent set S ∈ I 1 ∩ I 2 by making independence oracle queries of the form "Is S ∈ I 1 ?" or "Is S ∈ I 2 ?" for S ⊆ V . The goal is to minimize the number of queries.Beating the existing Õ(n 2 ) bound, known as the quadratic barrier, is an open problem that captures the limits of techniques from two lines of work. The first one is the classic Cunningham's algorithm [SICOMP 1986], whose Õ(n 2 )-query implementations were shown by CLS+ [FOCS 2019] and Nguy ễn [2019]. 1 The other one is the general cutting plane method of Lee, Sidford, and Wong [FOCS 2015]. The only progress towards breaking the quadratic barrier requires either approximation algorithms or a more powerful rank oracle query [CLS+ FOCS 2019]. No exact algorithm with o(n 2 ) independence queries was known.In this work, we break the quadratic barrier with a randomized algorithm guaranteeing Õ(n 9/5 ) independence queries with high probability, and a deterministic algorithm guaranteeing Õ(n 11/6 ) independence queries. Our key insight is simple and fast algorithms to solve a graph reachability problem that arose in the standard augmenting path framework [Edmonds 1968]. Combining this with previous exact and approximation algorithms leads to our results.1 More generally, these algorithms take Õ(nr) queries where r denotes the rank which can be as big as n.

Breaking O(nr) for Matroid Intersection

“…Plugging Theorem 1 in the framework of [BvdBMN21], we get an improved algorithm-more efficient than the previous state-of-the-art-for exact matroid intersection which we state next.…”

“…This question has been answered for large r (r = ω( √ n)), first by the Õ(n 1.5 /ε 1.5 )-query (1 − )-approximation algorithm of Chakrabarty-Lee-Sidford-Singla-Wong 2 [CLS + 19], and very recently by the randomized Õ(n 6/5 r 3/5 )-query exact algorithm of Blikstad-v.d.Brand-Mukhopadhyay-Nanongkai [BvdBMN21]. Whether we can break the O(nr)-query bound for the full range of r remained open even for approximation algorithms.…”

“…Exact algorithm. To obtain the exact algorithm (Theorem 2), we use the framework of Blikstadv.d.Brand-Mukhopadhyay-Nanongkai's Õ(n 6/5 r 3/5 )-query exact algorithm [BvdBMN21]. The main idea of this algorithm is to combine approximation algorithms-which can efficiently find a common independent set only εr away from the optimal-with a randomized Õ(n √ r)-query subroutine to find each of the remaining few, very long augmenting paths.…”

“…The main idea of this algorithm is to combine approximation algorithms-which can efficiently find a common independent set only εr away from the optimal-with a randomized Õ(n √ r)-query subroutine to find each of the remaining few, very long augmenting paths. The Õ(n 6/5 r 3/5 )-query exact algorithm [BvdBMN21] currently uses Chakrabarty-Lee-Sidford-Singla-Wong's Õ(n 1.5 /ε 1.5 ) approximation algorithm [CLS + 19] as a subroutine. Simply replacing it with our improved approximation algorithm (Theorem 1) yields our Õ(nr 3/4 )-query exact algorithm.…”

“…In this section, we prove Theorem 2 (restated below) by showing how our improved approximation algorithm leads to an improved exact algorithm when combined with the algorithms of [BvdBMN21].…”