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On the Congruency-Constrained Matroid Base

Siyue Liu, Chao Xu

Abstract

Consider a matroid where all elements are labeled with an element in $\mathbb{Z}$. We are interested in finding a base where the sum of the labels is congruent to $g \pmod m$. We show that this problem can be solved in $\tilde{O}(2^{4m} n r^{5/6})$ time for a matroid with $n$ elements and rank $r$, when $m$ is either the product of two primes or a prime power. The algorithm can be generalized to all moduli and, in fact, to all abelian groups if a classic additive combinatorics conjecture by Schrijver and Seymour holds true. We also discuss the optimization version of the problem.

On the Congruency-Constrained Matroid Base

Abstract

Consider a matroid where all elements are labeled with an element in . We are interested in finding a base where the sum of the labels is congruent to . We show that this problem can be solved in time for a matroid with elements and rank , when is either the product of two primes or a prime power. The algorithm can be generalized to all moduli and, in fact, to all abelian groups if a classic additive combinatorics conjecture by Schrijver and Seymour holds true. We also discuss the optimization version of the problem.
Paper Structure (13 sections, 16 theorems, 6 equations)

This paper contains 13 sections, 16 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Overview.
  4. Notation.
  5. Preliminaries
  6. Algorithms for GCOMB
  7. Properties of minimum counterexamples to \ref{['conj:feasibility']} and \ref{['conj:optimization']}
  8. $k$-closeness and isolation
  9. Congruency-constrained base with prime modulus
  10. General group constraints
  11. Strong $k$-closeness
  12. Strongly base orderable matroids
  13. Small groups

Key Result

Theorem 3.1

If $G$ is $k$-close or strongly $k$-close, then there is a $\tilde{O}(\binom{k+|G|-1}{k}^2 nr^{5/6})$ running time algorithm for $\operatorname{GCMB}(m)$ or $\operatorname{GCOMB}(m)$, respectively.

Theorems & Definitions (29)

  • Conjecture 2.1: Feasibility
  • Conjecture 2.2: Optimization
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • Lemma 4.1: brualdi_1969
  • Lemma 4.2
  • proof
  • Theorem 4.1
  • proof
  • ...and 19 more