Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Matching Composition and Efficient Weight Reduction in Dynamic Matching

Aaron Bernstein, Jiale Chen, Aditi Dudeja, Zachary Langley, Aaron Sidford, Ta-Wei Tu

TL;DR

A general reduction is provided that reduces the problem on graphs with a weight range of n to 1 to dynamic $(1-\varepsilon)-approximate maximum cardinality matching in bipartite graphs at the cost of a multiplicative $\mathrm{poly}(1/\varepsilon)$ in update time, thereby resolving an open problem in [GP'13; BDL'21].

Abstract

We consider the foundational problem of maintaining a $(1-\varepsilon)$-approximate maximum weight matching (MWM) in an $n$-node dynamic graph undergoing edge insertions and deletions. We provide a general reduction that reduces the problem on graphs with a weight range of $\mathrm{poly}(n)$ to $\mathrm{poly}(1/\varepsilon)$ at the cost of just an additive $\mathrm{poly}(1/\varepsilon)$ in update time. This improves upon the prior reduction of Gupta-Peng (FOCS 2013) which reduces the problem to a weight range of $\varepsilon^{-O(1/\varepsilon)}$ with a multiplicative cost of $O(\log n)$. When combined with a reduction of Bernstein-Dudeja-Langley (STOC 2021) this yields a reduction from dynamic $(1-\varepsilon)$-approximate MWM in bipartite graphs with a weight range of $\mathrm{poly}(n)$ to dynamic $(1-\varepsilon)$-approximate maximum cardinality matching in bipartite graphs at the cost of a multiplicative $\mathrm{poly}(1/\varepsilon)$ in update time, thereby resolving an open problem in [GP'13; BDL'21]. Additionally, we show that our approach is amenable to MWM problems in streaming, shared-memory work-depth, and massively parallel computation models. We also apply our techniques to obtain an efficient dynamic algorithm for rounding weighted fractional matchings in general graphs. Underlying our framework is a new structural result about MWM that we call the "matching composition lemma" and new dynamic matching subroutines that may be of independent interest.

Matching Composition and Efficient Weight Reduction in Dynamic Matching

TL;DR

A general reduction is provided that reduces the problem on graphs with a weight range of n to 1 to dynamic \mathrm{poly}(1/\varepsilon)$ in update time, thereby resolving an open problem in [GP'13; BDL'21].

Abstract

We consider the foundational problem of maintaining a -approximate maximum weight matching (MWM) in an -node dynamic graph undergoing edge insertions and deletions. We provide a general reduction that reduces the problem on graphs with a weight range of to at the cost of just an additive in update time. This improves upon the prior reduction of Gupta-Peng (FOCS 2013) which reduces the problem to a weight range of with a multiplicative cost of . When combined with a reduction of Bernstein-Dudeja-Langley (STOC 2021) this yields a reduction from dynamic -approximate MWM in bipartite graphs with a weight range of to dynamic -approximate maximum cardinality matching in bipartite graphs at the cost of a multiplicative in update time, thereby resolving an open problem in [GP'13; BDL'21]. Additionally, we show that our approach is amenable to MWM problems in streaming, shared-memory work-depth, and massively parallel computation models. We also apply our techniques to obtain an efficient dynamic algorithm for rounding weighted fractional matchings in general graphs. Underlying our framework is a new structural result about MWM that we call the "matching composition lemma" and new dynamic matching subroutines that may be of independent interest.

Paper Structure

This paper contains 60 sections, 61 theorems, 78 equations, 1 figure, 3 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Dynamic Weighted Matching.
  3. Our Contribution
  4. Techniques
  5. A Limitation of Our Reductions
  6. Overview of the Paper
  7. Additional Computational Models
  8. Our Results.
  9. Contrast to Previous Work
  10. Applications
  11. Applications to Bipartite Graphs
  12. Applications to Non-Bipartite Graphs
  13. Preliminaries
  14. General Notation
  15. Graphs and Matchings
  16. ...and 45 more sections

Key Result

theorem 1.1

Given any dynamic $(1-\varepsilon)$-approximate MCM algorithm in $n$-node $m$-edge bipartite graphs with update time $\mathcal{U}(n,m,\varepsilon)$, there is a transformation which produces a dynamic $(1-O(\varepsilon))$-approximate MWM algorithm for $n$-node bipartite graphs with amortized update t

Figures (1)

  • Figure 1: Gadget for answering \ref{['conjecture:weight partition']} in the negative

Theorems & Definitions (115)

  • theorem 1.1: Informal version of \ref{['red:WeightedtoUnweighted:new']}
  • theorem 1.2: Informal version of \ref{['thm:partial reduction:ultimate']}
  • theorem 1.3: Informal version of \ref{['thm:weighted-rounding']}
  • theorem 2.1: DuanP14
  • lemma 3.0: Matching Composition Lemma
  • lemma 3.0: Matching Substitution Lemma
  • theorem 3.1
  • theorem 3.2
  • lemma 4.0: Matching Substitution Lemma
  • proof
  • ...and 105 more