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Approximating Maximum Matching Requires Almost Quadratic Time

Soheil Behnezhad, Mohammad Roghani, Aviad Rubinstein

TL;DR

This paper settles a central question in sublinear graph algorithms by proving a near-quadratic lower bound for estimating the size of a maximum matching in $n$-vertex graphs from adjacency-list queries: for any fixed $\delta>0$ there exists $\varepsilon(\delta)>0$ such that achieving an additive error of $\varepsilon n$ requires $\Omega(n^{2-\delta})$ time, even under bipartite promises. The authors introduce a recursive hierarchy that camouflages a sizable core and show that the algorithm’s ability to identify camouflaged edges diminishes with each level, ultimately preventing accurate estimation at the base level. This breaks the cycle-discovery barrier that previous lower-bound frameworks relied on, showing that near-quadratic time is necessary irrespective of cycle discovery, within the adjacency-list model. The result solidifies the boundary between subquadratic estimators and trivial quadratic-time algorithms, and has implications for related dynamic problems by limiting how sublinear estimators can be leveraged in dynamic maintenance of maximum matchings.

Abstract

We study algorithms for estimating the size of maximum matching. This problem has been subject to extensive research. For $n$-vertex graphs, Bhattacharya, Kiss, and Saranurak [FOCS'23] (BKS) showed that an estimate that is within $\varepsilon n$ of the optimal solution can be achieved in $n^{2-Ω_\varepsilon(1)}$ time, where $n$ is the number of vertices. While this is subquadratic in $n$ for any fixed $\varepsilon > 0$, it gets closer and closer to the trivial $Θ(n^2)$ time algorithm that reads the entire input as $\varepsilon$ is made smaller and smaller. In this work, we close this gap and show that the algorithm of BKS is close to optimal. In particular, we prove that for any fixed $δ> 0$, there is another fixed $\varepsilon = \varepsilon(δ) > 0$ such that estimating the size of maximum matching within an additive error of $\varepsilon n$ requires $Ω(n^{2-δ})$ time in the adjacency list model.

Approximating Maximum Matching Requires Almost Quadratic Time

TL;DR

This paper settles a central question in sublinear graph algorithms by proving a near-quadratic lower bound for estimating the size of a maximum matching in -vertex graphs from adjacency-list queries: for any fixed there exists such that achieving an additive error of requires time, even under bipartite promises. The authors introduce a recursive hierarchy that camouflages a sizable core and show that the algorithm’s ability to identify camouflaged edges diminishes with each level, ultimately preventing accurate estimation at the base level. This breaks the cycle-discovery barrier that previous lower-bound frameworks relied on, showing that near-quadratic time is necessary irrespective of cycle discovery, within the adjacency-list model. The result solidifies the boundary between subquadratic estimators and trivial quadratic-time algorithms, and has implications for related dynamic problems by limiting how sublinear estimators can be leveraged in dynamic maintenance of maximum matchings.

Abstract

We study algorithms for estimating the size of maximum matching. This problem has been subject to extensive research. For -vertex graphs, Bhattacharya, Kiss, and Saranurak [FOCS'23] (BKS) showed that an estimate that is within of the optimal solution can be achieved in time, where is the number of vertices. While this is subquadratic in for any fixed , it gets closer and closer to the trivial time algorithm that reads the entire input as is made smaller and smaller. In this work, we close this gap and show that the algorithm of BKS is close to optimal. In particular, we prove that for any fixed , there is another fixed such that estimating the size of maximum matching within an additive error of requires time in the adjacency list model.
Paper Structure (41 sections, 36 theorems, 62 equations, 2 figures, 1 table)

This paper contains 41 sections, 36 theorems, 62 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work:
  3. Our contribution:
  4. Paper organization:
  5. Further Related Work: Dynamic Algorithms
  6. Our Techniques: Bypassing The Cycle Discovery Threshold
  7. Background on Existing Lower Bounds
  8. High degree dummy vertices.
  9. Camouflage the good matching.
  10. The Cycle Discovery Barrier
  11. Our Key Contribution: Bypassing the Cycle Discovery Barrier
  12. The base (due to BehnezhadRR-FOCS23):
  13. The recursion (new to this work):
  14. Decrease in the Advantage of the Algorithm in Discovering Camouflaged Edges
  15. Preliminaries
  16. ...and 26 more sections

Key Result

Theorem 1

For any $\delta > 0$ there is an $\varepsilon = \varepsilon(\delta) > 0$ such that any (randomized) algorithm that (with probability at least $2/3$) estimates the size of maximum matching of an $n$-vertex graph up to an additive error of $\varepsilon n$ has to make $\Omega(n^{2-\delta})$ adjacency l

Figures (2)

  • Figure 1: Construction of $G^\ell$ based on $G^{\ell-1}$.
  • Figure :

Theorems & Definitions (108)

  • Theorem 1: Main Result
  • Definition 3.1: Strongly Connected Component
  • Proposition 3.2: König Theorem
  • Proposition 3.3: Chernoff Bound
  • Definition 3.4: Negative Association kumarDevProschensaxenaKhursheedwajc2017negative
  • Proposition 3.5: Chernoff Bound Negatively Associated Variables
  • Proposition 3.6: Yao's Minimax Principle Yao77
  • Lemma 5.1
  • proof : Proof of \ref{['thm:main']}
  • Remark 1
  • ...and 98 more