[Paper Review] Improved Deterministic Distributed Matching via Rounding

Manuela Fischer|arXiv (Cornell University)|Jan 1, 2017

Complexity and Algorithms in Graphs23 citations

TL;DR

This paper presents a novel deterministic distributed rounding method for linear programs, enabling faster and more efficient algorithms for matching problems in distributed networks. It achieves the first O(log²∆·log n)-round deterministic maximal matching algorithm and exponentially improves query complexity for Local Computation Algorithms, while also providing faster (2+ε)-approximation algorithms for maximum and weighted b-matching problems.

ABSTRACT

We present improved deterministic distributed algorithms for a number of well-studied matching problems, which are simpler, faster, more accurate, and/or more general than their known counterparts. The common denominator of these results is a deterministic distributed rounding method for certain linear programs, which is the first such rounding method, to our knowledge. A sampling of our end results is as follows. - An O(log^2 Delta log n)-round deterministic distributed algorithm for computing a maximal matching, in n-node graphs with maximum degree Delta. This is the first improvement in about 20 years over the celebrated O(log^4 n)-round algorithm of Hanckowiak, Karonski, and Panconesi [SODA'98, PODC'99]. - A deterministic distributed algorithm for computing a (2+epsilon)-approximation of maximum matching in O(log^2 Delta log(1/epsilon) + log^* n) rounds. This is exponentially faster than the classic O(Delta + log^* n)-round 2-approximation of Panconesi and Rizzi [DIST'01]. With some modifications, the algorithm can also find an epsilon-maximal matching which leaves only an epsilon-fraction of the edges on unmatched nodes. - An O(log^2 Delta log(1/epsilon) + log^* n)-round deterministic distributed algorithm for computing a (2+epsilon)-approximation of a maximum weighted matching, and also for the more general problem of maximum weighted b-matching. These improve over the O(log^4 n log_(1+epsilon) W)-round (6+epsilon)-approximation algorithm of Panconesi and Sozio [DIST'10], where W denotes the maximum normalized weight. - A deterministic local computation algorithm for a (2+epsilon)-approximation of maximum matching with 2^O(log^2 Delta) log^* n queries. This improves almost exponentially over the previous deterministic constant approximations which have query-complexity of 2^Omega(Delta log Delta) log^* n.

Motivation & Objective

• To develop a deterministic distributed rounding method for fractional solutions of specific linear programs.
• To improve the round complexity of deterministic distributed algorithms for maximal matching and related problems.
• To achieve faster (2+ε)-approximation algorithms for maximum matching and weighted b-matching in distributed settings.
• To reduce the query complexity of Local Computation Algorithms (LCAs) for maximum matching.
• To close the gap between randomized and deterministic distributed algorithms for fundamental graph problems.

Proposed method

• Introduces a novel deterministic distributed rounding technique for fractional matchings in linear programs.
• Applies iterative rounding: repeatedly computes approximate b-matchings and removes matched edges and zero-capacity vertices.
• Uses a 2-decomposition of graphs to reduce degree and enable efficient local computation in O(1) rounds.
• Employs a reduction to bipartite graphs via node splitting to handle general graphs and b-matchings.
• Leverages a constant-approximation algorithm for auxiliary weighted graphs in each iteration to build a (2+ε)-approximate solution.
• Adapts the method to Local Computation Algorithms (LCAs) via standard simulation techniques with improved query complexity.

Experimental results

Research questions

• RQ1 Can a deterministic distributed rounding method be developed for fractional solutions of a class of linear programs relevant to matching problems?
• RQ2 What is the best possible round complexity for deterministic maximal matching in the LOCAL model?
• RQ3 Can (2+ε)-approximation algorithms for maximum matching be computed significantly faster than O(∆+log*n) rounds?
• RQ4 How can the query complexity of deterministic LCAs for maximum matching be reduced?
• RQ5 Can the same rounding framework be extended to weighted and b-matching problems?

Key findings

• The paper presents the first deterministic distributed rounding method for a class of linear programs, enabling new algorithmic advances.
• It achieves an O(log²∆·log n)-round deterministic algorithm for maximal matching, improving over the 20-year-old O(log⁴n) bound.
• The (2+ε)-approximation for maximum matching runs in O(log²∆·log(1/ε)+log*n) rounds, exponentially faster than the prior O(∆+log*n) method.
• For maximum weighted b-matching, the algorithm runs in O(log²∆·log(1/ε)+log*n) rounds, improving over the prior O(log⁴n·log¹⁺εW) bound.
• The LCA for (2+ε)-approximate maximum matching has a query complexity of 2^O(log²∆)·log*n, an exponential improvement over the prior 2^Ω(∆·log∆)·log*n bound.
• The method also yields a (2+ε)-approximation for edge dominating sets via ε-maximal matchings, with a small additive error in edge coverage.