[Paper Review] Local algorithms in (weakly) coloured graphs

Matti Åstrand, Valentin Polishchuk+3 more|arXiv (Cornell University)|Jan 31, 2010

Complexity and Algorithms in Graphs18 references17 citations

TL;DR

This paper presents local approximation algorithms for the minimum dominating set and maximum matching problems in (weakly) 2-coloured graphs, showing that a weak 2-colouring enables a (Δ+1)/2-approximation for both problems, matching the best possible approximation factor achievable. In 2-coloured graphs, a local approximation scheme achieves arbitrarily good approximation for maximum matching, while the dominating set problem remains bounded by (Δ+1)/2 even with full 2-colouring.

ABSTRACT

A local algorithm is a distributed algorithm that completes after a constant number of synchronous communication rounds. We present local approximation algorithms for the minimum dominating set problem and the maximum matching problem in 2-coloured and weakly 2-coloured graphs. In a weakly 2-coloured graph, both problems admit a local algorithm with the approximation factor $(Δ+1)/2$, where $Δ$ is the maximum degree of the graph. We also give a matching lower bound proving that there is no local algorithm with a better approximation factor for either of these problems. Furthermore, we show that the stronger assumption of a 2-colouring does not help in the case of the dominating set problem, but there is a local approximation scheme for the maximum matching problem in 2-coloured graphs.

Motivation & Objective

• To investigate how much symmetry-breaking information from 2-colouring or weak 2-colouring improves the approximability of fundamental graph problems in the local computation model.
• To determine whether weak 2-colouring provides sufficient structure for constant-factor local approximation of dominating set and matching.
• To establish tight upper and lower bounds on the best possible approximation factors achievable by deterministic local algorithms for these problems.
• To explore the limitations of local algorithms in weakly 2-coloured graphs, particularly for the independent set problem.

Proposed method

• The paper designs local algorithms that operate in a constant number of synchronous communication rounds, relying only on local neighbourhood information and node colouring.
• For maximum matching, the algorithm uses a recursive subroutine that finds and augments along odd-length augmenting paths (length 2i−1), repeating until no such paths exist.
• The flooding phase constructs disjoint augmenting trees of length h=2i−1 rooted at unmatched black nodes, using message passing to identify endpoints of length-h augmenting paths.
• The proposal phase selects one augmenting path per tree by propagating messages from leaves to roots, discarding duplicates when messages meet.
• The augmenting phase updates the matching in parallel along selected paths, ensuring no conflicts.
• The algorithm leverages the 2-colouring to ensure that augmenting paths alternate between black and white nodes, enabling efficient path discovery and parallel augmentation.

Experimental results

Research questions

• RQ1 Can weak 2-colouring enable a constant-factor local approximation for the minimum dominating set problem, and if so, what is the best possible approximation factor?
• RQ2 Does 2-colouring provide a significant advantage over weak 2-colouring for local approximation of the dominating set problem?
• RQ3 Can local algorithms achieve arbitrarily good approximation for maximum matching in 2-coloured graphs, and if so, how is this achieved?
• RQ4 Is there a local constant-factor approximation for maximum independent set in weakly 2-coloured graphs, or is the problem fundamentally unapproachable locally?
• RQ5 What are the tightest possible approximation factors for local algorithms on these problems, and can matching lower bounds be proven?

Key findings

• A weak 2-colouring enables a local (Δ+1)/2-approximation for the minimum dominating set problem, and this factor is optimal.
• The same (Δ+1)/2-approximation is achievable for maximum matching in weakly 2-coloured graphs, and this bound is tight.
• In 2-coloured graphs, a local approximation scheme achieves (1+ε)-approximation for maximum matching, with the approximation factor arbitrarily close to 1.
• No local constant-factor approximation exists for the maximum independent set problem in weakly 2-coloured graphs, even with bounded degree.
• For both dominating set and matching, the approximation factor of (Δ+1)/2 is optimal in weakly 2-coloured graphs, as proven by matching lower bounds.
• The 2-colouring assumption does not improve the approximation factor for the dominating set problem beyond what is achievable with weak 2-colouring.