[Paper Review] A Distributed $(2+\epsilon)$-Approximation for Vertex Cover in $O(\log{\Delta}/\epsilon\log\log{\Delta})$ Rounds

Reuven Bar-Yehuda, Keren Censor-Hillel+1 more|arXiv (Cornell University)|Feb 11, 2016

Complexity and Algorithms in Graphs19 references3 citations

TL;DR

This paper presents a deterministic distributed algorithm that achieves a (2 + ε)-approximation for the minimum weight vertex cover problem in O(log Δ / (ε log log Δ)) rounds, where Δ is the maximum degree. By introducing a novel two-part weight management system—'vault' and 'bank'—the algorithm decouples interdependent weight reductions, enabling efficient, scalable approximation without requiring global knowledge or vertex identifiers.

ABSTRACT

We present a simple deterministic distributed $(2+\epsilon)$-approximation algorithm for minimum weight vertex cover, which completes in $O(\log{\Delta}/\epsilon\log\log{\Delta})$ rounds, where $\Delta$ is the maximum degree in the graph, for any $\epsilon>0$ which is at most $O(1)$. For a constant $\epsilon$, this implies a constant approximation in $O(\log{\Delta}/\log\log{\Delta})$ rounds, which contradicts the lower bound of [KMW10].

Motivation & Objective

• To design a deterministic distributed algorithm for minimum weight vertex cover with improved round complexity.
• To adapt the sequential local ratio technique to the distributed setting in a way that avoids long dependency chains.
• To achieve a (2 + ε)-approximation in sub-logarithmic rounds relative to Δ, challenging prior lower bounds.
• To demonstrate that the algorithm works in anonymous networks and can be adapted to the CONGEST model with bounded-weight assumptions.

Proposed method

• Introduces a two-part weight system: a 'vault' for initiating reduction requests and a 'bank' for responding to them.
• Each vertex v maintains di(v) and wi(v), tracking the number of reductions and remaining weight in its vault.
• Uses a parameter Kv = d(v) + 1 for d(v) ≤ 16 and Kv = log d(v)/log log d(v) otherwise to bound iteration count.
• Reduces weight in response to neighbor requests only when sufficient vault capacity exists, ensuring no negative weights.
• Modifies the algorithm for the CONGEST model by encoding requests and budgets in O(log n)-bit messages using precomputed values.
• Proves correctness via local ratio invariance: the final set of vertices with zero remaining weight forms a (2 + ε)-approximation.

Experimental results

Research questions

• RQ1 Can the local ratio technique be effectively adapted to the distributed setting without incurring high communication overhead?
• RQ2 Is it possible to achieve a (2 + ε)-approximation for minimum weight vertex cover in o(log Δ) rounds?
• RQ3 Does the algorithm remain efficient and correct in anonymous networks without unique identifiers?
• RQ4 Can the algorithm be made efficient in the CONGEST model with bounded message size?
• RQ5 What is the true round complexity lower bound for constant-approximation vertex cover in distributed systems?

Key findings

• The algorithm achieves a (2 + ε)-approximation in O(log Δ / (ε log log Δ)) rounds, improving upon prior distributed algorithms.
• For constant ε, the round complexity is O(log Δ / log log Δ), contradicting the claimed Ω(log Δ) lower bound from [KMW10].
• The algorithm completes in O(1/ε) rounds for vertices of degree ≤16, and in O(log d(v)/(ε log log d(v))) rounds otherwise.
• The algorithm works in anonymous networks and does not require synchronized start times.
• The authors identify a flaw in the lower bound of [KMW10], showing it only applies under restrictive assumptions on k (i.e., k = O((log n)^{1/3}) and k = O(√log Δ)).
• The true lower bound is revised to Ω(log Δ / log log Δ), making the algorithm asymptotically tight.