Combinatorial Approximation Algorithms for MaxCut using Random Walks

We give the first combinatorial approximation algorithm for Maxcut that beats the trivial 0.5 factor by a constant. The main partitioning procedure is very intuitive, natural, and easily described. It essentially performs a number of random walks and aggregates the information to provide the partition. We can control the running time to get an approximation factor-running time tradeoff. We show that for any constant b > 1.5, there is an O(n^{b}) algorithm that outputs a (0.5+delta)-approximation for Maxcut, where delta = delta(b) is some positive constant. One of the components of our algorithm is a weak local graph partitioning procedure that may be of independent interest. Given a starting vertex $i$ and a conductance parameter phi, unless a random walk of length ell = O(log n) starting from i mixes rapidly (in terms of phi and ell), we can find a cut of conductance at most phi close to the vertex. The work done per vertex found in the cut is sublinear in n.

💡 Research Summary

The paper introduces the first purely combinatorial algorithm for the Max‑Cut problem that provably exceeds the trivial 0.5 approximation ratio by a constant amount. The authors’ approach is built around a simple, intuitive partitioning routine that repeatedly launches short random walks from selected vertices, aggregates the walk outcomes, and uses the aggregated information to decide on a bipartition of the vertex set. By carefully controlling the length of the walks (ℓ = O(log n)) and the number of repetitions, the algorithm can be tuned to trade off running time against the achieved approximation factor.

The main technical contribution is a “weak local graph partitioning” subroutine. Given a start vertex i and a conductance parameter φ, the subroutine runs a random walk of length ℓ. If the walk does not mix rapidly—i.e., its distribution after ℓ steps is still far from the stationary distribution in a way that depends on φ—then the algorithm can extract a set S containing i whose edge boundary has conductance at most φ. The work required per discovered vertex is sub‑linear in the total number of vertices, making the procedure efficient even on massive graphs. This local partitioning tool is of independent interest for community detection, clustering, and other graph‑analysis tasks.

The overall Max‑Cut algorithm proceeds in three phases. First, the local partitioning subroutine is invoked from many vertices to collect a collection of low‑conductance “candidate” sets. Second, for each vertex the algorithm aggregates the outcomes of multiple random walks that start from that vertex or from vertices in its candidate sets, producing a real‑valued bias score b(v). Intuitively, b(v) measures how much the random walk mass tends to stay on one side of a cut versus the other; a positive bias suggests that v belongs to one side, a negative bias to the opposite side. Finally, the vertices are partitioned according to the sign of their bias scores.

The authors prove that, for any constant b > 1.5, the algorithm runs in O(n^b) time and returns a cut whose expected weight is at least (0.5 + δ)·OPT, where δ = δ(b) > 0 is a constant that depends only on the chosen exponent b. The proof hinges on several classic tools from the analysis of Markov chains: a relationship between conductance and mixing time, a variant of Cheeger’s inequality, and concentration bounds for the bias scores. By showing that the bias scores are positively correlated with the indicator of an optimal cut, they establish that the expected cut weight exceeds the random‑cut baseline by a fixed additive factor.

A notable aspect of the work is that the approximation guarantee does not rely on semidefinite programming or spectral embeddings; all operations are combinatorial (random walk steps, local edge counting, and simple arithmetic). Consequently, the algorithm scales much better than SDP‑based methods, which typically require Ω(n^3) time and substantial memory. The authors discuss how, by increasing b (and therefore the number of walk repetitions and candidate sets examined), one can obtain larger δ values at the cost of higher polynomial running time.

Beyond Max‑Cut, the weak local partitioning routine itself offers a sublinear‑time method for finding low‑conductance cuts near a given vertex, provided the local random walk fails to mix quickly. This result may be directly applied to problems such as graph clustering, local community detection, and conductance‑based graph sparsification.

The paper also outlines several directions for future research. The constant δ obtained from the analysis is likely very small; tightening the analysis or designing refined walk‑aggregation schemes could increase the practical approximation factor. Exploring the trade‑off between running time and approximation quality on specific graph families (e.g., planar, bounded‑degree, or expander graphs) may yield stronger guarantees. Finally, empirical evaluation on large real‑world networks would help assess the practical performance of the algorithm compared to both the classic Goemans‑Williamson SDP approach and more recent combinatorial heuristics.

In summary, this work demonstrates that random walks—when combined with a clever local conductance test—are sufficient to break the 0.5 barrier for Max‑Cut using only combinatorial operations. It opens a new line of research that seeks to replace heavy algebraic machinery with simple probabilistic processes, potentially leading to faster, more scalable approximation algorithms for a broad class of graph optimization problems.