[Paper Review] Local algorithms for Maximum Cut and Minimum Bisection on locally treelike regular graphs of large degree

A. El Alaoui, Andrea Montanari+1 more|arXiv (Cornell University)|Nov 12, 2021

Graph theory and applications4 citations

TL;DR

This paper presents a local message-passing algorithm for computing near-optimal maximum cut and minimum bisection partitions in locally treelike regular graphs of large degree. The algorithm runs in O(nkL) time and achieves a cut value of nk/4 + nP⋆√(k/4) + o(n) + no(k)(√k) + n√k oL(1), where P⋆ ≈ 0.763166 is the Parisi constant, matching the theoretical optimum for random k-regular graphs and establishing near-optimality in the local algorithm regime.

ABSTRACT

Given a graph $G$ of degree $k$ over $n$ vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth $2L$, we develop a local message passing algorithm whose complexity is $O(nkL)$, and that achieves near optimal cut values among all $L$-local algorithms. Focusing on max-cut, the algorithm constructs a cut of value $nk/4+ n\\mathsf{P}_\\star\\sqrt{k/4}+\\mathsf{err}(n,k,L)$, where $\\mathsf{P}_\\star\\approx 0.763166$ is the value of the Parisi formula from spin glass theory, and $\\mathsf{err}(n,k,L)=o_n(n)+no_k(\\sqrt{k})+n \\sqrt{k} o_L(1)$ (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, i.e., graphs whose girth becomes $2L$ after removing a small fraction of vertices. Earlier work established that, for random $k$-regular graphs, the typical max-cut value is $nk/4+ n\\mathsf{P}_\\star\\sqrt{k/4}+o_n(n)+no_k(\\sqrt{k})$. Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max-cut, and nearly maximum min-bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near-Ramanujan property of random regular graphs.

Motivation & Objective

• To design a polynomial-time local algorithm for maximum cut and minimum bisection on locally treelike regular graphs with large degree.
• To achieve cut values that match the theoretical optimum predicted by the Parisi formula for random k-regular graphs.
• To generalize results from random k-regular graphs to the broader class of locally treelike graphs, where girth is large after removing a small fraction of vertices.
• To establish that the algorithm's performance is nearly optimal among all L-local algorithms, with error terms decaying in n, k, and L.
• To demonstrate that random regular graphs have nearly minimum max-cut and nearly maximum min-bisection, providing a combinatorial analog of the near-Ramanujan property.

Proposed method

• The algorithm uses a message-passing framework on the k-ary tree of depth L rooted at each vertex, simulating belief propagation with non-linearities derived from the Parisi functional.
• It computes local estimates zv^L and zv^L for each vertex v using recursive updates based on the previous layer's messages, with non-linearities Fℓ and Hℓ designed to approximate the optimal solution to the Sherrington-Kirkpatrick model.
• The algorithm applies a rounding procedure to convert continuous message outputs into a ±1 assignment σi ∈ {−1, +1}, ensuring the resulting partition is nearly balanced for min-bisection and feasible for max-cut.
• Error bounds are derived via the Lindeberg exchange method and comparison with the continuous-time Parisi PDE, showing convergence to the Parisi constant P⋆ as k → ∞.
• The analysis leverages the no-overlap-gap assumption and properties of the Parisi functional to bound the difference between the algorithm’s output and the theoretical optimum.
• The final solution is adjusted to enforce exact balance in the bisection case by flipping εn signs, with a controlled error of order kε.

Experimental results

Research questions

• RQ1 Can a local algorithm achieve near-optimal maximum cut and minimum bisection values on large-degree regular graphs?
• RQ2 Does the performance of such an algorithm match the theoretical limit predicted by the Parisi formula for the Sherrington-Kirkpatrick spin glass model?
• RQ3 How does the algorithm’s performance scale with the graph’s girth and local tree-likeness, particularly in the limit of large degree k?
• RQ4 To what extent do random k-regular graphs achieve nearly minimum max-cut and nearly maximum min-bisection, and is this a consequence of their local tree-likeness?
• RQ5 Can the algorithm’s error be quantified in terms of n, k, and L, and does it converge to the optimal value as L increases?

Key findings

• The algorithm achieves a maximum cut value of nk/4 + nP⋆√(k/4) + o(n) + no(k)(√k) + n√k oL(1), matching the known asymptotic optimum for random k-regular graphs.
• For minimum bisection, the algorithm produces a cut value within O(√k) of the theoretical optimum, with the same error structure, confirming near-optimality.
• The Parisi constant P⋆ ≈ 0.763166 emerges as the key scaling factor, linking the graph partitioning problem to spin glass theory and the SK model.
• The algorithm’s error decays as o(n) in n, no(k)(√k) in k, and n√k oL(1) in L, showing convergence to the optimal value as L increases.
• The result implies that random k-regular graphs have nearly minimum max-cut and nearly maximum min-bisection, providing a combinatorial counterpart to their near-Ramanujan spectral properties.
• The algorithm’s complexity is O(nkL), making it efficient for large graphs with large degree and moderate girth.