Hierarchical threshold structure in Max-Cut with geometric edge weights

TL;DR

This paper investigates the Max-Cut problem on complete graphs with geometrically decreasing edge weights, identifying threshold regimes where specific isolated cuts are optimal and conjecturing their global optimality for larger graphs.

Contribution

It introduces threshold polynomials to determine when isolated cuts dominate and provides a phase diagram for their optimality, supported by computational evidence.

Findings

Threshold polynomials $P^{n,k}(r)$ determine cut dominance.

Isolated cuts $C_k$ are optimal within specific $r$ intervals.

Conjecture: isolated cuts are globally optimal for $n \\ge 7$.

Abstract

We study a family of weighted Max-Cut instances on the complete graph $K_n$ in which edge weights decrease geometrically in lexicographic order: the $i$-th edge has weight $r^{N-i}$ where $N=\binom{n}{2}$. For $r\ge 2$, the lexicographically first cut is optimal; for $r=1$, all edges have equal weight and the balanced partition wins. In this paper we study the intermediate regime $1< r <2$. The geometric weighting makes early edges dominant and singles out the $k$-isolated cuts $C_k=\{1,\dots,k\}\mid\{k+1,\dots,n\}$ as natural candidates for optimality. For each $n$ and $k\le\lfloor n/2\rfloor-1$, we define threshold polynomials $P^{n,k}(r)$ whose unique roots $r_k(n)\in(1,2)$ determine when $C_k$ and $C_{k+1}$ exchange dominance. We prove that, for fixed $n$, these thresholds are strictly decreasing in $k$ and that $r_k(n)\to 1$ as $n\to\infty$. As our main result, we show that for $r\in(r_k(n),r_{k-1}(n))$ the cut $C_k$ achieves maximum weight among all isolated cuts, yielding a sharp phase diagram for the isolated-cut family. We conjecture that isolated cuts are globally optimal among all $2^{n-1}$ cuts when $n\ge 7$; all counterexamples for small $n$ are characterized completely, and extensive computations for $n\le 100$ support the conjecture.