🤖 AI Summary
This study investigates the structure of optimal Max-Cut solutions in complete graphs with geometrically weighted edges, focusing on the parameter regime \(1 < r < 2\). For instances where edge weights decay geometrically in lexicographic order, the authors analyze the optimality transitions within the family of “k-isolated cuts.” By integrating combinatorial optimization, root analysis of polynomials, and extensive numerical verification, they establish—for the first time—a strictly decreasing hierarchy of threshold values \(r_k(n)\) that govern the dominance of these cuts. These thresholds decrease strictly with \(k\) and converge to 1 as \(n \to \infty\). The resulting phase diagram precisely delineates the regions of dominance among competing cuts, and computational evidence for \(n \leq 100\) supports the conjecture that k-isolated cuts are globally optimal for all \(n \geq 7\).
📝 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.