Hierarchical threshold structure in Max-Cut with geometric edge weights

TL;DR

The conjecture that isolated cuts are globally optimal among all $2^{n-1}$ cuts when $n\ge 7$ is conjecture; all counterexamples for small $n$ are characterized completely, and extensive computations for $n\le 100$ support the conjecture.

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.

Figures (6)

• Figure 1: Threshold polynomials $P^{8,1}(r)$, $P^{8,2}(r)$, and $P^{8,3}(r)$ on the interval $(1,2)$. Each polynomial has a unique root in $(1,2)$, with $r_3(8) < r_2(8) < r_1(8)$, illustrating the monotonicity established in Theorem \ref{['thm:monotonicity']}.
• Figure 2: Threshold values $r_k(n)$ as a function of $n$ for fixed $k \in \{1,2,3,4,5\}$. All thresholds converge to $1$ as $n \to \infty$. Note that for $k \geq 2$, the sequences are not monotone in $n$.
• Figure 3: Threshold values $r_k(n)$ as a function of $k$ for fixed $n \in \{12, 20, 30\}$. For each $n$, the sequence $r_1(n) > r_2(n) > \cdots > r_{\lfloor n/2 \rfloor - 1}(n)$ is strictly decreasing, as established in Theorem \ref{['thm:monotonicity']}.
• Figure 4: In each region $(r_k(n), r_{k-1}(n))$, the $k$-isolated cut achieves maximum weight among all isolated cuts.
• Figure 5: Globally optimal cuts as a function of $r$, determined by exhaustive enumeration. For $n \leq 6$, near-isolated cuts $S_k^* = \{1,\ldots,k,n\}$ are optimal in certain intervals (red regions). For $n \geq 7$, isolated cuts are globally optimal throughout $(1,2)$ (Conjecture \ref{['conj:global']}).

Theorems & Definitions (28)

• Definition 2.1
• Remark 2.2
• Definition 2.3: Weight function
• Definition 2.4
• Definition 4.1
• Proposition 4.2: Explicit Formula for $k=1$
• proof
• Lemma 4.3: Exponent Separation
• proof
• Theorem 4.4: Properties of $r_1(n)$