Judicious Partitions in Edge-Weighted Graphs with Bounded Maximum Weighted Degree

TL;DR

This paper studies optimal partitions of edge-weighted graphs to minimize the maximum induced subgraph weight and cut weight, providing tight bounds for specific cases and proposing conjectures for general cases.

Contribution

It establishes tight bounds for judicious k-partitions in weighted graphs, including the first tight bound for k=3, and explores limitations for larger k.

Findings

Tight bounds for k=2 partitions.

First tight bound for k=3 partitions.

Existence of partitions with controlled maximum subgraph weight and cut weight.

Abstract

In this paper, we investigate bounds for the following judicious $k$-partitioning problem: Given an edge-weighted graph $G$, find a $k$-partition $(V_1,V_2,\dots ,V_k)$ of $V(G)$ such that the total weight of edges in the heaviest induced subgraph, $\max_{i=1}^k w(G[V_i])$, is minimized. In our bounds, we also take into account the weight $w(V_1,V_2,\dots,V_k)$ of the cut induced by the partition (i.e., the total weight of edges with endpoints in different parts) and show the existence of a partition satisfying tight bounds for both quantities simultaneously. We establish such tight bounds for the case $k=2$ and, to the best of our knowledge, present the first (even for unweighted graphs) completely tight bound for $k=3$. We also show that, in general, these results cannot be extended to $k \geq 4$ without introducing an additional lower-order term, and we propose a corresponding conjecture. Moreover, we prove that there always exists a $k$-partition satisfying $\max \left\{ w(G[V_i]) : i \in [k] \right\} \leq \frac{w(G)}{k^2} + \frac{k - 1}{2k^2} \Delta_w(G),$ where $\Delta_w(G)$ denotes the maximum weighted degree of $G$. This bound is tight for every integer $k\geq 2$.