Judicious Partitions in Edge-Weighted Graphs with Bounded Maximum Weighted Degree
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 $\max \left\{ w(G[V_i]) : i \in [k] \right\} \leq \frac{w(G)}{k^2} + \frac{k - 1}{2k^2} Δ_w(G),$ where $Δ_w(G)$ denotes the maximum weighted degree of $G$. This bound is tight for every integer $k\geq 2$.