Quadratic Degree Sequence Optimization and the Critical Roots of a Graph

2026-08-06

AI Summary

This study addresses the subgraph selection problem on graphs, aiming to maximize the sum of vertex degrees transformed by a quadratic function. Recognizing this as an NP-hard problem, the work introduces the notion of “critical roots” to characterize how the optimal value varies with the roots of the quadratic function. It proves that the optimal value function exhibits a convex piecewise-affine structure over intervals defined by these critical roots. By integrating techniques from combinatorial optimization, graph theory, and piecewise convex analysis, the paper establishes a theoretical framework for quadratic degree sequence optimization, revealing its intrinsic structural properties and offering a novel analytical perspective and methodological foundation for tackling related challenging problems.

Abstract

The degree sequence optimization problem is to find a subgraph of a given graph which maximizes the sum over all vertices of a given function evaluated at the subgraph degree of that vertex. Here we study this problem and its complexity for quadratic functions. In particular, we introduce the critical roots of a graph, and show they define intervals over which the optimal value of the problem, as the quadratic root varies, is convex piecewise affine.

Problem

• degree sequence optimization
• quadratic functions
• critical roots
• graph theory
• subgraph

Innovation

• quadratic degree sequence optimization
• critical roots
• convex piecewise affine
• graph theory
• combinatorial optimization