Optimization over Degree Sequences of Graphs

TL;DR

This paper studies a graph optimization problem involving subgraph degrees and convex functions, providing polynomial-time solutions in specific cases and highlighting open complexity questions.

Contribution

It introduces polynomial-time algorithms for certain convex function cases and connects the problem to known graph factor problems, expanding understanding of its complexity.

Findings

Polynomial-time solutions for convex functions on any graph.

Efficient algorithms for bipartite graphs with one fixed side.

Open complexity questions for arbitrary functions on complete graphs.

Abstract

We consider the problem of finding a subgraph of a given graph minimizing the sum of given functions at vertices evaluated at their subgraph degrees. While the problem is NP-hard already for bipartite graphs when the functions are convex on one side and concave on the other, we show that when all functions are convex, the problem can be solved in polynomial time for any graph. We also provide polynomial time solutions for bipartite graphs with one side fixed for arbitrary functions, and for arbitrary graphs when all but a fixed number of functions are either nondecreasing or nonincreasing. We note that the general factor problem and the (l,u)-factor problem over a graph are special cases of our problem, as well as the intriguing exact matching problem. The complexity of the problem remains widely open, particularly for arbitrary functions over complete graphs.