# Optimization over Degree Sequences

## TL;DR

This paper investigates the computational complexity of optimizing functions over degree sequences of hypergraphs and multihypergraphs, revealing polynomial solvability in some cases and NP-completeness in others, and connects these results to broader combinatorial optimization theories.

## Contribution

It proves NP-completeness of recognizing degree sequences for 3-hypergraphs, solving a 30-year open problem, and explores polynomial-time solutions for specific graph cases and optimization types.

## Key findings

- Polynomial-time solution for multihypergraph degree sequence optimization.
- NP-completeness of recognizing 3-hypergraph degree sequences.
- Positive results for convex optimization over graphs and multihypergraphs.

## Abstract

We introduce and study the problem of optimizing arbitrary functions over degree sequences of hypergraphs and multihypergraphs. We show that over multihypergraphs the problem can be solved in polynomial time. For hypergraphs, we show that deciding if a given sequence is the degree sequence of a 3-hypergraph is NP-complete, thereby solving a 30 year long open problem. This implies that optimization over hypergraphs is hard already for simple concave functions. In contrast, we show that for graphs, if the functions at vertices are the same, then the problem is polynomial time solvable. We also provide positive results for convex optimization over multihypergraphs and graphs and exploit connections to degree sequence polytopes and threshold graphs. We then elaborate on connections to the emerging theory of shifted combinatorial optimization.