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Degree Sequence Optimization and Extremal Degree Enumerators

Shmuel Onn

TL;DR

This work introduces degree enumerators as nonlinear transforms of degree sequences and defines degree enumerator polytopes to study subgraph optimization. It characterizes the vertices of the degree enumerator polytope for complete graphs and, in the bipartite setting, the degree bi-enumerator polytope for $K_{m,n}$ (notably $m=1$ and $m=2$). The main results show that extremal enumerators correspond to (almost) regular subgraphs and yield compact, efficiently searchable vertex sets, enabling $O(n^2)$-time algorithms for degree optimization on $K_n$ and $K_{2,n}$. These findings provide structural insights and practical speedups for specialized graph classes, while clarifying the polyhedral structure underlying degree-based optimization problems.

Abstract

The degree sequence optimization problem is to find a subgraph of a given graph which maximizes the sum of given functions evaluated at the subgraph degrees. Here we study this problem by replacing degree sequences, via suitable nonlinear transformations, by suitable degree enumerators, and we introduce suitable degree enumerator polytopes. We characterize their vertices, that is, the extremal degree enumerators, for complete graphs and some complete bipartite graphs, and use these characterizations to obtain simpler and faster algorithms for optimization over degree sequences for such graphs.

Degree Sequence Optimization and Extremal Degree Enumerators

TL;DR

This work introduces degree enumerators as nonlinear transforms of degree sequences and defines degree enumerator polytopes to study subgraph optimization. It characterizes the vertices of the degree enumerator polytope for complete graphs and, in the bipartite setting, the degree bi-enumerator polytope for (notably and ). The main results show that extremal enumerators correspond to (almost) regular subgraphs and yield compact, efficiently searchable vertex sets, enabling -time algorithms for degree optimization on and . These findings provide structural insights and practical speedups for specialized graph classes, while clarifying the polyhedral structure underlying degree-based optimization problems.

Abstract

The degree sequence optimization problem is to find a subgraph of a given graph which maximizes the sum of given functions evaluated at the subgraph degrees. Here we study this problem by replacing degree sequences, via suitable nonlinear transformations, by suitable degree enumerators, and we introduce suitable degree enumerator polytopes. We characterize their vertices, that is, the extremal degree enumerators, for complete graphs and some complete bipartite graphs, and use these characterizations to obtain simpler and faster algorithms for optimization over degree sequences for such graphs.
Paper Structure (6 sections, 7 theorems, 34 equations)

This paper contains 6 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Graphs
  3. Bipartite graphs
  4. Overview
  5. Extremal enumerators of complete graphs
  6. On extremal bi-enumerators of complete bipartite graphs

Key Result

Theorem 2.1

The vertices of the degree enumerator polytope ${\cal E}_n={\cal E}_{K_n}$ are as follows. The number of vertices for even $n$ is $n$ and for odd $n$ is $({n+1\over2})^2$, so grows quadratically in $n$.

Theorems & Definitions (7)

  • Theorem 2.1
  • Corollary 2.2
  • Proposition 2.3
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Corollary 3.4