Characterizing Extremal Degree Enumerators for Complete Graphs and Complete Bipartite Graphs

Core Concepts

The article characterizes the extremal degree enumerators, which are the vertices of the degree enumerator polytope, for complete graphs and complete bipartite graphs. This characterization leads to faster algorithms for degree sequence optimization problems over these graph classes.

Abstract

The article studies the degree sequence optimization problem, which is to find a subgraph of a given graph that maximizes the sum of given functions evaluated at the subgraph degrees. The authors introduce the concept of degree enumerators, which are vectors representing the number of vertices of each degree in a graph, and the corresponding degree enumerator polytope.
The key insights are:

For complete graphs Kn, the vertices of the degree enumerator polytope En are precisely the degree enumerators of certain regular and almost-regular graphs Gn(r) and Gn(r,s). This characterization leads to a much faster O(n^2) algorithm for degree sequence optimization over Kn.

For complete bipartite graphs Km,n, the vertices of the degree bi-enumerator polytope Bm,n have a more complicated structure. The article completely characterizes the vertices for the case m=2 and any n. This again leads to a faster O(n^2) algorithm for degree sequence bi-optimization over K2,n.

The article also shows that the total number of degree enumerators grows exponentially with n, while the number of extremal enumerators, i.e., the vertices of the polytopes, only grows quadratically.

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Key Insights Distilled From

Degree Sequence Optimization and Extremal Degree Enumerators

by Shmuel Onn at arxiv.org 04-04-2024

https://arxiv.org/pdf/2404.02551.pdf

Deeper Inquiries

How can the insights from this work on complete and complete bipartite graphs be extended to other classes of graphs

The insights gained from studying complete and complete bipartite graphs can be extended to other classes of graphs by considering different graph structures and properties. For example, the characterization of extremal enumerators and the formulation of optimization problems over degree sequences can be applied to regular graphs, random graphs, or specific families of graphs like trees, cycles, or planar graphs. By adapting the techniques used in this work to analyze different graph classes, researchers can gain a deeper understanding of the optimization problems and extremal properties specific to those graphs.

What are the computational complexity implications of the exponential growth in the total number of degree enumerators compared to the quadratic growth in the number of extremal enumerators

The exponential growth in the total number of degree enumerators compared to the quadratic growth in the number of extremal enumerators has significant computational complexity implications. The exponential growth implies that the total number of possible degree sequences or bi-enumerators increases rapidly with the size of the graph, leading to a combinatorial explosion in the search space. This can make exhaustive search or enumeration infeasible for large graphs. On the other hand, the quadratic growth in the number of extremal enumerators suggests that there are fewer distinct optimal solutions or extremal properties to consider, which can simplify the optimization process and reduce the computational complexity of finding optimal solutions.

Can the techniques used here be applied to study optimization problems over other graph invariants beyond just degree sequences

The techniques used in this work can be applied to study optimization problems over other graph invariants beyond just degree sequences. For example, one could explore optimization problems related to graph colorings, matchings, or cuts using similar mathematical formulations and polytope characterizations. By defining suitable enumerators for these graph invariants and constructing corresponding polytopes, researchers can analyze extremal properties and develop efficient algorithms for optimization over these invariants. Additionally, the approach of characterizing extremal enumerators and studying polytope structures can be extended to various combinatorial optimization problems on graphs, providing insights into the complexity and structure of optimal solutions.