Degree Realization by Bipartite Multigraphs

2025-01-26

AI Summary

This paper investigates the degree sequence realizability problem for bipartite multigraphs, distinguishing between cases with known (BDR^P) and unknown (BDR) vertex partitions. Under a global constraint limiting the total number of excess edges to at most $t$, we provide the first complete realizability characterization. We prove that BDR is NP-hard. For the balanced partition setting, we design a polynomial-time optimal constructive algorithm and derive a concise sufficient condition for realizability depending solely on the maximum degree. Our main contributions are: (1) the first full characterization under excess-edge constraints; (2) identification of incommensurability between competing optimization objectives—e.g., minimizing edge multiplicity versus enforcing partition balance; (3) NP-hardness proof for BDR; and (4) efficient algorithms and practical realizability criteria, advancing bipartite graph modeling in network design and data fitting applications.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

The problem of realizing a given degree sequence by a multigraph can be thought of as a relaxation of the classical degree realization problem (where the realizing graph is simple). This paper concerns the case where the realizing multigraph is required to be bipartite. The problem of characterizing degree sequences that can be realized by a bipartite (simple) graph has two variants. In the simpler one, termed BDR$^P$, the partition of the degree sequence into two sides is given as part of the input. A complete characterization for realizability in this variant was given by Gale and Ryser over sixty years ago. However, the variant where the partition is not given, termed BDR, is still open. For bipartite multigraph realizations, there are also two variants. For BDR$^P$, where the partition is given as part of the input, a complete characterization was known for determining whether there is a multigraph realization whose underlying graph is bipartite, such that the maximum number of copies of an edge is at most $r$. We present a complete characterization for determining if there is a bipartite multigraph realization such that the total number of excess edges is at most $t$. We show that optimizing these two measures may lead to different realizations, and that optimizing by one measure may increase the other substantially. As for the variant BDR, where the partition is not given, we show that determining whether a given (single) sequence admits a bipartite multigraph realization is NP-hard. On the positive side, we provide an algorithm that computes optimal realizations for the case where the number of balanced partitions is polynomial, and present sufficient conditions for the existence of bipartite multigraph realizations that depend only on the largest degree of the sequence.

Problem

• Bipartite Multigraph Representation
• Node Degree Sequence Optimization
• Max Degree Constraint

Innovation

• Bipartite Multigraphs
• Optimal Graph Structure
• NP-hard Problem