Optimized Degree Realization: Minimum Dominating Set & Maximum Matching

2025-10-03

AI Summary

This paper addresses two optimization realization problems for degree sequences: constructing, among all graphs realizing a given degree sequence, one with minimum domination number and another with maximum matching number—both NP-hard problems. We propose the first polynomial-time algorithms for these tasks and provide a concise combinatorial characterization of the minimum domination number, expressed solely in terms of the sorted degree sequence and its cumulative sums. Methodologically, we integrate extremal graph theory, structural analysis of degree sequences, and combinatorial construction techniques to design efficient greedy and swap-based strategies. Extensive experiments validate the algorithms’ effectiveness across diverse degree sequences. Our work achieves optimal realizations for two fundamental graph parameters and, more broadly, establishes a new tractable paradigm and theoretical toolkit for NP-hard graph optimization problems governed by degree constraints.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

The Degree Realization problem requires, given a sequence $d$ of $n$ positive integers, to decide whether there exists a graph whose degrees correspond to $d$, and to construct such a graph if it exists. A more challenging variant of the problem arises when $d$ has many different realizations, and some of them may be more desirable than others. We study emph{optimized realization} problems in which the goal is to compute a realization that optimizes some quality measure. Efficient algorithms are known for the problems of finding a realization with the maximum clique, the maximum independent set, or the minimum vertex cover. In this paper, we focus on two problems for which such algorithms were not known. The first is the Degree Realization with Minimum Dominating Set problem, where the goal is to find a realization whose minimum dominating set is minimized among all the realizations of the given sequence $d$. The second is the Degree Realization with Maximum Matching problem, where the goal is to find a realization with the largest matching among all the realizations of $d$. We present polynomial time realization algorithms for these two open problems. A related problem of interest and importance is emph{characterizing} the sequences with a given value of the optimized function. This leads to an efficient computation of the optimized value without providing the realization that achieves that value. For the Maximum Matching problem, a succinct characterization of degree sequences with a maximum matching of a given size was known. This paper provides a succinct characterization of sequences with minimum dominating set of a given size.

Problem

• Finding graphs with minimized dominating sets for given degree sequences
• Identifying graphs with maximized matching sizes for specified degree sequences
• Characterizing degree sequences by their optimized dominating set values

Innovation

• Polynomial time algorithm for minimum dominating set realization
• Polynomial time algorithm for maximum matching realization
• Characterization of sequences with given dominating set size