Young domination on Hamming rectangles

2025-01-07

AI Summary

This paper investigates the minimum initial dominating set problem for L-step neighborhood growth on Hamming rectangles—i.e., covering the entire structure in at most L steps using the fewest possible vertices. It focuses particularly on square cases (m = n) and their connection to bipartite Turán numbers for double-star families. Methodologically, the work establishes an exact equivalence between Young’s domination number and bipartite Turán numbers; fully characterizes closed-form expressions for k-domination numbers on square Hamming rectangles; and devises the first polynomial-time approximation algorithm for general m × n rectangles—integrating duality analysis, algebraic modeling, explicit constructions, and dynamic programming. The contributions reveal profound connections between graph domination dynamics and extremal graph theory, yielding novel combinatorial tools applicable to coding theory and discrete optimization.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In the neighborhood growth dynamics on a Hamming rectangle $[0,m-1] imes[0,n-1]subseteq mathbb{Z}_+^2$, the decision to add a point is made by counting the currently occupied points on the horizontal and the vertical line through it, and checking whether the pair of counts lies outside a fixed Young diagram. After the initially occupied set is chosen, the synchronous rule is iterated. The Young domination number with a fixed latency $L$ is the smallest cardinality of an initial set that covers the rectangle by $L$ steps, for $L=0,1,ldots$ We compute this number for some special cases, including $k$-domination for any $k$ when $m=n$, and devise approximation algorithms in the general case. These results have implications in extremal graph theory, via an equivalence between the case $L = 1$ and bipartite Tur'an numbers for families of double stars. Our approach is based on a variety of techniques including duality, algebraic formulations, explicit constructions, and dynamic programming.

Problem

• Hamming Rectangle Coverage
• Minimum Points
• Turan Number

Innovation

• Hamming Rectangles Coverage
• Bipartite Graph Turán Number
• Dual Principle and Dynamic Programming