Young domination on Hamming rectangles

TL;DR

This work studies Young domination on Hamming rectangles, where growth is driven by a downward-closed zero-set ${\mathcal Z}$ (a Young diagram) and latency $L$ dictates the number of steps needed to occupy the whole grid. It derives exact formulas in key cases (notably $a$-domination for $m=n$ with $T_a$, and $L$-shaped/rectangular zero-sets) and provides polynomial-time approximation algorithms for general ${\mathcal Z}$ and fixed latency via enhanced dynamics and duality with extremal bipartite Turán problems. A central contribution is the exact resolution of several $a$-domination instances on Hamming squares, including even/odd dichotomies, plus a 3-approximation and a constant-factor approximation framework that place these problems in APX. The paper also establishes a deep duality between domination numbers and bipartite Turán numbers for families of double stars, linking nucleation-like growth to classical extremal graph theory and enabling transfer of techniques across disciplines.

Abstract

In the neighborhood growth dynamics on a Hamming rectangle $[0,m-1]\times[0,n-1]\subseteq \mathbb{Z}_+^2$, 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 $m=n$, thereby proving a conjecture from 2009 due to Burchett, Lachniet, and Lane, 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 Turán numbers for families of double stars. Our approach is based on a variety of techniques including duality between Young diagrams, algebraic formulations, explicit constructions, and dynamic programming.

Figures (5)

• Figure 2.1: Left: a zero-set ${\mathcal{Z}}$; the points of ${\mathcal{Z}}$ are the centers of the dark squares. Its concave corners (see Section \ref{['sec-prelim']}) are $(0,3)$, $(1,2)$, $(3,1)$, and $(4,0)$ and are labeled by $\square$. Right: a ${\mathcal{Z}}$-dominating set $D$ (this time, with its members indicated by $\bullet$) on the graph $K_4\Box K_5$ with vertex set $R_{5,4}=[0,4]\times [0,3]$, of (minimal) size 10. Note that the vertices of the product graph are represented so that the first coordinate is vertical and the second coordinate is horizontal, in line with the definitions of rows and columns, and row and column neighborhoods of a vertex. Accordingly, the row counts and column counts are given, respectively, along the vertical and horizontal axis. For every point in $z\in R_{5,4}\setminus D$ the point $(r,c)$ with coordinates given by its row and column counts lies outsize ${\mathcal{Z}}$, therefore $(r,c)$ is greater than or equal to a concave corner of ${\mathcal{Z}}$ in the partial order of ${\mathbb Z}_+^2$.
• Figure 2.2: A Young diagram ${\mathcal{Y}}$ (dark gray squares) within $R_{8,5}$ (outlined) is given on the left. Its four concave corners, labeled by $\square$, are reflected through both of the bisecting lines, giving the four sites labeled by $\times$. These are reproduced on the right, and determine ${ 0-0 {\ooalign{\hbox{[-1]{}} \cr \hbox{\m@th${\mathcal{Y}}$}}}}$, whose four concave corners are labeled by $\bullet$. Observe that the double reflection of the four $\bullet$'s gives the three points diagonally adjacent to the convex corners of ${\mathcal{Y}}$, as required, because the dual map is an involution. Observe also that ${ 0-0 {\ooalign{\hbox{[-1]{}} \cr \hbox{\m@th${\mathcal{Y}}$}}}}$ equals $R_{8,5}\setminus {\mathcal{Y}}$, reflected through the centroid of $R_{8,5}$, as defined in Roh.
• Figure 5.1: An example of the construction of $B$ for $m=8$, $k=1$, $c=3$, $d=4$, which correspond to $b=11$ and $a=23$, the largest one with $m = 8$ for which $k=1$. The first column of $B_{12}$ is the only one with $4$ 1s, so the corresponding diagonal element of $B_{22}$ switches to 0. Points in the $a$-dominating set are represented by dark squares.
• Figure 6.1: A zero-set ${\mathcal{Z}}$ and its concave corners.
• Figure 7.1: A trigger ${\mathcal{Y}}$ for the point $(a,b)=(8,7)$. For this example, we assume that $(r_7+5, c_8+2)$ is a concave corner of ${\mathcal{Z}}$. Vertices that touch the shaded area comprise ${\mathcal{Y}}$.

Theorems & Definitions (57)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4