Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1902.01228

Stabilization Time in Weighted Minority Processes

Pál András Papp·Roger Wattenhofer
View original source ↗Read source document ↗Export metadata ↓

Abstract

A minority process in a weighted graph is a dynamically changing coloring. Each node repeatedly changes its color in order to minimize the sum of weighted conflicts with its neighbors. We study the number of steps until such a process stabilizes. Our main contribution is an exponential lower bound on stabilization time. We first present a construction showing this bound in the adversarial sequential model, and then we show how to extend the construction to establish the same bound in the benevolent sequential model, as well as in any reasonable concurrent model. Furthermore, we show that the stabilization time of our construction remains exponential even for very strict switching conditions, namely, if a node only changes color when almost all (i.e., any specific fraction) of its neighbors have the same color. Our lower bound works in a wide range of settings, both for node-weighted and edge-weighted graphs, or if we restrict minority processes to the class of sparse graphs.

Explore related subjects

cs.DM

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pál András Papp, Roger Wattenhofer. 2019-02-04. Stabilization Time in Weighted Minority Processes. https://arxiv.org/abs/1902.01228

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

UTVPI-representable integer point sets: discrete convexity, polymorphisms, and pairwise closure

We study subsets of the integer lattice represented by single-variable-per-inequality (SVPI), difference-constraint (DC), unit two-variable-per-inequality (UTVPI), and two-variable-per-inequality (TVPI) systems. We relate five viewpoints: inequality representation, discrete convexity, polymorphisms, reconstruction from two-coordinate projections, and fixed points of closure operators. Our central result completely characterizes UTVPI-representability. For every set $S\subseteq\mathbb Z^n$ with $n>1$, \[ \begin{aligned} &S\text{ is UTVPI-representable}\\ &\;\Longleftrightarrow\; S\text{ is closed under the directed midpoint and median operations}\\ &\;\Longleftrightarrow\; S\text{ is integrally convex and $2$-decomposable}. \end{aligned} \] The median condition may instead be replaced by closedness under some majority operation, and the same class is the fixed-point class of a pairwise directed-midpoint closure operator. Thus, all five viewpoints yield equivalent characterizations of UTVPI-representability. In particular, $2$-decomposability is exactly the global condition needed to lift the known two-dimensional equivalence between integral convexity and UTVPI-representability to arbitrary dimension. This theorem is embedded in a broader pairwise-closure theory. For a family $F$ of operations, we define a closure operator by closing every two-coordinate projection under $F$ and joining the resulting sets. Its fixed points are precisely the sets that are both $2$-decomposable and $F$-closed, and we establish a local-to-global criterion for such characterizations. A closed-convex-hull analogue characterizes TVPI-representability. We also characterize SVPI-representability by natural multioperations, prove limitations of operation-based characterizations for several related classes, and determine the complete inclusion hierarchies in the general, Boolean, and two-dimensional settings.

cs.DM↗
arXiv2026

Integrality gap preserving reductions

We propose a framework for the systematic study of integrality gaps of combinatorial optimization problems with respect to a fixed linear programming formulation. The method, called \emph{integrality gap preserving reduction}, consists of iteratively shrinking the input universe of the problem while guaranteeing that gap-maximizing instances remain selected. When the subset of remaining instances becomes specific enough, we calculate the integrality gap explicitly. Besides applying integrality gap preserving reductions to three well-known optimization problems via their standard linear programming formulations (weighted vertex cover problem, multiple knapsack problem, and unrelated machine scheduling problem), we analyse the restricted assignment problem via its configuration LP relaxation. We prove that the integrality gap is equal to $1$ for three ``easy'' subclasses of the problem that are either solvable in polynomial time or admit a PTAS (e.g., the all-one processing time case). For some remaining cases, we improve the current lower bound using our technique.

cs.DM↗
arXiv2026

Paired Disjunctive Domination Number of Middle Graphs

The concept of domination in graphs plays a central role in understanding structural properties and applications in network theory. In this study, we focus on the paired disjunctive domination number in the context of middle graphs, a transformation that captures both adjacency and incidence relations of the original graph. We begin by investigating this parameter for middle graphs of several special graph classes, including path graphs, cycle graphs, wheel graphs, complete graphs, complete bipartite graphs, star graphs, friendship graphs, and double star graphs. We then present general results by establishing lower and upper bounds for the paired disjunctive domination number in middle graphs of arbitrary graphs, with particular emphasis on trees. Additionally, we determine the exact value of the parameter for middle graphs obtained through the join operation. These findings contribute to the broader understanding of domination-type parameters in transformed graph structures and offer new insights into their combinatorial behavior.

cs.DM↗