An integer program for Open Locating Dominating sets and its results on the hexagon-triangle infinite grid and other graphs

Sweigart, D. Blair; Presnell, Julia; Kincaid, Rex K.

doi:10.1109/sieds.2014.6829887

Cited by 7 publications

(4 citation statements)

References 4 publications

(2 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Sweigart et al [29] showed that, for any two vertices u and v if d(u, v) ≥ 3, then both u and v have no common neighbors. This implies that we do not need to check the set N(u) ∩ S = N(v) ∩ S for equivalence, since it permits us to reduce the number of constraints that the locating requirements generate.…”

Section: An Integer Linear Programming Modelmentioning

confidence: 99%

Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

2018

View full text Add to dashboard Cite

A convex polytope or simply polytope is the convex hull of a finite set of points in Euclidean space R d . Graphs of convex polytopes emerge from geometric structures of convex polytopes by preserving the adjacency-incidence relation between vertices. In this paper, we study the problem of binary locating-dominating number for the graphs of convex polytopes which are symmetric rotationally. We provide an integer linear programming (ILP) formulation for the binary locating-dominating problem of graphs. We have determined the exact values of the binary locating-dominating number for two infinite families of convex polytopes. The exact values of the binary locating-dominating number are obtained for two rotationally-symmetric convex polytopes families. Moreover, certain upper bounds are determined for other three infinite families of convex polytopes. By using the ILP formulation, we show tightness in the obtained upper bounds.

show abstract

Section: An Integer Linear Programming Modelmentioning

confidence: 99%

Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

2018

View full text Add to dashboard Cite

show abstract

“…In [28] it was noted that if d(u, v) ≥ 3 then u, v has no neighbors in common, therefore, N (u)∩S = N (v)∩S need not be checked for equivalence. This becomes computationally important for large graphs as it allows us to minimize the number of constraints generated by the locating requirement.…”

Section: A Modified Integer Linear Programming Formulationmentioning

confidence: 99%

The binary locating-dominating number of some convex polytopes

2017

View full text Add to dashboard Cite

show abstract

“…An ILP formulation for the (unweighted) OLD-set problem can be found in [77]. We extend the OLD-set ILP formulation in [77] to include mixed-weight OLD-sets.…”

Section: Mixed-weight Old-setsmentioning

confidence: 99%

“…An ILP formulation for the (unweighted) OLD-set problem can be found in [77]. We extend the OLD-set ILP formulation in [77] to include mixed-weight OLD-sets. We check the branch and bound results for correctness and consider the utility of the solution to We also use the ILP model to find minimum-sized mixed-weight OLD-sets in large random geometric graphs.…”

Section: Mixed-weight Old-setsmentioning

confidence: 99%

Mixed-weight open locating-dominating sets

Givens

Kincaid

Mao

et al. 2017

2017 51st Annual Conference on Information Sciences and Systems (CISS)

View full text Add to dashboard Cite

The detection and location of issues in a network is a common problem encompassing a wide variety of research areas. Location-detection problems have been studied for wireless sensor networks and environmental monitoring, microprocessor fault detection, public utility contamination, and finding intruders in buildings. Modeling these systems as a graph, we want to find the smallest subset of nodes that, when sensors are placed at those locations, can detect and locate any anomalies that arise. One type of set that solves this problem is the open locating-dominating set (OLD-set), a set of nodes that forms a unique and nonempty neighborhood with every node in the graph. For this work, we begin with a study of OLD-sets in circulant graphs. Circulant graphs are a group of regular cyclic graphs that are often used in massively parallel systems. We prove the optimal OLD-set size for two circulant graphs using two proof techniques: the discharging method and Hall's Theorem. Next we introduce the mixed-weight open locating-dominating set (mixed-weight OLD-set), an extension of the OLD-set. The mixed-weight OLD-set allows nodes in the graph to have different weights, representing systems that use sensors of varying strengths. This is a novel approach to the study of location-detection problems. We show that the decision problem for the minimum mixed-weight OLD-set, for any weights up to positive integer d, is NP-complete. We find the size of mixed-weight OLD-sets in paths and cycles for weights 1 and 2. We consider mixed-weight OLD-sets in random graphs by providing probabilistic bounds on the size of the mixed-weight OLD-set and use simulation to reinforce the theoretical results. Finally, we build and study an integer linear program to solve for mixed-weight OLD-sets and use greedy algorithms to generate mixed-weight OLD-set estimates in random geometric graphs. We also extend our results for mixed-weight OLD-sets in random graphs to random geometric graphs by estimating the probabilistic upper bound for the size of the set.

show abstract

An integer program for Open Locating Dominating sets and its results on the hexagon-triangle infinite grid and other graphs

Cited by 7 publications

References 4 publications

Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

Binary Locating-Dominating Sets in Rotationally-Symmetric Convex Polytopes

The binary locating-dominating number of some convex polytopes

Mixed-weight open locating-dominating sets

Contact Info

Product

Resources

About