An integer linear program for mixed-weight open locating-dominating sets

Givens, Robin M.; Kincaid, Rex K.; Mao, Weizhen; Yu, Gexin

doi:10.1109/ciss.2018.8362259

2018 52nd Annual Conference on Information Sciences and Systems (CISS) 2018

DOI: 10.1109/ciss.2018.8362259

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An integer linear program for mixed-weight open locating-dominating sets

Robin M. Givens

Rex K. Kincaid

Weizhen Mao

et al.

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Cited by 1 publication

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“…In Section 7.4 we conclude our discussion of the ILP formulation. The majority of the work presented in this chapter has been published in [35] c ⃝ 2018 IEEE.…”

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confidence: 99%

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Mixed-weight open locating-dominating sets

Givens

Kincaid

Mao

et al. 2017

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

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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.

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“…In Section 7.4 we conclude our discussion of the ILP formulation. The majority of the work presented in this chapter has been published in [35] c ⃝ 2018 IEEE.…”

mentioning

confidence: 99%

“…We conclude our study of greedy algorithms for mixed-weight OLD-sets in Section 8.4. Most of the work presented in this chapter has been published in [35] c ⃝ 2018 IEEE.…”

mentioning

confidence: 99%

Mixed-weight open locating-dominating sets

Givens

Kincaid

Mao

et al. 2017

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

Self Cite

View full text Add to dashboard Cite

show abstract

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An integer linear program for mixed-weight open locating-dominating sets

Cited by 1 publication

References 14 publications

Mixed-weight open locating-dominating sets

Mixed-weight open locating-dominating sets

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