On minimum clique partition and maximum independent set on unit disk graphs and penny graphs: complexity and approximation

Cerioli, Márcia R.; Faria, Luérbio; Ferreira, Talita O.; Protti, Fábio

doi:10.1016/j.endm.2004.06.012

Cited by 20 publications

(17 citation statements)

References 7 publications

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“…It is shown to be MaxSNP-hard for cubic graphs and NP-complete for planar cubic graphs (see Cerioli et al [5]); they also give a 5/4-approximation algorithm for graphs with maximum degree at most 3. MCP is NPhard for a subclass of UDGs, called unit coin graphs, where the interiors of the associated disks are pairwise disjoint (see Cerioli et al [6]). Good approximations, however, are possible on UDGs.…”

Section: Introductionmentioning

confidence: 99%

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A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

Pirwani

Salavatipour

2011

Algorithmica

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We consider the problem of partitioning the set of vertices of a given unit disk graph (UDG) into a minimum number of cliques. The problem is NP-hard and various constant factor approximations are known, with the current best ratio of 3. Our main result is a weakly robust polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths that either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) it computes a clique partition having size that is guaranteed to be within (1 + ε) of the optimum size if the input is UDG; however if the input is not a UDG it either computes a clique partition as in case (i) with no guarantee on the quality of the clique partition or detects that it is not a UDG. Noting that recognition of UDG's is NP-hard even if we are given edge lengths, our PTAS is a weakly-robust algorithm. Our algorithm can be transformed into an O( log * n ε O(1) ) time distributed PTAS. We consider a weighted version of the clique partition problem on vertex-weighted UDGs that generalizes the problem. We note some key distinctions with the unweighted version, where ideas useful in obtaining a PTAS break down. Yet, surprisingly, it admits a (2 + ε)-approximation algorithm for the weighted case where the graph is expressed, say, as an adjacency matrix. This improves on the best known 8-approximation for the unweighted case for UDGs expressed in standard form.

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Section: Introductionmentioning

confidence: 99%

“…Good approximations, however, are possible on UDGs. The best known approximation is due to [6] who give a 3-approximation via a partitioning the vertices into co-comparability graphs, and solving the problem exactly on them. They present a 2-approximation algorithm for coin graphs.…”

Section: Introductionmentioning

confidence: 99%

A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

Pirwani

Salavatipour

2011

Algorithmica

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“…This problem was shown to be NP-hard in [8]. Our reduction relies on the following simple but important property of the most likely hull in the point model, whose proof is included in Appendix D.…”

Section: Hardness Of the 3-dimensional Most Likely Hullmentioning

confidence: 99%

On the Most Likely Convex Hull of Uncertain Points

Suri

Verbeek

Yıldız

2013

Lecture Notes in Computer Science

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Abstract. Consider a set of d-dimensional points where the existence or the location of each point is determined by a probability distribution. The convex hull of this set is a random variable distributed over exponentially many choices. We are interested in finding the most likely convex hull, namely, the one with the maximum probability of occurrence. We investigate this problem under two natural models of uncertainty: the point (also called the tuple) model where each point (site) has a fixed position si but only exists with some probability πi, for 0 < πi ≤ 1, and the multipoint model where each point has multiple possible locations or it may not appear at all. We show that the most likely hull under the point model can be computed in O(n 3 ) time for n points in d = 2 dimensions, but it is NP-hard for d ≥ 3 dimensions. On the other hand, we show that the problem is NP-hard under the multipoint model even for d = 2 dimensions. We also present hardness results for approximating the probability of the most likely hull. While we focus on the most likely hull for concreteness, our results hold for other natural definitions of a probabilistic hull.

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“…However for UDGs, the maximum clique problem is polynomial time solvable while the maximum independent set problem is NP-hard [7], which is in curious analogy to planar graphs. The related minimum graph coloring problem [7], and the minimum clique partitioning problem [6] on UDGs are also known to be NP-hard. Hence, the research in this area has emphasized approximation algorithms and distributed heuristics.…”

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confidence: 99%

Approximation algorithms for finding and partitioning unit-disk graphs into co-k-plexes

2009

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This article studies a degree-bounded generalization of independent sets called co-k-plexes. Constant factor approximation algorithms are developed for the maximum co-k-plex problem on unit-disk graphs. The related problem of minimum co-k-plex coloring that generalizes classical vertex coloring is also studied in the context of unit-disk graphs. We extend several classical approximation results for independent sets in UDGs to co-k-plexes, and settle a recent conjecture on the approximability of co-k-plex coloring in UDGs.

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On minimum clique partition and maximum independent set on unit disk graphs and penny graphs: complexity and approximation

Cited by 20 publications

References 7 publications

A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

A Weakly Robust PTAS for Minimum Clique Partition in Unit Disk Graphs

On the Most Likely Convex Hull of Uncertain Points

Approximation algorithms for finding and partitioning unit-disk graphs into co-k-plexes

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