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 {\em weakly robust} polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within $(1+\eps)$ ratio of the optimum 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 $O(\frac{\log^* n}{\eps^{O(1)}})$ 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 breakdown. Yet, surprisingly, it admits a $(2+\eps)$-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 {\em unweighted} case for UDGs expressed in standard form.
Deep Dive into A Weakly-Robust PTAS for Minimum Clique Partition in Unit Disk Graphs.
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 {\em weakly robust} polynomial time approximation scheme (PTAS) for UDGs expressed with edge-lengths, it either (i) computes a clique partition or (ii) gives a certificate that the graph is not a UDG; for the case (i) that it computes a clique partition, we show that it is guaranteed to be within $(1+\eps)$ ratio of the optimum 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 $O(\frac{\log^* n}{\eps^{O(1)}})$ time distributed PTA
A standard network model for homogeneous networks is the unit disk graph (UDG). A graph G = (V, E) is a UDG if there is a mapping f : V → R 2 such that f (u)f (v) 2 ≤ 1 ⇔ {u, v} ∈ E; f (u) 1 models the position of the node u while the unit disk centered at f (u) models the range of radio communication. Two nodes u and v are said to be able to directly communicate if they lie in the unit disks placed at each others' centers. There is a vast collection of literature on algorithmic problems studied on UDGs. See the survey [2].
Clustering of a set of points is an important subroutine in many algorithmic and practical applications and there are many kinds of clusterings depending upon the application. A typical objective in clustering is to minimize the number of “groups” such that each “group” (cluster) satisfies a set of criteria. Mutual proximity of points in a cluster is one such criterion, while points in a cluster forming a clique in the underlying network is an extreme form of mutual proximity. We study an optimization problem related to clustering, called the minimum clique partition problem on this UDGs.
Minimum clique partition on unit disk graphs (MCP): Given a unit disk graph, G = (V, E), partition V into a smallest number of cliques.
Despite being theoretically interesting, MCP has been useful for other problems. For example, [17] shows how to use a small-sized clique partition of a UDG to construct a large collection of disjoint (almost) dominating sets. They [18] also show how to obtain a good quality realization of UDGs, and an important ingredient in their technique was to construct a small-sized clique partition of the graph. It is shown [12] how to use a small-sized clique partition to obtain sparse spanners with bounded dilation, which also permit guaranteed geographic routing on a related class of graphs. [14] employ MCP to obtain an O(log * n) time distributed algorithm which is an O(log n)-approximation for the facility location problem on UDGs without geometry; they also give an O(1) time distributed O(1)-approximation to the facility location problem on UDGs with geometry also using MCP. Recently, [15] shows how to obtain a first O(1) approximation to the domatic partition problem on UDGs using MCP.
On general graphs, the clique-partition problem is equivalent to the minimum graph coloring on the complement graph which is not approximable within n 1-ε , for any ε > 0, unless P=NP [22]. MCP has been studied for special graph classes. It is shown to be MaxSNP-hard for cubic graphs and NP-complete for planar cubic graphs [5]; they also give a 5/4-approximation algorithm for graphs with maximum degree at most 3. MCP is NP-hard for a subclass of UDGs, called unit coin graphs, where the interiors of the associated disks are pairwise disjoint [6]. 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 give a 2-approximation algorithm for coin graphs. MCP has also been studied on UDGs expressed in standard form. For UDGs expressed in general form [18] give an 8-approximation algorithm.
In this paper we present a weakly-robust 2 PTAS for MCP on a given UDG. For ease of exposition, first we prove this (in Section 2.1) when the UDG is given with a realization, f (.). The holy-grail is a PTAS when the UDG is expressed in standard form, say, as an adjacency matrix. However, falling short of proving this, we show (in Section 2) how to get a PTAS when the input UDG is expressed in standard form along with associated edge-lengths corresponding to some (unknown) realization. The algorithm is weakly-robust in the sense that it either (i) computes a clique partition of the input graph or (ii) gives a certificate that the input graph is not a UDG. If the input is indeed a UDG then the algorithm returns a clique partition (case (i)) which is a (1 + ε)-approximation (for a given ǫ > 0). However, if the input is not a UDG, the algorithm either computes a clique partition but with no guarantee on the quality of the solution or returns that it is not a UDG. Therefore, this algorithm should be seen as a weakly-robust PTAS. The generation of a polynomial-sized certificate which proves why the input graph is not a UDG should be seen in the context of the negative result of [1] which says that even if edge lengths are given, UDG recognition is NP-hard. We show (in Section 4) how this algorithm can be modified to run in O( log * n ε O(1) ) distributed rounds. In Section 3 we explore a weighted version of MCP where we are given a vertex weighted UDG. In this formulation, the weight of a clique is the weight of a heaviest vertex in it, and the weight of a clique partition is the sum of the weights of the cliques in it. We note some key distinctions between the weighted and the unweighted versions of the problem and show that the ideas that he
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