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 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.Comment: 21 pages, 9 figure

Decomposition by maxclique separators

We present a minimal counterexample to an O(|V||E|) algorithm for decomposing a graph G=(V,E) by maximal cliques proposed by R. Tarjan. We also present a modification to this algorithm which is correct while retaining its O(|V||E|) complexity

On the (in)dependence of the Dedekind-Peano axioms for natural numbers

We present a direct proof that the Dedekind-Peano axioms for the sequence of natural numbers are not completely independent, as well as a new completely independent set of axioms based on the same set of primitives as the one originally proposed by R. Dedekind

Loop Graphs And Asteroidal Sets

[No abstract available]22179183Alcón, L., de Figueiredo, C.H., Cerioli, M., Gutierrez, M., Meidanis, J., Tree Loop Graphs (2004) Electronic Notes in Discrete Mathematics, 18, pp. 17-23. , Proceedings of LACGA 2004 Full paper accepted for publication in Discrete Applied Mathematics Series on Computational Molecular BiologyAlcón, L., de Figueiredo, C.H., Cerioli, M., Gutierrez, M., Meidanis, J., Loop Graphs with induced cycles (2005) Electronic Notes in Discrete Mathematics, 19, pp. 289-295. , Proceedings of GRACO 2005Setubal, J.C., Meidanis, J., (1997) Introduction to Computational Molecular Biology, , PWS, BostonWalter, J.R., Representations of Chordal Graphs as subtrees of a tree (1978) J. of Graph Theory, 2, pp. 265-26

On the most likely convex hull of uncertain points

Consider a set of points in d dimensions 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 s i but only exists with some probability p i , for 0¿\u3cp/\u3