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

Max-coloring paths: Tight bounds and extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G = (V, E) with vertex weights w such that ∑k i=1 maxv∈Ci w(vi) is minimized, where C1,..., Ck are the various color classes. For general graphs, max-coloring is as hard as the classical vertex coloring problem, a special case of the former where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring skinny trees, a broad class of trees that includes paths and spiders. For these graphs, we show that max-coloring can be solved in time O(|V | + time for sorting the vertex weights). When vertex weights are real numbers, we show a matching lower bound of Ω(|V | log |V |) in the algebraic computation tree model.

Long-term prognosis in patients with vasculopathic sixth nerve palsy.

PURPOSE: To better define the long-term prognosis in patients with a vasculopathic sixth nerve palsy (6NP), specifically addressing the degree of recovery and incidence of recurrent similar episodes. DESIGN: Observational case series. METHODS: Retrospective chart review. SETTING: An outpatient neuroophthalmic practice. STUDY POPULATION: Patients with one or more vascular risk factors and an acute, isolated 6NP that spontaneously recovered. OBSERVATION PROCEDURE: Information regarding resolution of the 6NP, subsequent vascular events and recurrent ocular motor nerve palsy was obtained from chart review of follow-up clinic visits, mailed questionnaires and telephone interviews. The duration of follow-up ranged from 2 to 13 years. MAIN OUTCOME MEASURES: Resolution of 6NP (complete or incomplete) and incidence of recurrent ocular motor nerve palsy. RESULTS: Fifty-nine patients were identified with a mean age of 65.3 years +/- 11.6 (range 34-90 years). Fifty-one patients (86%) experienced complete resolution of their first episode of vasculopathic 6NP and eight patients (14%) had incomplete resolution. A subsequent episode of ocular motor mononeuropathy occurred in 18 of 59 (31%) patients. The number of recurrences ranged from one (in 14 patients) to four (in one patient). There was no association between any risk factor and recurrence of ocular motor nerve palsy. Similarly, incomplete resolution of the vasculopathic 6NP was not associated with any risk factor. CONCLUSIONS: Patients with a vasculopathic 6NP usually have complete resolution of their ophthalmoplegia, but nearly one third of patients in our study later experienced at least one episode of recurrent vasculopathic ocular motor nerve palsy