Renato Carmo

Searching in random partially ordered sets

We consider the problem of searching for a given element in a partially ordered set. More precisely, we address the problem of computing efficiently near-optimal search strategies for typical partial orders under two classical models for random partial orders, the random graph model and the uniform model.We shall show that the problem of determining an optimal strategy is NP-hard, but there are simple, fast algorithms able to produce near-optimal search strategies for typical partial orders under the two models of random partial orders that we consider. We present a (1 + o(1))- approximation algorithm for typical partial orders under the random graph model (constant p) and present a 6.34-approximation algorithm for typical partial orders under the uniform model. Both algorithms run in polynomial time.

Branch and bound algorithms for the maximum clique problem under a unified framework

In this paper we review branch and bound-based algorithms proposed for the exact solution of the maximum clique problem and describe them under a unifying conceptual framework. As a proof of concept, we actually implemented eight of these algorithms as parametrized versions of one single general branch and bound algorithm.The purpose of the present work is double folded. In the one hand, the implementation of several different algorithms under the same computational environment allows for a more precise assessment of their comparative performance at the experimental level. On the other hand we see the unifying conceptual framework provided by such description as a valuable step toward a more fine grained analysis of these algorithms.

Finding stable cliques of PlanetLab nodes

Users of large scale network testbeds often execute experiments that require a set of nodes that behave and communicate among themselves in a reasonably stable pattern. In this work we call such a set of nodes a stable clique, and introduce a monitoring strategy that allows their detection in PlanetLab, a non-trivial task for such a large scale dynamic network. Nodes monitor each other by sampling the RTT (Round-Trip-Time) and computing its variation. Based on this data and a threshold, pairs of nodes are classified as stable or unstable. A set of graphs is generated, on which maximum sized cliques are computed. Three experiments were conducted in which hundreds of nodes were monitored for several days. Results show the unexpected behavior of some nodes, and the size of the maximum stable clique for different time windows and different thresholds.

Querying Priced Information in Databases: The Conjunctive Case

Query optimization that involves expensive predicates have received considerable attention in the database community. Typically, the output to a database query is a set of tuples that satisfy certain conditions, and, with expensive predicates, these conditions may be computationally costly to verify. In the simplest case, when the query looks for the set of tuples that simultaneously satisfy k expensive predicates, the problem reduces to ordering the evaluation of the predicates so as to minimize the time to output the set of tuples comprising the answer to the query.

Searching in Random Partially Ordered Sets

We consider the problem of searching for a given element in a partially ordered set. More precisely, we address the problem of computing efficiently near-optimal search strategies for typical partial orders. We consider two classical models for random partial orders, the random graph model and the uniform model.We shall show that certain simple, fast algorithms are able to produce nearly-optimal search strategies for typical partial orders under the two models of random partial orders that we consider. For instance, our algorithm for the random graph model produces, in linear time, a search strategy that makes O((log n)1/2 log log n) more queries than the optimal strategy, for almost all partial orders on n elements. Since we need to make at least lg n = log2 n queries for any n-element partial order, our result tells us that one may efficiently devise near-optimal search strategies for almost all partial orders in this model (the problem of determining an optimal strategy is NP-hard, as proved recently in [1]).

Upper Bounds for the Total Chromatic Number of Join Graphs and Cobipartite Graphs.

We concern ourselves with the combinatorial optimisation problem of determining a minimum total colouring of a graph G for the case wherein G is a join graph G = G1 ∗G2 or a cobipartite graph G = (V1 ∪V2,E(G)). We present algorithms for computing a feasible, not necessarily optimal, solution for this problem, providing the following upper bounds for the total chromatic numbers of these graphs (let ni := |Vi| and i := (Gi) for i ∈ {1,2} and ∈ {∆,χ,χ′,χ′′}): χ′′(G) 6 max{n1,n2}+ 1+P(G1,G2) if G is a join graph, wherein P(G1,G2) :=min{∆1+∆2+1,max{χ1,χ 2}}; χ(G)6max{n1,n2}+2(max{∆1 ,∆2}+1) if G is cobipartite, wherein ∆i := maxu∈Vi dG[∂G(Vi)](u) for i∈ {1,2}. Our algorithm for the cobipartite graphs runs in polynomial time. Our algorithm for the join graphs runs in polynomial time if P(G1,G2) is replaced by ∆1 +∆2 +1 or if it can be computed in polynomial time. We also prove the Total Colouring Conjecture for some subclasses of join graphs, such as some joins of indifference (unitary interval) graphs.

On comparing algorithms for the maximum clique problem

Abstract Several algorithms for the exact solution of the maximum clique problem are available in the literature. Some have been proposed with the aim of bounding the worst case complexity of the problem, while others focus on practical performance as evaluated experimentally. These two groups of works are somewhat independent, in the sense that little experimental investigation is available in the former group, and little theoretical analysis exists for the latter. Moreover, the experimental results seem to be much better than could be expected from the theoretical results. We show that a broad class of branch and bound algorithms for the maximum clique problem display sub-exponential average running time behavior, and also show how this helps to explain the apparent discrepancy between the theoretical and experimental results. We also propose a more structured methodology for the experimental analysis of algorithms for the maximum clique problem, which takes into account the peculiarities of cliques in random graphs, bringing the theoretical and experimental approaches closer together in the search for better algorithms. As a proof of concept, we apply the proposed methodology to thirteen algorithms from the literature.