Prefetching constitutes a valuable tool toward efficient Web surfing. As a result, estimating the amount of resources that need to be preloaded during a surfer's browsing becomes an important task. In this regard, prefetching can be modeled as a two-player combinatorial game [Fomin et al., Theoretical Computer Science 2014], where a surfer and a marker alternately play on a given graph (representing the Web graph). During its turn, the marker chooses a set of $k$ nodes to mark (prefetch), whereas the surfer, represented as a token resting on graph nodes, moves to a neighboring node (Web resource). The surfer's objective is to reach an unmarked node before all nodes become marked and the marker wins. Intuitively, since the surfer is step-by-step traversing a subset of nodes in the Web graph, a satisfactory prefetching procedure would load in cache all resources lying in the neighborhood of this growing subset. Motivated by the above, we consider the following problem to which we refer to as the Maximum Rooted Connected Expansion (MRCE) problem. Given a graph $G$ and a root node $v_0$, we wish to find a subset of vertices $S$ such that $S$ is connected, $S$ contains $v_0$ and the ratio $|N[S]|/|S|$ is maximized, where $N[S]$ denotes the closed neighborhood of $S$, that is, $N[S]$ contains all nodes in $S$ and all nodes with at least one neighbor in $S$. We prove that the problem is NP-hard even when the input graph $G$ is restricted to be a split graph. On the positive side, we demonstrate a polynomial time approximation scheme for split graphs. Furthermore, we present a $\frac{1}{6}(1-\frac{1}{e})$-approximation algorithm for general graphs based on techniques for the Budgeted Connected Domination problem [Khuller et al., SODA 2014]. Finally, we provide a polynomial-time algorithm for the special case of interval graphs.

Abstract We consider a probabilistic model for the Steiner Tree problem. Under this model, the problem is defined in a two‐stage setting over a first‐stage complete weighted graph having its vertices associated with a probability of presence (independently each from another) in the second stage. A first‐stage feasible solution on the input graph might become infeasible in the second stage, when certain vertices of the graph fail. Therefore, a well defined modification strategy is devised for modifying the remainders of a first‐stage solution to render it second‐stage feasible. The objective is to minimize the expected weight of the second‐stage solution over the distribution of all possible second‐stage materializable subgraphs of the input graph. We recognize two complementary computational problems in this setting, one being the a priori computation of first‐stage decisions given a particular modification strategy, and the second being the cost‐efficient modification of a first‐stage feasible solution. We prove that both these problems are NP‐hard for the Steiner Tree problem under this setting. We design and analyze probabilistically an efficient modification strategy and derive tight approximation results for both aforementioned problems. We show that our techniques can be extended to the case of the more general Steiner Forest problem in the same probabilistic setting. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010

A commonly employed abstraction for studying the object placement problem for the purpose of Internet content distribution is that of a distributed replication group. In this work, the initial model of the distributed replication group of Leff et al. [CHECK END OF SENTENCE] is extended to the case that individual nodes act selfishly, i.e., cater to the optimization of their individual local utilities. Our main contribution is the derivation of equilibrium object placement strategies that 1) can guarantee improved local utilities for all nodes concurrently as compared to the corresponding local utilities under greedy local object placement, 2) do not suffer from potential mistreatment problems, inherent to centralized strategies that aim at optimizing the social utility, and 3) do not require the existence of complete information at all nodes. We develop a baseline computationally efficient algorithm for obtaining the aforementioned equilibrium strategies and then extend it to improve its performance with respect to fairness. Both algorithms are realizable, in practice, through a distributed protocol that requires only a limited exchange of information.

The extended Hopfield neural network proposed by Abe et al. for solving combinatorial optimization problems with equality and/or inequality constraints has the drawback of being frequently stabilized in states with neurons of ambiguous classification as active or inactive. We introduce in the model a competitive activation mechanism and we derive a new expression of the penalty energy allowing us to reduce significantly the number of neurons with intermediate level of activations. The new version of the model is validated experimentally on the set covering problem. Our results confirm the importance of instituting competitive activation mechanisms in Hopfield neural-network models.

A Boltzmann machine architecture to solve the problems of maximum independent set, set partitioning, clique, minimum vertex cover, minimum set cover, and maximum set packing is described. The authors evaluate the maximum and the average error of the method where the error is defined as the ratio of the cardinality of the obtained solution for an instance with respect to the optimal one. The results are compared with those obtained from the implementation of the heuristic described by D.S. Johnson (1974). The model treats the general case of all these problems that is the case when costs are associated with the data (vertices or subsets). The unweighted case becomes a particular case in this approach. It is shown that the model finds optimal solutions for a large percentage of the treated instances and provides a good performance ratio for the rest.>