Complexity of increasing the secure connectivity in wireless ad hoc networks

We consider the problem of maximizing the secure connectivity in wireless ad hoc networks, and analyze complexity of the post-deployment key establishment process constrained by physical layer properties such as connectivity, energy consumption and interference. Two approaches, based on graph augmentation problems with nonlinear edge costs, are formulated. The first one is based on establishing a secret key using only the links that are already secured by shared keys. This problem is in NP-hard and does not accept polynomial time approximation scheme PTAS since minimum cutsets to be augmented do not admit constant costs. The second one extends the first problem by increasing the power level between a pair of nodes that has a secret key to enable them physically connect. This problem can be formulated as the optimal key establishment problem with interference constraints with bi-objectives: (i) maximizing the concurrent key establishment flow, (ii) minimizing the cost. We prove that both problems are NP-hard and MAX-SNP with a reduction to MAX3SAT problem

Shapley Meets Shapley

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.Comment: 17 page

Approximating minimum cost connectivity problems

We survey approximation algorithms of connectivity problems. The survey presented describing various techniques. In the talk the following techniques and results are presented. 1)Outconnectivity: Its well known that there exists a polynomial time algorithm to solve the problems of finding an edge k-outconnected from r subgraph [EDMONDS] and a vertex k-outconnectivity subgraph from r [Frank-Tardos] . We show how to use this to obtain a ratio 2 approximation for the min cost edge k-connectivity problem. 2)The critical cycle theorem of Mader: We state a fundamental theorem of Mader and use it to provide a 1+(k-1)/n ratio approximation for the min cost vertex k-connected subgraph, in the metric case. We also show results for the min power vertex k-connected problem using this lemma. We show that the min power is equivalent to the min-cost case with respect to approximation. 3)Laminarity and uncrossing: We use the well known laminarity of a BFS solution and show a simple new proof due to Ravi et al for Jain\u27s 2 approximation for Steiner network

Faster graph algorithms via switching classes

2012 Summer.Includes bibliographical references.The runtime of an algorithm is intimately related to how an instance is represented. Recall that the runtimes of the first generation of graph algorithms were expressed as functions of n := |V|. This analysis was natural since at this time graphs were represented in n2 space via their adjacency matrix. It was soon noticed that if m := |E| = o(n2), then a variety of graph algorithms could be sped-up by computing the adjacency-list from the adjacency matrix, then running the algorithm on the more efficient adjacency-list representation. This motivated the introduction of m to the runtime of graph algorithms and it is now customary in algorithm design to assume that a graph instance is given in the form of its adjacency-list. For instance, a graph algorithm is not considered to run in linear time unless it runs in O(n + m) time. An O(n2) bound is not considered linear, even though the two bounds are the same in the worst case. Let m͂ be the size of the minimum representative of a graph G's switching class (w.r.t. to some switching operation). It is shown that better bounds for several classical graph algorithms can be obtained by modifying them so that their running time is a function of n+m͂ rather than of n+m. This is significant because m͂ is O(m) but m is not O(m͂). This is accomplished by first computing the so-called partially complemented adjacency list (pc-list) from an adjacency list, then designing an algorithm that is amenable to the more efficient pc-list representation. The pc-list data-structure is generalization of the adjacency list that has a natural correspondence to switching classes. Using this approach, better bounds are obtained for bipartite maximum matching, graph diameter, and vertex-weighted all-pairs shortest path