The Homogeneous Broadcast Problem in Narrow and Wide Strips

Let $P$ be a set of nodes in a wireless network, where each node is modeled as a point in the plane, and let $s\in P$ be a given source node. Each node $p$ can transmit information to all other nodes within unit distance, provided $p$ is activated. The (homogeneous) broadcast problem is to activate a minimum number of nodes such that in the resulting directed communication graph, the source $s$ can reach any other node. We study the complexity of the regular and the hop-bounded version of the problem (in the latter, $s$ must be able to reach every node within a specified number of hops), with the restriction that all points lie inside a strip of width $w$. We almost completely characterize the complexity of both the regular and the hop-bounded versions as a function of the strip width $w$.Comment: 50 pages, WADS 2017 submissio

A Faster Distributed Single-Source Shortest Paths Algorithm

We devise new algorithms for the single-source shortest paths (SSSP) problem with non-negative edge weights in the CONGEST model of distributed computing. While close-to-optimal solutions, in terms of the number of rounds spent by the algorithm, have recently been developed for computing SSSP approximately, the fastest known exact algorithms are still far away from matching the lower bound of $\tilde \Omega (\sqrt{n} + D)$ rounds by Peleg and Rubinovich [SIAM Journal on Computing 2000], where $n$ is the number of nodes in the network and $D$ is its diameter. The state of the art is Elkin's randomized algorithm [STOC 2017] that performs $\tilde O(n^{2/3} D^{1/3} + n^{5/6})$ rounds. We significantly improve upon this upper bound with our two new randomized algorithms for polynomially bounded integer edge weights, the first performing $\tilde O (\sqrt{n D})$ rounds and the second performing $\tilde O (\sqrt{n} D^{1/4} + n^{3/5} + D)$ rounds. Our bounds also compare favorably to the independent result by Ghaffari and Li [STOC 2018]. As side results, we obtain a $(1 + \epsilon)$-approximation $\tilde O ((\sqrt{n} D^{1/4} + D) / \epsilon)$-round algorithm for directed SSSP and a new work/depth trade-off for exact SSSP on directed graphs in the PRAM model.Comment: Presented at the the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2018

On the Approximability of External-Influence-Driven Problems

Domination problems in general can capture situations in which some entities have an effect on other entities (and sometimes on themselves). The usual goal is to select a minimum number of entities that can influence a target group of entities or to influence a maximum number of target entities with a certain number of available influencers. In this work, we focus on the distinction between \textit{internal} and \textit{external} domination in the respective maximization problem. In particular, a dominator can dominate its entire neighborhood in a graph, internally dominating itself, while those of its neighbors which are not dominators themselves are externally dominated. We study the problem of maximizing the external domination that a given number of dominators can yield and we present a 0.5307-approximation algorithm for this problem. Moreover, our methods provide a framework for approximating a number of problems that can be cast in terms of external domination. In particular, we observe that an interesting interpretation of the maximum coverage problem can capture a new problem in elections, in which we want to maximize the number of \textit{externally represented} voters. We study this problem in two different settings, namely Non-Secrecy and Rational-Candidate, and provide approximability analysis for two alternative approaches; our analysis reveals, among other contributions, that an earlier resource allocation algorithm is, in fact, a 0.462-approximation algorithm for maximum external domination in directed graphs

Structural Parameters, Tight Bounds, and Approximation for (k,r)-Center

In (k,r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically: - For any r>=1, we show an algorithm that solves the problem in O*((3r+1)^cw) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm\u27s performance. As a corollary, for r=1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw. - We strengthen previously known FPT lower bounds, by showing that (k,r)-Center is W[1]-hard parameterized by the input graph\u27s vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs. - We show that the complexity of the problem parameterized by tree-depth is 2^Theta(td^2) by showing an algorithm of this complexity and a tight ETH-based lower bound. We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth which work efficiently independently of the values of k,r. In particular, we give algorithms which, for any epsilon>0, run in time O*((tw/epsilon)^O(tw)), O*((cw/epsilon)^O(cw)) and return a (k,(1+epsilon)r)-center, if a (k,r)-center exists, thus circumventing the problem\u27s W-hardness