Approximation Algorithms and Hardness for Domination with Propagation

Abstract

The power dominating set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes S that power dominates all the nodes, where a node v is power dominated if (1) v is in S or v has a neighbor in S, or (2) v has a neighbor w such that w and all of its neighbors except v are power dominated. We show a hardness of approximation threshold of 2 log 1−ǫ n in contrast to the logarithmic hardness for the dominating set problem. We give an O( √ n) approximation algorithm for planar graphs, and show that our methods cannot improve on this approximation guarantee. Finally, we initiate the study of PDS on directed graphs, and show the same hardness threshold of 2 log 1−ǫ n for directed acyclic graphs. Also we show that the directed PDS problem can be solved optimally in linear time if the underlying undirected graph has bounded tree-width.

Improved Approximation Algorithms for Label Cover Problems

“…We will prove two things: (1) first, that the chance that, in the perfect matching graph M , S contains an edge, is at least 1 − e −1/54 > 0, and (2) second, the chance that S 1 contains an edge in G is at least as large as the chance that S contains an edge in the perfect matching graph M . Now we prove (1), that both endpoints of some edge in M are chosen with constant probability. Consider one fixed edge d = (c A , c B ) carrying a flowf d .…”

“…First, we construct a bipartite graph M = (A , B , E ), for the given i, j, in which for each vertex a ∈ A i (resp., b ∈ B j ), we put r vertices a 1 We now consider a random process in which we build a set S by selecting each vertex (clone) c in M independently with probability √ q times thef flow of the unique edge incident to c in M . Let the subset of A i ∪B j chosen by MinRepAlg be called S 1 .…”

Power Domination in Circular-Arc Graphs

“…For ease of reference, the vertices in V i \ V i−1 , i > 0 are called the i th generation descendants (i-descendants for short) of those in V 0 . Note that the Induced Observation Rules are equivalent to the original observation rules [1,2,8,15,21,31]. In addition, given a graph G = (V, E), the observed graph of some kernel V 0 can be computed in O(|V | + |E|) time by the Induced Observation Rules [8].…”

“…Liao and Lee [23] proposed a different NP-completeness proof for the power domination problem in split graphs. Subsequently, Aazami and Stilp [2] separated the approximation hardness of domination and power domination. They proved that, in contrast to the logarithmic threshold of the domination problem, the power domination problem cannot be approximated within the ratio 2 log 1− n , unless NP ⊆ DTIME(n poly log(n) ).…”

The Robust Set Problem: Parameterized Complexity and Approximation

“…The subject of optimization problems that involve a diffusion process through a network is a large and well-studied topic [15,8,1,12,3]. Such problems share a common idea of selecting an initial subset of vertices to activate in a graph such that, according to a propagation rule, all vertices are activated once the propagation process stops.…”