Parameterized Optimization in Uncertain Graphs—A Survey and Some Results

Abstract

We present a detailed survey of results and two new results on graphical models of uncertainty and associated optimization problems. We focus on two well-studied models, namely, the Random Failure (RF) model and the Linear Reliability Ordering (LRO) model. We present an FPT algorithm parameterized by the product of treewidth and max-degree for maximizing expected coverage in an uncertain graph under the RF model. We then consider the problem of finding the maximal core in a graph, which is known to be polynomial time solvable. We show that the Probabilistic-Core problem is polynomial time solvable in uncertain graphs under the LRO model. On the other hand, under the RF model, we show that the Probabilistic-Core problem is W[1]-hard for the parameter d, where d is the minimum degree of the core. We then design an FPT algorithm for the parameter treewidth.

Budgeted Dominating Sets in Uncertain Graphs

“…In particular, for trees there exists a linear running time algorithm. PBDS was studied as Max-Exp-Cover-1-RF in the survey paper [46], and given a dynamic programming algorithm on a nice tree decomposition with runtime 2 O(w•∆) n O (1) , where ∆ is the maximum degree of G. The question whether PBDS has a treewidth parameterized FPT algorithm remained unresolved; it is settled in the negative in this work.…”