A target set selection model is a graph G with threshold function τ : V(G) → N upper-bounded by the vertex degree. For given model, S0 ⊆ if can be partitioned into non-empty subsets S0, S1,....,St such that, for all i∈{1,....,t}, Si contains exactly every v having at least τ(v) neighbors in S0∪⋯∪Si−1. We say that t activation time tτ(S0) of S0. The problem of, finding minimum size has been exte...

An instance of the graph-constrained max-cut (GCMC) problem consists of (i) an undirected graph G = (V,E) and (ii) edge-weights c : ( V 2 ) → R+ on a complete undirected graph. The objective is to find a subset S ⊆ V of vertices satisfying some graph-based constraint in G that maximizes the weight ∑ u∈S,v 6∈S cuv of edges in the cut (S, V \ S). The types of graph constraints we can handle inclu...

We improve the running time of the general algorithmic technique known as Baker’s approach (1994) on H-minor-free graphs from O(n) to O(f(|H |)n) showing that it is fixed-parameter tractable w.r.t. the parameter |H |. The numerous applications include e.g. a 2-approximation for coloring and PTASes for various problems such as dominating set and max-cut, where we obtain similar improvements. On ...

We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2−ǫ)m time, we show that for any ǫ > 0; • INDEPENDENT SET cannot be solved in (2 − ǫ)tw(G)|V (G)| time, • DOMINATING SET cannot be s...

It is shown that for essentially all MAX SNP-hard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max c-Sat, and a number of problems on bounded degree graphs such as Dominating Set and Independent Set, unless W [1] = FPT . Our results a...