Let G be a graph and τ : V (G) → N ∪ {0} be an assignment of thresholds to the vertices of G. A subset of vertices D is said to be a dynamic monopoly corresponding to (G, τ ) if the vertices of G can be partitioned into subsets D 0 , D 1 , . . . , D k such that D 0 = D and for any i ∈ {0, . . . , k − 1}, each vertex v in D i+1 has at least τ (v) neighbors in D 0 ∪ . . . ∪ D i . Dynamic monopolies are in fact modeling the irreversible spread of influence in social networks. In this paper we first obtain a lower bound for the smallest size of any dynamic monopoly in terms of the average threshold and the order of graph. Also we obtain an upper bound in terms of the minimum vertex cover of graphs. Then we derive the upper bound |G|/2 for the smallest size of any dynamic monopoly when the graph G contains at least one odd vertex, where the threshold of any vertex v is set as ⌈(deg(v) + 1)/2⌉ (i.e. strict majority threshold). This bound improves the best known bound for strict majority threshold. We show that the latter bound can be achieved by a polynomial time algorithm. We also show that α ′ (G) + 1 is an upper bound for the size of strict majority dynamic monopoly, where α ′ (G) stands for the matching number of G. Finally, we obtain a basic upper bound for the smallest size of any dynamic monopoly, in terms of the average threshold and vertex degrees. Using this bound we derive some other upper bounds.Mathematics Subject Classification: 91D30, 05C85, o5C69.
Abstract. Let G be a graph and τ be an assignment of nonnegative integer thresholds to the vertices of G. A subset of vertices, D is said to be a τ -dynamic monopoly, ifDenote the size of smallest τ -dynamic monopoly by dyn τ (G) and the average of thresholds in τ by τ . We show that the values of dyn τ (G) over all assignments τ with the same average threshold is a continuous set of integers. For any positive number t, denote the maximum dyn τ (G) taken over all threshold assignments τ with τ ≤ t, by Ldyn t (G). In fact, Ldyn t (G) shows the worst-case value of a dynamic monopoly when the average threshold is a given number t. We investigate under what conditions on t, there exists an upper bound for Ldyn t (G) of the form c|G|, where c < 1. Next, we show that Ldyn t (G) is coNP-hard for planar graphs but has polynomial-time solution for forests.
Optimization problems consist of either maximizing or minimizing an objective function.
Instead of looking for a maximum solution (resp. minimum solution), one can find a minimum maximal solution (resp. maximum minimal solution).
Such "flipping" of the objective function was done for many classical optimization problems.
For example, ${\rm M{\small INIMUM}}$ ${\rm V{\small ERTEX}}$ ${\rm C{\small OVER}}$ becomes ${\rm M{\small AXIMUM}}$ ${\rm M{\small INIMAL}}$ ${\rm V{\small ERTEX}}$ ${\rm C{\small OVER}}$, ${\rm M{\small AXIMUM}}$ ${\rm I{\small NDEPENDENT}}$ ${\rm S{\small ET}}$ becomes ${\rm M{\small INIMUM}}$ ${\rm M{\small AXIMAL}}$ ${\rm I{\small NDEPENDENT}}$ ${\rm S{\small ET}}$ and so on. In this paper, we propose to study the weighted version of Maximum Minimal Edge Cover called ${\rm U{\small PPER}}$ ${\rm E{\small DGE}}$ ${\rm C{\small OVER}}$, a problem having application in genomic sequence alignment.
It is well-known that ${\rm M{\small INIMUM}}$ ${\rm E{\small DGE}}$ ${\rm C{\small OVER}}$ is polynomial-time solvable and the "flipped" version is NP-hard, but constant approximable. We show that the weighted ${\rm U{\small PPER}}$ ${\rm E{\small DGE}}$ ${\rm C{\small OVER}}$ is much more difficult
than ${\rm U{\small PPER}}$ ${\rm E{\small DGE}}$ ${\rm C{\small OVER}}$ because it is not $O(\frac{1}{n^{1/2-\varepsilon}})$ approximable, nor $O(\frac{1}{\Delta^{1-\varepsilon}})$ in edge-weighted graphs of size $n$ and maximum degree $\Delta$ respectively.
Indeed, we give some hardness of approximation results for some special restricted graph classes such as bipartite graphs, split graphs and $k$-trees.
We counter-balance these negative results by giving some positive approximation results in specific graph classes.
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