Because of its flexibility, intuitiveness, and expressivity, the graph edit distance (GED) is one of the most widely used distance measures for labeled graphs. Since exactly computing GED is NP-hard, over the past years, various heuristics have been proposed. They use techniques such as transformations to the linear sum assignment problem with error-correction, local search, and linear programming to approximate GED via upper or lower bounds. In this paper, we provide a systematic overview of the most important heuristics. Moreover, we empirically evaluate all compared heuristics within an integrated implementation.
We study an online model for the maximum k-coverage problem, where given a universe of elements E = {e 1 , e 2 ,. .. , e m }, a collection of subsets of E, S = {S 1 , S 2 ,. .. , S n }, and an integer k, we ask for a subcollection A ⊆ S, such that |A| = k and the number of elements of E covered by A is maximized. In our model, at each step i, a new set S i is revealed, and we have to decide whether we will keep it or discard it. At any time of the process, only k sets can be kept in memory; if at some point the current solution already contains k sets, any inclusion of any new set in the solution must entail the irremediable deletion of one set of the current solution (a set not kept when revealed is irremediably deleted). We first propose an algorithm that improves upon former results for the same model. We next settle a graph-version of the problem, called maximum k-vertex coverage problem. Here also we propose non-trivial improvements of the competitive ratio for natural classes of graphs (mainly regular and bipartite).
We address the max min vertex cover problem, which is the maximization version of the well studied min independent dominating set problem, known to be NP-hard and highly inapproximable in polynomial time. We present tight approximation results for this problem on general graphs, namely a polynomial approximation algorithm which guarantees an n −1/2 approximation ratio, while showing that unless P = NP, the problem is inapproximable within ratio n ε−(1/2) for any strictly positive ε. We also analyze the problem on various restricted classes of graph, on which we show polynomiality or constant-approximability of the problem. Finally, we show that the problem is fixed-parameter tractable with respect to the size of an optimal solution, to tree-width and to the size of a maximum matching.
We address the max min vertex cover problem, which is the maximization version of the well studied min independent dominating set problem, known to be NP-hard and highly inapproximable in polynomial time. We present tight approximation results for this problem on general graphs, namely a polynomial approximation algorithm which guarantees an n −1/2 approximation ratio, while showing that unless P = NP, the problem is inapproximable within ratio n ε−(1/2) for any strictly positive ε. We also analyze the problem on various restricted classes of graph, on which we show polynomiality or constant-approximability of the problem. Finally, we show that the problem is fixed-parameter tractable with respect to the size of an optimal solution, to tree-width and to the size of a maximum matching.
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