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).

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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.

We study reoptimization versions of the minimum spanning tree problem. The reoptimization setting can generally be formulated as follows: given an instance of the problem for which we already know some optimal solution, and given some "small" perturbations on this instance, is it possible to compute a new (optimal or at least near-optimal) solution for the modified instance without ex nihilo computation? We focus on two kinds of modifications: node-insertions and node-deletions. When k new nodes are inserted together with their incident edges, we mainly propose a fast strategy with complexity O (kn) which provides a max{2, 3 − (2/(k − 1))}-approximation ratio, in complete metric graphs and another one that is optimal with complexity O (n log n). On the other hand, when k nodes are deleted, we devise a strategy which in O (n) achieves approximation ratio bounded above by 2 |L max |/2 in complete metric graphs, where L max is the longest deleted path and |L max | is the number of its edges. For any of the approximation strategies, we also provide lower bounds on their approximation ratios.

The reoptimization issue studied in this paper can be described as follows: given an instance I of some problem Π, an optimal solution OPT for Π in I and an instance I ′ resulting from a local perturbation of I that consists of insertions or removals of a small number of data, we wish to use OPT in order to solve Π in I ′ , either optimally or by guaranteeing an approximation ratio better than that guaranteed by an ex nihilo computation and with running time better that that needed for such a computation. We use this setting in order to study weighted versions of several representatives of a broad class of problems known in the literature as maximum induced hereditary subgraph problems. The main problems studied are max independent set, max k-colorable subgraph, max P k -free subgraph, max split subgraph and max planar subgraph. We also show, how the techniques presented allow us to handle also bin packing.

This survey presents major results and issues related to the study of NPO problems in dynamic environments, that is, in settings where instances are allowed to undergo some modifications over time. In particular, the survey focuses on two complementary frameworks. The first one is the reoptimization framework, where an instance I that is already solved undergoes some local perturbation. The goal is then to make use of the information provided by the initial solution to compute a new solution. The second framework is probabilistic optimization, where the instance to optimize is not fully known at the time when a solution is to be proposed, but results from a determined Bernoulli process. Then, the goal is to compute a solution with optimal expected value.