Pivot Selection for Median String Problem The Median String Problem is W1Hard under the Levenshtein distance, thus, approximation heuristics are used. Perturbationbased heuristics have been proved to be very competitive as regards the ratio approximation accuracyconvergence speed. However, the computational burden increase with the size of the set. In this paper, we explore the idea of reducing the size of the problem by selecting a subset of representative elements, i.e. pivots, that are used to compute the approximate median instead of the whole set. We aim to reduce the computation time through a reduction of the problem size while achieving similar approximation accuracy. We explain how we find those pivots and how to compute the median string from them. Results on commonly used test data suggest that our approach can reduce the computational requirements (measured in computed edit distances) by 8% with approximation accuracy as good as the state of the art heuristic.

Complexity Issues of String to Graph Approximate Matching. The problem of matching a query string to a directed graph, whose vertices are labeled by strings, has application in different fields, from data mining to computational biology. Several variants of the problem have been considered, depending on the fact that the match is exact or approximate and, in this latter case, which edit operations are considered and where are allowed. In this paper we present results on the complexity of the approximate matching problem, where edit operations are symbol substitutions and are allowed only on the graph labels or both on the graph labels and the query string. We introduce a variant of the problem that asks whether there exists a path in a graph that represents a query string with any number of edit operations and we show that is is NPcomplete, even when labels have length one and in the case the alphabet is binary. Moreover, when it is parameterized by the length of the input string and graph labels have length one, we show that the problem is fixedparameter tractable and it is unlikely to admit a polynomial kernel. The NPcompleteness of this problem leads to the inapproximability (within any factor) of the approximate matching when edit operations are allowed only on the graph labels. Moreover, we show that the variants of approximate string matching to graph we consider are not fixedparameter tractable, when the parameter is the number of edit operations, even for graphs that have distance one from a DAG. The reduction for this latter result allows us to prove the inapproximability of the variant where edit operations can be applied both on the query string and on graph labels.