Computing Optimal Assignments in Linear Time for Approximate Graph Matching Finding an optimal assignment between two sets of objects is a fundamental problem arising in many applications, including the matching of u0027bagofwordsu0027 representations in natural language processing and computer vision. Solving the assignment problem typically requires cubic time and its pairwise computation is expensive on large datasets. In this paper, we develop an algorithm which can find an optimal assignment in linear time when the cost function between objects is represented by a tree distance. We employ the method to approximate the edit distance between two graphs by matching their vertices in linear time. To this end, we propose two tree distances, the first of which reflects discrete and structural differences between vertices, and the second of which can be used to compare continuous labels. We verify the effectiveness and efficiency of our methods using synthetic and realworld datasets.

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.