Hardness of edge-modification problems

Abstract

For a graph property P consider the following computational problem. Given an input graph G, what is the minimum number of edge modifications (additions and/or deletions) that one has to apply to G in order to turn it into a graph that satisfies P? Namely, what is the edit distance ∆(G, P) of a graph G from satisfying P. Clearly, the computational complexity of such a problem strongly depends on P. For over 30 years this family of computational problems has been studied in several contexts and various algorithms, as well as hardness results, were obtained for specific graph properties. Alon, Shapira and Sudakov studied in [3] the approximability of the computational problem for the family of monotone graph properties, namely properties that are closed under removal of edges and vertices. They describe an efficient algorithm that achieves an o(n 2) additive approximation to ∆(G, P) for any monotone property P, where G is an n-vertex input graph, and show that the problem of achieving an O(n 2−ε) additive approximation is N P-hard for most monotone proeprties. The methods in [3] also provide a polynomial time approximation algorithm which computes ∆(G, P)±o(n 2) for the broader family of hereditary graph properties (which are closed under removal of vertices). In this work we introduce two approaches for showing that improving upon the additive approximation achieved by this algorithm is N Phard for several sub-families of hereditary properties. In addition, we state a conjecture on the hardness of computing the edit distance from being induced H-free for any forbidden graph H.

Correction of Weighted Orthology and Paralogy Relations - Complexity and Algorithmic Results

“…preserving at least half as many relations as the optimal). Using Lemma 6 and results from [1], one can devise a PTAS for MaxES in the case that every gene belongs to a distinct species. Let OP T (R) be the value of an optimal solution on R, and let c be such that OP T (R) = cn 2 .…”

“…The algorithm, however, offers no guarantees in the case of weighted graphs, as there are weighted instances on which it is arbitrarily far from optimal. It is shown in [1] that the minimum edge editing problem cannot be approximated within an "additive" factor of n 2− , for any > 0. Yet, the authors give a class of polynomial time algorithms that are approximable within an additive factor of n 2 , for any > 0.…”

On the computation of edit distance functions

“…Given K, a p-core, there is a unique optimum weight vector, x, with all entries positive, that is a solution to (5).…”

“…So, we may assume that p < 1/2 and K has only black vertices and only white or gray edges. Let x be the weight function that is the optimal solution to (5).…”

Stability‐type results for hereditary properties

“…In a subsequent work [6] we describe how this extremal result may be used to show that it is N P -hard to approximate the edit distance ∆(G, P * Kr(s) ) of a graph G on n vertices within an additive error of n 2−η for any positive η and pair (r, s) (s.t. r + s > 2).…”