Quasimetric Graph Edit Distance as a Compact Quadratic Assignment Problem

Abstract

The graph edit distance (GED) is a widely used distance measure for attributed graphs. It has recently been shown that the problem of computing GED, which is a NPhard optimization problem, can be formulated as a quadratic assignment problem (QAP). This formulation is useful, since it allows to derive well performing approximative heuristics for GED from existing techniques for QAP. In this paper, we focus on the case where the edit costs that underlie GED are quasimetric. This is the case in many applications of GED. We show that, for quasimetric edit costs, it is possible to reduce the size of the corresponding QAP formulation. An empirical evaluation shows that this reduction significantly speeds up the QAP-based approximative heuristics for GED.

Comparing heuristics for graph edit distance computation

“…is the complexity of multiplying two matrices with n rows and n columns. The asymptotically fastest matrix multiplication algorithms achieve ω < 2.38 [40]; the fastest practically useful matrix multiplication algorithm runs in O(n log 2 (7) ) ≈ O(n 2.81 ) time [63].…”

“…Slightly different versions of IPFP that use LSAP instead of LSAPE as a linear model have been presented in [14] and [7]. The main advantage of these versions w.r.t.…”

“…the one presented in this survey is that they are easier to implement: Unlike LSAPE, LSAP is a standard combinatorial optimization problem and solvers are available for all major programming languages. The drawback of the version presented in [14] is that is uses a significantly larger quadratic matrix Q, while the drawback of the version presented in [7] is that it can be used only for quasimetric edit cost functions.…”

Improved local search for graph edit distance

“…The algorithm IPFP [19] is a variant of the Frank-Wolfe algorithm [18] for cases where an integer solution is required. Its adaptation to GED suggested in [15,16,17]…”

“…Given one or several initial node maps, local search based algorithms explore suitably defined neighborhoods to find improved node maps that induce cheaper edit * * Work supported by Region Normandie under project RIN AGAC paths. Several algorithms have been proposed that instantiate this paradigm: REFINE [4] varies the initial node maps via binary swaps; IPFP [15,16,17] computes locally optimal node maps by using a variant of the classical Frank-Wolfe algorithm [18,19]; BP-BEAM [20] uses beam search to improve the initial node maps; and IBP-BEAM [21] further improves BP-BEAM by iteratively running it with different processing orders.…”