Fast linear sum assignment with error-correction and no cost constraints

Abstract

We propose an algorithm that efficiently solves the linear sum assignment problem with error-correction and no cost constraints. This problem is encountered for instance in the approximation of the graph edit distance. The fastest currently available solvers for the linear sum assignment problem require the pairwise costs to respect the triangle inequality. Our algorithm is as fast as these algorithms, but manages to drop the cost constraint. The main technical ingredient of our algorithm is a cost-dependent factorization of the node substitutions.

Ring Based Approximation of Graph Edit Distance

“…For constructing a set of node operations that induces a cheap edit path, a suitably defined instance of LSAPE is solved. LSAPE is defined as follows [5]:…”

Comparing heuristics for graph edit distance computation

“…Definition 6 (LSAPE -first definition [15]) Given a matrix C ∈ R (n+1)×(m+1) with c n+1,m+1 = 0, the linear sum assignment problem with error-correction (LSAPE) consists in the task to minimize C(π) := ∑ (i,k)∈π c i,k over all relations . We write π(i) = k if (i, k) ∈ π and i = n + 1; and π −1 (k) = i if (i, k) ∈ π and k = m + 1.…”

“…Given a matrix C ∈ R (n+1)×(m+1) , an optimal solution π ∈ Π (C) can be computed in O(min{n, m} 2 max{n, m}) time [15], using variants of the famous Hungarian Algorithm [38,46]. Once one optimal solution has been found, for each s ∈ [|Π (C)|], a solution set Π s (C) ⊆ Π (C) of size s can be enumerated in O(nm √ n + m + s log (n + m)) time [64,65].…”

GEDLIB: A C++ Library for Graph Edit Distance Computation

“…The task is to compute a mapping π from rows to columns, such that each row except for n + 1 and each column expect for m + 1 is covered exactly once and C(π) := (i,k)∈π c i,k is minimized. LSAPE can be solved optimally in cubic time [10]; in GEDLIB, we use the LSAPE toolbox [8] for solving LSAPE.…”