Fast Approximate Quadratic Programming for Graph Matching

Abstract

Quadratic assignment problems arise in a wide variety of domains, spanning operations research, graph theory, computer vision, and neuroscience, to name a few. The graph matching problem is a special case of the quadratic assignment problem, and graph matching is increasingly important as graph-valued data is becoming more prominent. With the aim of efficiently and accurately matching the large graphs common in big data, we present our graph matching algorithm, the Fast Approximate Quadratic assignment algorithm. We empirically demonstrate that our algorithm is faster and achieves a lower objective value on over 80% of the QAPLIB benchmark library, compared with the previous state-of-the-art. Applying our algorithm to our motivating example, matching C. elegans connectomes (brain-graphs), we find that it efficiently achieves performance.

A Graduated Non-Convexity Relaxation for Large Scale Seriation

“…Furthermore, recent work [30,24] on solving general QAP problems suggests that convex relaxations do not always outperform indefinite relaxations. To this direction, Lim and Wright [22] present a new framework for approximating general QAP problems formulated in terms of sorting networks, and use a continuation procedure [26,23], where they start solving a convex relaxation of the problem and then gradually convert it to a concave one, yielding a local optimum to the original discrete problem.…”

“…• Fast approximate QAP (FAQ) [30]: approximate QAP-solver based on relaxation to the Birkhoff polytope using the Frank-Wolfe algorithm.…”

“…For small-scale problems, exact solutions such as branch-and-bound can be applied [5]. Besides generic approximation algorithms, such as simulated annealing, there has been recent interest in relaxation-based approaches for solving QAPs in the context of graph matching [30,24]. This is particularly alluring for the 2-SUM case as it corresponds to the minimization of a convex objective function, and recent works [8,21] have shown how the relaxed 2-SUM problem can be exactly solved using interior-point methods, in either a matrix or vector-based formulation.…”

Approximate Graph Edit Distance by Several Local Searches in Parallel

“…2 have been investigated in . Initially developed for the graph matching problem (Zaslavskiy et al, 2009;Leordeanu et al, 2009;Liu and Qiao, 2014;Vogelstein et al, 2015), they relax the set of solutions to the set of doubly-stochastic matrices (values in [0, 1]). Then the integer projected fixed point (IPFP) procedure (Leordeanu et al, 2009), or the fast approximate quadratic programming procedure (Vogelstein et al, 2015), proceed as follows.…”

On spectral properties for graph matching and graph isomorphism problems

“…As such, there are several and diverse techniques addressing the graph matching problem, including spectral methods [8] and relaxations techniques [4,9,10].…”