Clique and Constraint Models for Maximum Common (Connected) Subgraph Problems

Abstract

Abstract. The maximum common subgraph problem is to find the largest subgraph common to two given graphs. This problem can be solved either by constraint-based search, or by reduction to the maximum clique problem. We evaluate these two models using modern algorithms, and see that the best choice depends mainly upon whether the graphs have labelled edges. We also study a variant of this problem where the subgraph is required to be connected. We introduce a filtering algorithm for this property and show that it may be combined with a restricted branching technique for the constraint-based approach. We show how to implement a similar branching technique in clique-inspired algorithms. Finally, we experimentally compare approaches for the connected version, and see again that the best choice depends on whether graphs have labels.

On Maximum Weight Clique Algorithms, and How They Are Evaluated

“…Graphs of the latter kind may very well be dense. This is true even when the question being answered is regarding a sparse graph: for example, when solving the maximum common subgraph problem via reduction to clique, the encoding of two sparse graphs gives a dense graph [39]. Similarly, microstructure graphs for non-trivial problems are usually reasonably dense.…”

“…Meanwhile, the randomly generated instances are useful because they provide a challenge: although being able to solve crafted hard instances is not the primary goal of developing clique algorithms, working on these instances has led to better solvers. For example, Depolli et al [18] use a maximum clique algorithm for new instances from a biochemistry application, and note that although their instances are reasonably easy for a modern algorithm, they are challenging for earlier algorithms that predate experiments on these instances; a similar conclusion holds for clique-based solvers for maximum common subgraph problems [39]. For the weighted problem, standard practice is to take these same instances, and to follow a convention usually ascribed to Pullan [45]:…”

“…The reduction of constraint satisfaction problems to finding a clique in a corresponding microstructure graph is primarily studied for its theoretical properties [13][14][15][16]28]. For problems with a special objective function, a reduction to the maximum clique problem which preserves the objective value is possible-indeed, recent experimental work shows that this encoding, rather than conventional constraint programming, is the best practical approach for solving the maximum common subgraph problem when vertex or edge labels are present [39]. To tackle other problems this way (such as graph edit distance problems with complex scoring schemes), we would like to be able to relax the restrictions on the objective function, by reducing to the maximum weight clique problem instead.…”

Observations from Parallelising Three Maximum Common (Connected) Subgraph Algorithms

“…A clique in a graph is a subgraph where every vertex is adjacent to every other. There is a well-known reduction from the maximum common subgraph problem to the problem of finding a maximum clique in an association graph [21,33,25]; this reduction resembles the microstructure encoding [17] of the constraint programming approach described below. When combined with a modern maximum clique solver [35], this is the current best approach for solving the problem on labelled graphs [25].…”

Enumerating common molecular substructures

“…In this setting, common molecular substructures correspond to common connected induced subgraphs, which gives rise to the computational problem of finding a maximum common connected induced subgraph (MCCIS) of two input molecular graphs. Many heuristics (Rahman et al, 2009;Englert and Kovács, 2015) and exact approaches (McCreesh et al, 2016;Droschinsky et al, 2017) have been proposed for MCCIS. A frequent variant of MCCIS is to enumerate all maximal common connected induced subgraphs, which we refer to as MCCIS-E. Koch (2001) proposed an exact algorithm for MCCIS-E.…”