maximum common induced subgraph

In graph theory and theoretical computer science, a maximum common induced subgraph of two graphs G and H is a graph that is an induced subgraph of both G and H,

and that has as many vertices as possible.

Finding this graph is NP-hard.

In the associated decision problem, the input is two graphs G and H and a number k. The problem is to decide whether G and H have a common induced subgraph with at least k vertices. This problem is NP-complete.{{citation|author = Michael R. Garey and David S. Johnson | year = 1979 | title = Computers and Intractability: A Guide to the Theory of NP-Completeness | publisher = W.H. Freeman | isbn = 0-7167-1045-5}} A1.4: GT48, pg.202. It is a generalization of the induced subgraph isomorphism problem, which arises when k equals the number of vertices in the smaller of G and H, so that this entire graph must appear as an induced subgraph of the other graph.

Based on hardness of approximation results for the maximum independent set problem, the maximum common induced subgraph problem is also hard to approximate.{{citation

| last = Kann | first = Viggo

| contribution = On the approximability of the maximum common subgraph problem

| doi = 10.1007/3-540-55210-3_198

| pages = 375–388

| publisher = Springer Science $\mathplus$ Business Media

| series = Lecture Notes in Computer Science

| title = STACS 92: 9th Annual Symposium on Theoretical Aspects of Computer Science Cachan, France, February 13–15, 1992, Proceedings

| volume = 577

| isbn = 978-3-540-55210-9

| year = 1992}}. This implies that, unless P = NP, there is no approximation algorithm that, in polynomial time on $n$-vertex graphs, always finds a solution within a factor of $n^\{1-\backslash epsilon\}$ of optimal, for any $\backslash epsilon\; >\; 0$.{{citation |first=D. |last=Zuckerman |title=Proc. 38th ACM Symp. Theory of Computing |pages=681–690 |year=2006 |doi=10.1145/1132516.1132612 |id={{ECCC|2005|05|100}}|chapter=Linear degree extractors and the inapproximability of max clique and chromatic number |isbn=1-59593-134-1|s2cid=5713815 }}.

One possible solution for this problem is to build a modular product graph of G and H.

In this graph, the largest clique corresponds to a maximum common induced subgraph of G and H. Therefore, algorithms for finding maximum cliques can be used to find the maximum common induced subgraph.{{citation

| last1 = Barrow | first1 = H.

| last2 = Burstall | first2 = R. | author2-link = Rod Burstall

| doi = 10.1016/0020-0190(76)90049-1

| issue = 4

| journal = Information Processing Letters

| pages = 83–84

| title = Subgraph isomorphism, matching relational structures and maximal cliques

| volume = 4

| year = 1976}}. Moreover, a modified maximum-clique algorithm can be used to find a maximum common connected subgraph.{{citation

| last1=McCreesh | first1=Ciaran

| last2=Ndiaye | first2=Samba Ndojh

| last3=Prosser | first3=Patrick

| last4=Solnon | first4=Christine

| title=Principles and Practice of Constraint Programming - 22nd International Conference, CP 2016, Toulouse, France, September 5-9, 2016, Proceedings

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

| series=Lecture Notes in Computer Science

| publisher=Springer International Publishing

| year=2016

| volume=9892

| pages=350–368

| doi=10.1007/978-3-319-44953-1_23| isbn=978-3-319-44952-4

| s2cid=215812381

| chapter-url=http://eprints.gla.ac.uk/120631/7/120631.pdf

}}

The McSplit algorithm (along with its McSplit↓ variant) is a forward checking algorithm that does not use the clique encoding, but uses a compact data structure to keep track of the vertices in graph H to which each vertex in graph G may be mapped. Both versions of the McSplit algorithm outperform the clique encoding for many graph classes.{{citation

| last1=McCreesh | first1=Ciaran

| last2=Prosser | first2=Patrick

| last3=Trimble | first3=James

| title=Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, {IJCAI} 2017, Melbourne, Australia, August 19-25, 2017

| chapter=A Partitioning Algorithm for Maximum Common Subgraph Problems

| publisher=ijcai.org

| year=2017

| pages=712–719

| doi=10.24963/ijcai.2017/99| isbn=9780999241103

| doi-access=free

}} A more efficient implementation of McSplit is McSplitDAL+PR, which combines a Reinforcement Learning agent with some heuristic scores computed with the PageRank algorithm.{{Cite book |last1=Calabrese |first1=Andrea |last2=Cardone |first2=Lorenzo |last3=Licata |first3=Salvatore |last4=Porro |first4=Marco |last5=Quer |first5=Stefano |date=2023 |title=A Web Scraping Algorithm to Improve the Computation of the Maximum Common Subgraph |publisher=SCITEPRESS - Science and Technology Publications |pages=197–206 |doi=10.5220/0012130800003538 |isbn=978-989-758-665-1|doi-access=free }}