Mining for trees in a graph is NP-complete

Mining for trees in a graph is NP-complete Jan V an den Bussc he Univ ersiteit Hasselt The problem of mining patterns in gr aph-structured data has received con- siderable attent ion in recent years, as it has many interesting applicatio ns in such div ers e areas as biology , the life sciences, the W o rld Wide W eb, and so cial sciences. Kuramo chi en Karypis [2 ] ident ified the following pr oblem as funda- men tal to gr aph mining: Problem: Disjoint Occurrences of a Graph P parameter: a s mall gr aph P , c a lled the p attern . input: a la rge gr aph G a nd a natura l num b er k . decide: are there k edge-disjoint subgraphs of G that are all iso mo rphic to P ? Kuramo chi and Karypis relate this pr oblem to Indep endent Set, a well-known NP-complete problem [1]. They do not go a s far, howev er , as actually pr oving that Disjoint Oc currences itself can also b e NP -complete. The purpo se of this short note is to confirm this, even in the very s imple case where P is the tre e on four no de s , co nsisting of a no de with thr ee children. W e denote this tree by T 3 : T 3 = As an immediate conseq ue nc e , the following more general pr oblem, where the pattern is a tree no t fixed in adv ance but pa r t of the input, is NP -complete as well: Problem: Disjoint Occurrences of a T r ee input: a tr ee T , a graph G , and a natural num b er k . decide: are there k edge-disjoint s ubgraphs of G that ar e all isomor phic to T ? Of course, the even more general problem where the pattern is a gra ph is alrea dy well-kno wn to be NP-complete, as it contains the well-kno wn NP-complete Sub- graph Isomor phism problem as the sp ecial case k = 1. W e thus see her e that restricting to tr e e patterns do es not low er the worst-cas e co mplexit y . The NP-completeness of Disjoin t Occurrences of T 3 follows immediately fro m the NP-co mpletenes s of the following pr o blem: (a g raph is called cubic if every no de has degree 3, i.e., has e dg es to precisely 3 other no des) 1 Problem: Independent Set for Cubic Graphs input: a c ubic g raph G and a natural num b er k decide: are ther e k no des in G such that no edge runs b et ween these no des ? As a matter of fact, for cubic g raphs, Indep ent Set is precisely the same problem as Disjoint O c currences of T 3 ! T o see this, let G b e an arbitr a ry cubic gr aph. W e ca ll a subgr aph of G isomorphic to T 3 an o c curr enc e of T 3 in G . W e call the unique node of T 3 that has three c hildren the c ent er of T 3 . Since G is c ubic, we c an ident ify the o ccurr ences of T 3 in G with their centers. Indeed, every o ccurrence has a unique center, and every no de is the cent er of a unique o ccurrence. W e now easily observe: Two distinct o c curr enc es of T 3 , with c en ters x and y , have an e dge in c ommon, if and only if ther e is an e dge b etwe en x and y . Consequently: G c ontains k no des without e dges in b etwe en, if and only if ther e ar e k e dge-disjoi nt o c cu rr enc es of T 3 in G . In other words, Disjoin t O ccurrences of T 3 is precisely the same problem a s Independent Set, re stricted to cubic graphs . T o conclude, we p oint o ut that the whole reaso n for NP-completeness is that we wan t to count disjoint o ccurrenc e s . Indeed, just counting the o ccurr ences of a fixed pattern P in a graph can b e done in p olynomia l time. Unfortu- nately , allowing non-disjoint o ccurr ences has pr oblems of its own, as discuss ed by K uramo chi and Ka rypis. References [1] M.R. Garey and D.S. Johnson. Comp uters and Intr actability: A Guide to the The ory of NP-Completeness . F reeman, 19 79. [2] M. K uramo chi and G. Karypis. Finding fre q uen t pa tterns in a lar ge spars e graph. In Pr o c e e dings 4th SIAM International Confer enc e on Data Mining , 2004. 2