A conjecture on independent sets and graph covers

A conjectur e on independ en t sets and graph co vers Y usuke W atanab e a a The Institute of Statistic al Mathematics 10-3 Midori-cho, T achikawa , T okyo 190-85 62, Jap an Phone: +81-(0)50-5533-8500 Abstract In this a rticle, I present a s imple conjecture on the num b er of indep endent sets on gra ph cov ers. The co njecture implies that the pa rtition function of a bina r y pairwise a ttractive mo del is grea ter than that of the Bethe a pproximation. Key wor ds: graph cover, independent set, Bethe approximation, 1. T erminologies Throughout this article, G = ( V , E ) is a finite graph with vertices V and undirected edges E . F or each undirected edge of G, w e ma ke a pair of opp ositely directed e dges, which form a set of directed edges ~ E . Thus, | ~ E | = 2 | E | . An M -c over of a graph G is its M -fold co vering space 1 . All M -cov ers are explicitly constr ucted using per mutation voltage assig nment as fo llows [1]. A p ermutation voltage assignment of G is a map α : ~ E → S M s.t. α ( u → v ) = α ( v → u ) − 1 ∀ uv ∈ E , (1) where S M is the p ermutation g roup o f { 1 , . . . , M } . Then an M -cov er ˜ G = ( ˜ V , ˜ E ) of G is given by ˜ V := V × { 1 , . . . , M } and ( v , k )( u, l ) ∈ ˜ E ⇔ uv ∈ E a nd l = α ( v → u )( k ) . (2) If an M -cover is M co pie s o f G then it is ca lled trivial M -c over and deno ted by G ⊕ M . T his is o btained by identit y p ermutations. The natur al pr oje ction , π , from a cov er ˜ G to G is obtained by for g etting the “lay er num b er” . Tha t is, π : ˜ V → V is g iven b y π ( u, i ) = u and π : ˜ E → E is g iven by π (( v , k )( u, l )) = v u . An indepe ndent set I of a g r aph G is a subs e t o f V such that no ne of the elements in I are a dja c ent in G . F ormally , I is an indep endent set iff u, v ∈ I ⇒ uv 6∈ E . The m ultiv ar ia te indep endent set p olynomial of G is defined by p ( G ) := X I : indep endent set Y v ∈ I x v , (3) with indeterminates x v ( v ∈ V ). URL: watay@ism.ac .jp (Y usuke W atanabe) 1 In terpret graphs as topological spaces. Novemb er 23, 2018 2. The conjecture W e extend the definition of the pro jection map π ov e r the multiv ar iate poly- nomial ring. First, let us define a map Π for ea ch indeterminate b y Π( x v ) = x Π( v ) , where v ∈ ˜ V . Then this is uniquely ex tended to the po lynomial ring as a ring ho momorphism; for example Π( x v + x v ′ ) = Π( x v ) + Π( x v ′ ) and Π( x v x v ′ ) = Π( x v )Π( x v ′ ). Conjecture 1. 2 F or any bip artite gr aph G and it s M - c over ˜ G , we c onje ct u r e the fol lowing r elation: Π( p ( ˜ G ))  p ( G ) M , (4) wher e Π is define d as ab ove and the symb ol  me ans the ine qualities for al l c o efficients of monomials. W e can int erpret the conjecture mo r e explicitly a s follows. F or a subs e t U of V , define I ( ˜ G, U ) := { I ⊂ ˜ V | I is indep endent set , π ( I ) = U } , where π ( I ) is the image of I . Since p ( G ) M = Π( p ( G ⊕ M )), the conjecture is equiv alent to the following statement : |I ( ˜ G, U ) | ≤ |I ( G ⊕ M , U ) | for all U ⊂ V . (5) Example 1. Let G b e a cycle graph of length four and let ˜ G be its 3-cover that is iso morphic to the cycle of length tw e lve. Then, p ( G ) = 1 + x 1 + x 2 + x 3 + x 4 + x 1 x 3 + x 2 x 4 , (6) p ( ˜ G ) = 1 + 4 X v =1 3 X m =1 x ( v, m ) + . . . , (7) Π( p ( ˜ G )) = 1 + 3( x 1 + x 2 + x 3 + x 4 ) + . . . . (8) It takes time a nd effor t to chec k the conjectur e , how e ver, it is true in this ca se. Remark. The abov e conjecture is claimed for the pair (bi parti te gra ph, indepe ndent set) . I also co njecture a nalogo us prop er ties for (bipa rtite graph, matchi ng) , (graph with even number of vertices , pe rfect matchi ng) and ( graph , Eulerian s et 3 ) . 3. Implication o f the conjecture The conjecture o riginates from the theory of the Bethe a pproximation. The p artition function of a binary pairw is e mo del on a gra ph G is Z ( G ; J , h ) := X s ∈{ 0 , 1 } V exp( X uv ∈ E J uv s u s v + X v ∈ V h v s v ) , (9) 2 I ha ve che ck ed the conjecture f or m an y examples by computer. 3 A subset of edges i s Eulerian if it induces a s ubgraph that only has vertices of degree tw o and zero. 2 where the weigh ts ( J , h ) ar e called int er actions . 4 It is called attr active if J vu ≥ 0 for all v u ∈ E . The Bethe p artition function 5 Z B is defined by [2 ] Z B : = exp  − min q F B ( q )  (10) = lim sup M →∞ < Z ( ˜ G ) > 1 / M , (11) where F B is the Bethe free ener gy and < · > is the mean with r esp ect to the M ! | E | cov er s. (Details a re omitted. See [2].) Theorem 1. If Conje ctur e 1 holds, then Z ≥ Z B (12) holds for any binary p airwise attra ctive mo dels. 6 Pr o of. F r om (1 1), the a ssertion of the theorem is proved if we show that Z ( G ) M ≥ Z ( ˜ G ) (13) for any M -cover ˜ G of G . In the following, we see that the partition function can b e written by the indep endent set p olyno mial and thus the above inequa lity holds under the a ssumption of Conjecture 1. Z ( G ) = X s ∈{ 0 , 1 } V exp( X uv ∈ E J uv s u s v + X v ∈ V h v s v ) (14) = X s ∈{ 0 , 1 } V Y uv ∈ E (1 + A uv s u s v ) Y v ∈ V exp( h v s v ) (15) = X S ⊂ E  Y uv ′ ∈ S A uv ′  Y v ∈ V ( X s v =0 , 1 s d v ( S ) v exp( h v s v )) (16) = Y v ∈ V exp( h v ) X S ⊂ E ,U ⊂ V S and U are not “ adjacent ” Y uv ∈ S A uv Y v ∈ U B v (17) = Y v ∈ V exp( h v ) p ( G ′ ; A , B ) , (18) where d v ( S ) is the num b er of edg e s in S c o nnecting to v , A uv = e J uv − 1 , B v = e − h v and G ′ is a bipar tite gra ph obtained by adding a new vertex on each edge of G . 4 In the following, for a co ver ˜ G of G , we think that interac tions are naturally induced fr om G . 5 This is computed fr om the absolute mi nimum of the Bethe free energy; other Bethe appro ximations of the partition function c orresp onding to local m inima are sm aller than Z B . 6 In a quite l imited s ituation, t he inequalit y is pro ved in [3]. 3 Remark. The Bethe appr oximation can also be a pplied to the computation of the p ermanent of non-neg a tive matrices. F rom the combinatorial viewp oint, this problem is related to the (w eighted) p er fect matc hing problem o n complete bipartite gra phs. V ontobel a nalyzed this pr oblem and p ose a conjecture analo- gous to Eq.(5) [4]. This co njectur e implies the inequality betw een the per manent and its Bethe approximation, Z ≥ Z B , given his fo rmula Eq.(11). T he state- men t Z ≥ Z B is, how ever, directly proved by Gurvits, gener alizing Schrijv er fs per manental inequality [5]. References [1] W. Gros s Thomas and L. Jona tha n. Gener ating all graph cov erings by per- m utation voltage assignments. Discr ete Mathematics , 18 (3):273– 283, 1977. [2] P .O. V ontobel. Counting in gr aph cov er s: a co mbinatorial characterization of the b ethe entropy function. submitte d to IEEE T r ans. Inf. The ory,availa ble at arXiv:101 2.0065 v1 . [3] E.B. Sudderth, M.J. W a inwright, and A.S. Willsky . Lo op series and b ethe v ariationa l b ounds in attractiv e gr a phical mo dels. A dvanc es in neur al infor- mation pr o c essing systems , 20:142 5–14 32, 2 008. [4] P .O. V o ntob el. The b ethe p ermanent of a non-neg ative matrix. available at arXiv:110 7.4196v 1 . [5] L. Gurvits. Unharnessing the power of sch rijver’s p er manental inequality . availab le at arXiv:11 06.2844 v2 . 4