Symmetric Graphs with respect to Graph Entropy
aa r X i v : . [ m a t h . C O ] O c t Symmetric Graphs with respect to Graph Entropy
Seyed Saeed Changiz RezaeiDepartment of MathematicsSimon Fraser UniversityBurnaby, BC, Canada [email protected]
Ehsan ChiniforooshanGoogle Inc.Waterloo, ON, Canada [email protected]
October 13, 2018
Abstract
Let F G ( P ) be a functional defined on the set of all the probability distributions on thevertex set of a graph G . We say that G is symmetric with respect to F G ( P ) if the uniformdistribution on V ( G ) maximizes F G ( P ). Using the combinatorial definition of the entropy of agraph in terms of its vertex packing polytope and the relationship between the graph entropyand fractional chromatic number, we characterize all graphs which are symmetric with respectto graph entropy. We show that a graph is symmetric with respect to graph entropy if and onlyif its vertex set can be uniformly covered by its maximum size independent sets. Furthermore,given any strictly positive probability distribution P on the vertex set of a graph G , we showthat P is a maximizer of the entropy of graph G if and only if its vertex set can be uniformlycovered by its maximum weighted independent sets. We also show that the problem of decidingif a graph is symmetric with respect to graph entropy, where the weight of the vertices is givenby probability distribution P , is co-NP-hard. The entropy of a graph is an information theoretic functional which is defined on a graph witha probability distribution on its vertex set. This functional was originally proposed by J. K¨ornerin 1973 to study the minimum number of codewords required for representing an informationsource [7].Let
V P ( G ) be the vertex packing polytope of a given graph G which is the convex hull of thecharacteristic vectors of its independent sets. Let n := | V ( G ) | and P be a probability distributionon V ( G ). Then the entropy of G with respect to the probability distribution P is defined as H ( G, P ) = min a ∈ V P ( G ) X v ∈ V ( G ) p v log(1 /a v ) . J. K¨orner investigated the basic properties of the graph entropy in several papers from 1973 till1992 [7, 8, 10, 9, 12, 13, 11].Let F and G be two graphs on the same vertex set V . Then the union of graphs F and G isthe graph F ∪ G with vertex set V and its edge set is the union of the edge set of graph F and the1dge set of graph G . That is V ( F ∪ G ) = V,E ( F ∪ G ) = E ( F ) ∪ E ( G ) . Perhaps, the most important property of the entropy of a graph is that it is sub-additive withrespect to the union of graphs, that is H ( F ∪ G, P ) ≤ H ( F, P ) + H ( G, P ) . This leads to the application of graph entropy for graph covering problem as well as the problemof perfect hashing.The graph covering problem can be described as follows. Given a graph G and a family ofgraphs G where each graph G i ∈ G has the same vertex set as G , we want to cover the edge set of G with the minimum number of graphs from G . Using the sub-additivity of graph entropy one canobtain lower bounds on this number.Graph entropy was used implicitly in a paper by Fredman and Koml´os for the minimum numberof perfect hash functions of a given range that hash all k -element subsets of a set of a given size [4].Simonyi showed that the maximum of the graph entropy of a given graph over the probabilitydistribution on its vertex set is equal to the logarithm of its fractional chromatic number [15]. Inthis paper, we characterize all strictly positive probability distributions which maximize the entropyof a given graph.Let S be a multi-set of independent sets of a graph G . We say S is uniform over a subsetof vertices W of the vertex set of G if each vertex v ∈ W is covered by a constant number ofindependent sets in S . Then, our main result can be stated as follows. Theorem.
For every graph G and every probability distribution P over V ( G ) , we have H ( G, P ) =lg χ f ( G [ { v ∈ V ( G ) : p v > } ]) if and only if there exists a multi-set of independent sets S such that1. S is uniform over { v ∈ V ( G ) : p v > } , and2. every independent set I ∈ S is a maximum weighted independent sets with respect to P . We say a graph is symmetric with respect to graph entropy if the uniform probability distributionmaximizes its entropy. It is worth noting that the notion of a symmetric graph with respect to afunctional was already defined by G. Greco [6]. Furthermore, S.S. C. Rezaei and C. Godsil studiedsome classes of graphs which are symmetric with respect to graph entropy [1, 2]. A corollary of theabove-mentioned theorem is the following characterization for symmetric graphs.
Theorem.
A graph G is symmetric if and only if χ f ( G ) = nα ( G ) . Finally we consider the complexity of deciding whether a graph is symmetric with respect toits entropy by proving the following theorem.
Theorem.
It is co-NP-hard to decide whether a given graph G is symmetric. Preliminaries
Here we recall some properties of entropy of graphs. The following lemma shows the monotonicityfor graph entropy.
Lemma 2.1. (J. K¨orner).
Let F be a spanning subgraph of a graph G . Then for any probabilitydistribution P we have H ( F, P ) ≤ H ( G, P ) . The notion of substitution is defined as follows. Let F and G be two vertex disjoint graphsand v be a vertex of G . We substitute F for v by deleting v and joining all vertices of F to thosevertices of G which have been adjacent with v . Let G v ← F be the resulting graph.We extend the notion of substitution to distributions. Let P and Q be the probability distri-butions on V ( G ) and V ( F ), respectively. Then the probability distribution P v ← Q on V ( G v ← F ) isgiven by P v ← Q ( x ) = P ( x ) if x ∈ V ( G ) \ { v } and P v ← Q ( x ) = P ( x ) Q ( x ) if x ∈ V ( F ).Now we state the following lemma which was proved in J. K¨orner, et. al. [11]. Lemma 2.2. (J. K¨orner, G. Simonyi, and Zs. Tuza).
Let F and G be two vertex disjoint graphs, v a vertex of G , while P and Q are probability distributions on V ( G ) and V ( F ) , respectively. Thenwe have H ( G v ← F , P v ← Q ) = H ( G, P ) + P ( v ) H ( F, Q ) . Notice that the entropy of an empty graph (a graph with no edges) is always zero (regardlessof the distribution on its vertices).
G. Simonyi established the relationship between the entropy of a graph H ( G, P ) with its fractionalchromatic number χ f ( G ) by showing the following [14, 15].max P H ( G, P ) = log χ f ( G )Here we characterize strictly positive probability distributions which maximize H ( G, P ).First we recall a characterization of uniform independent set covers using fractional chromaticnumber. A b -fold coloring of the vertices of a graph G is an assignment of b -subsets of a set with a elements such that adjacent vertices get disjoint b -subsets. The least a such that G admits a b -foldcoloring is called the b -fold chromatic number of G and is denoted by χ b ( G ). Theorem 3.1. ([16]) For every graph G and integer bχ f ( G ) ≤ χ b ( G ) b . Furthormore, there exists an integer that realizes the equality.
As a corollary to the above theorem we have
Corollary 3.2.
Let S be a multi-set of independent sets of G . Then each element of S induces amaximum independent set of G with size α ( G ) and S is a uniform cover the vertices of G if andonly if χ f ( G ) = nα ( G ) . roof. First, assuming χ f = nα , we prove that there exists a uniform independent set cover whoseelements are maximum independent sets. From Theorem 3.1, there exists a b -fold coloring suchthat χ b ( G ) = a and we have χ f ( G ) = nα = ab . (1)Now we construct a uniform maximum independent cover S as follows. Let A be the set ofcolors of size a and B be the function that assigns a b -subset of A to every vertex. For every x ∈ A , I x = { v ∈ V ( G ) : x ∈ B ( v ) } is an independent set. So, S = { I x : x ∈ A } is a uniform independentset cover of size a . Equation (1) tells us that the average size of independent sets in S is α and sothey all must be maximum independent sets.Conversely, assume that G admits a uniform maximum independent set cover S such that eachvertex v ∈ V ( G ) lies in exactly b elements of S , and so |S| b = nα . Then, B ( v ) = { S ∈ S : v ∈ S } . isa b -fold coloring. Let b ′ be an integer such that χ f ( G ) = χ b ′ ( G ) b ′ and B ′ be the b ′ -fold coloring thatachieves this. Then nα ≤ χ b ′ ( G ) b ′ = χ f ( G ) ≤ χ b ( G ) b ≤ |S| b = nα . The left-most inequality comes from the fact that αχ b ′ ( G ) is an over-estimation of P v ∈ V ( G ) | B ′ ( v ) | = nb ′ .We prove our main result in two steps. Theorem 3.3.
For every graph G and every probability distribution P over V ( G ) , if H ( G, P ) =lg χ f ( G [ { v ∈ V ( G ) | p v > } ]) , then there exists a multi-set of independent sets S such that1. S is uniform over { v ∈ V ( G ) : p v > } , and2. every independent set I ∈ S is a maximum weighted independent sets with respect to P .Proof. Consider a fractional coloring ( I , w ) of G , where I is the family of independent sets of G and w : I → Q + is a weight function such that X I ∈I w I = χ f ( G ) , (2)and X I ∈I ,v ∈ I w I ≥ , (3)for all v ∈ V ( G ). We define x ∗ = P I ∈I w I χ f ( G ) · I , where I is the characteristic vector of I . Clearly, x ∗ ∈ VP( G ) due to (2), and x ∗ v ≥ χ f ( G ) , for all v ∈ V ( G ), due to (3).We turn the set I to a multi-set S by setting the multiplicity of any independent set I to r · w I ,where r is an integer such that r · w I ∈ N for all I ∈ I . If for some v ∈ D P = { v ∈ V ( G ) : p v > } , x ∗ v > χ f ( G ) , then H ( G, P ) ≤ − P v ∈ V ( G ) p v lg x ∗ v < lg χ f ( G ), which is a contradiction. So, every v ∈ D P is in exactly r · x ∗ v · χ f ( G ) = r independent sets of S , which means S is uniform over D P .We also claim that P I = α P for all I ∈ I , where α P is the weight of maximum weightedindependent set of G with respect to P . Indeed, if there exists an I ∈ I such that p I < α p ,then we define x ǫ = x ∗ − ǫ I + ǫ M , where M is the characteristic vector of an arbitrary maximumweighted independent set. Note that x ǫ ∈ VP( G ) for all 0 ≤ ǫ ≤ w I .4ow consider d ( ǫ ) = X v ∈ I \ M p v lg x ǫv x ∗ v + X v ∈ M \ I p v lg x ǫv x ∗ v = lg(1 − ǫχ f ( G )) X v ∈ I \ M p v + lg(1 + ǫχ f ( G )) X v ∈ M \ I p v . We have d (0) = 0, d is differentiable at zero, and d ′ (0) = ( P v ∈ M \ I p v − P v ∈ I \ M p v ) χ f ( G ) > ǫ + such that d ( ǫ ) is positive for all ǫ in (0 , ǫ + ). Then, foran arbitrary ǫ ∈ (0 , min( w I , ǫ + )), H ( G, P ) ≤ − P v ∈ V ( G ) p v lg x ǫv = − P v ∈ V ( G ) p v lg x ∗ v − d ( ǫ ) < − P v ∈ V ( G ) p v lg x ∗ v = lg χ f ( G ), which is a contradiction.It is easy to see that the converse of Theorem 3.3 holds for uniform distributions. We call agraph G symmetric if H ( G, U ) ≥ H ( G, P ) for all probability distributions P on V ( G ), where U isthe uniform distribution. Corollary 3.4.
A graph G is symmetric if and only if χ f ( G ) = nα ( G ) .Proof. If χ f ( G ) = nα , then for every x ∈ VP( G ) we have X v ∈ V ( G ) x v ≤ α ⇒ Y v ∈ V ( G ) x v ≤ (cid:16) αn (cid:17) n ⇒ − n X v ∈ V ( G ) lg x v ≥ − n lg (cid:16) αn (cid:17) n = − lg αn = lgχ f ( G ) . This means H ( G, u ) = lg χ f ( G ), and so G is symmetric.On the other hand, if G is symmetric, then, according to Theorem 3.3, there exists a uniformmaximum independent set cover. Thus from Theorem 3.2, we have χ f ( G ) = nα .Now we prove the converse of Theorem 3 . Theorem 3.5.
Let G be a graph with probability distribution P on its vertex set. Suppose thatthere exists a multi-set of independent sets S such that1. S is uniform over { v ∈ V ( G ) : p v > } , and2. every independent set I ∈ S is a maximum weighted independent sets with respect to P .Then H ( G, P ) = log χ f ( G [ { v ∈ V ( G ) : p v > } ]) .Proof. First we assume that the probability of every vertex v ∈ V ( G ), i.e., p v is equal to n v m forsome n v and m ∈ N . We then construct the graph G ′ by blowing each vertex v up n v times andmaking the corresponding vertices adjacent to the neighbours of v . We consider each set of verticessubstituted for each vertex v as an independent set F v of size n v with uniform distribution n v . Then G ′ is a probabilistic graph with uniform distribution m on its vertex set, and repeated applicationof Lemma 2.2 leads to H ( G ′ , m ) = H ( G, P ) + X v n v m H ( F v , n v ) . Noting that F v is an independent set, we have H ( F v , n v ) = 0 , ∀ v ∈ V ( G ) , H ( G ′ , m ) = H ( G, P ) . (4)Note that the vertex set of G is uniformly covered by maximum weighted independent set withrespect to p . Therefore, due to the construction of G ′ , graph G ′ is uniformly covered by its maximumindependent sets. Thus using Theorem 3.4 and equation (4), we have H ( G ′ , m ) = log χ f ( G ′ ) = H ( G, P ) ≤ log χ f ( G ) . (5)Noting that there is a homomorphism from G to G ′ , we get χ f ( G ) ≤ χ f ( G ′ ) . This along with (5) implies that χ f ( G ) = χ f ( G ′ ), and hence, probability distribution p over thevertex set of G maximizes H ( G, P ).Now, suppose that P is a real probability distribution over the vertex set of G . Now we showthat graph entropy H ( G, P ) is a continuous function of P . Let P be a strictly positive probabilitydistribution over V ( G ). Then for every ǫ > δ = 12 min v ∈ V p v × min (cid:18) , ǫ | V ( G ) | H ( G, P ) (cid:19) . Therefore, we have k P − P ′ k < δ → k H ( G, P ′ ) − H ( G, P ) k < ǫ. First we show H ( G, P ′ ) < H ( G, P ) + ǫ . Let a ∗ ∈ V P ( G ) achieves H ( G, P ), that is H ( G, P ) = X v p v log 1 a ∗ v . Thus (cid:16) min v p v (cid:17) × X v log 1 a ∗ v ≤ H ( G, P ) ⇒ X log 1 a ∗ v ≤ H ( G, P )min v p v ⇒ log 1 a ∗ v ≤ H ( G, P )min v p v , ∀ v ∈ V. (6)On the other hand, setting δ v = p ′ v − p v , we have H ( G, P ′ ) = min a ∈ V P ( G ) X v ( p v + δ v ) log 1 a v ≤ X v ( p v + δ v ) log 1 a ∗ v = H ( G, P ) + X v δ v log 1 a ∗ v ≤ H ( G, P ) + | V ( G ) | × δ × H ( G, P )min v p v < H ( G, P ) + ǫ . (7)Now we show that H ( G, P ′ ) > H ( G, P ) − ǫ . Let b ∗ ∈ V P ( G ) achieves H ( G, P ′ ), that is H ( G, P ′ ) = X v p ′ v log 1 b ∗ v . (8)6f H ( G, P ) ≤ H ( G ′ , P ), we are done. Thus we may assume H ( G, P ) > H ( G, P ′ ). Then using thevalue for δ defined above we havelog (cid:18) b ∗ v (cid:19) ≤ H ( G, P ′ )min v p ′ v ≤ H ( G, P )min v p ′ v , ∀ v ∈ V ( G ) . Now using the above equation and equation (8), we get H ( G, P ′ ) = X v p v log 1 b ∗ v + X v log 1 b ∗ v ≥ H ( G, P ) + X −| δ i | log 1 b ∗ v ≥ H ( G, P ) − | V ( G ) | × × H ( G, P )min v p v × δ ≥ H ( G, P ) − ǫ. (9)Thus H ( G, P ) is a continuous function. Now note that there exists a sequence of rational probabilitydistribution P k which tends to P as k → ∞ . Consequently, there exists a sequence of graphs G ′ k with the corresponding sequence of uniform probability distributions m k constructed as explainedabove. Then, noting that H ( G, P ) is a continuous function with respect to P and using (5), wehave lim k →∞ H ( G ′ k , m k ) = lim k →∞ log χ f ( G ′ k ) = lim k →∞ H ( G, P k ) = H ( G, P ) ≤ log χ f ( G ) . (10)Since there is a homomorphism from G to G ′ k for every k , we get χ f ( G ) ≤ lim k →∞ χ f ( G ′ k ) . (11)Therefore, (10) and (11) imply that probability distribution P over the vertex set of G maximizes H ( G, P ). In this section, we discuss the complexity of computing graph entropy by proving the followingtheorem.
Theorem 3.6.
It is co-NP-hard to decide whether a given graph G is symmetric.Proof. We use a reduction from the maximum independent set problem. Assume we are given agraph F and an integer k , and let A and B be two disjoint sets of size k − V ( F ).Then, graph F has an independent set of size at least k if and only if the graph G , defined below,is not symmetric: V ( G ) = A ∪ V ( F ) × B,E ( G ) = {{ a, ( v, b )) } : a ∈ A, v ∈ V ( F ) , b ∈ B } ∪{{ ( v, b ) , ( v ′ , b ′ ) } : v, v ′ ∈ V ( F ) , b, b ′ ∈ B, v = v ′ , b = b ′ } ∪{{ ( v, b ) , ( v ′ , b ) } : ( v, v ′ ) ∈ E ( F ) , b ∈ B }
7o see this, it is enough to note that α ( G ) = max { α ( F ) , k − } and G has a uniform independent setcover whose independents sets are all of size k −
1. If α ( F ) ≤ k −
1, then α ( G ) = k − G the vertex set of G is covered uniformly by its maximum size independent sets.Therefore, using Theorem 3.4, graph G is symmetric with respect to graph entropy. Conversely, if α ( F ) ≥ k , then since the vertices of G in A are adjacent to all vertices in V ( G ) \ A , the vertex setof G is not covered uniformly by its maximum size independent sets. Consequently, from Theorem3.4, graph G is not symmetric with respect to graph entropy. References [1] Seyed Saeed Changiz Rezaei, “Entropy and Graphs”, Master of Math Thesis, Combinatoricsand Optimization Department of the University of Waterloo, 2013.[2] Seyed Saeed Changiz Rezaei, Chris Godsil, “ Entropy of Symmetric Graphs”, available athttp://arxiv.org/abs/1311.6561v1.[3] I. Csisz´ar, J. K¨orner, L. Lov´as, K. Marton, and G. Simonyi, “Entropy Splitting for antiblock-ing corners and perfect graphs,”
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