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The Quasi-Randomness of Hypergraph Cut Properties

The Quasi-Randomness of Hypergraph Cut Properties. Asaf Shapira Georgia Institute of Technology Joint work with Raphael Yuster ( University of Haifa ). Background. Abstract Question: When can we say that a single object behaves like a random one?.

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The Quasi-Randomness of Hypergraph Cut Properties

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  1. The Quasi-Randomness of Hypergraph Cut Properties Asaf Shapira Georgia Institute of Technology Joint work with Raphael Yuster (University of Haifa)

  2. Background Abstract Question: When can we say that a single object behaves like a random one? “Concrete” problem: Which (hyper)graph properties “force” a (hyper)graph to behave like a “truly” random one. Chung-Graham-Wilson ’89, Thomason ‘87: 1. Defined the notion of a p-quasi-random graph= A graph that “behaves” like a typical graph generated by G(n,p). 2. Proved that several “natural” properties that hold w.h.p in G(n,p), “force” a graph to be p-quasi-random.

  3. The CGW Theorem • Theorem [CGW ‘89]: Fix any 0<p<1, and let G=(V,E)be a • graph on n vertices. The following are equivalent: • Any set U  V spans  ½p|U|2 edges • Any set U  V of size n spans  ½p2|U|2 edges • G contains  ½pn2edges and p4n4 copies of C4 • Most pairs u,v have co-degree p2n Note: All the above hold whp in G(n,p). Definition: A graph that satisfies any (and therefore all) the above properties is p-quasi-random, or just quasi-random.

  4. Quasi-Random Properties Definition: Say that a graph property is quasi-random if it is equivalent to the properties in the CGW theorem. “The” Question: Which graph properties are quasi-random? No! Any (reasonable) property that holds in G(n,p)whp? Example 1:Having  ½pn2 edges and p3n3 copies of K3 is not a quasi-random property. Example 2:Having degrees pn is not a quasi-random prop. Replacing K3with C4 gives a quasi-random property. …but having co-degreesp2nis a quasi-random property.

  5. The Chung-Graham Theorem p 2p 0 Theorem [Chung-Graham ’89]: 1. Having  ¼ pn2 edges crossing all (½n,½n)-cuts is not a quasi-random property. 2. For any 0<<½, having  (1-)pn2 edges crossing all (n,(1-)n)-cutsis a quasi-random property. To get (1), take an Indep-Set on n/2 vertices, a G(n/2,2p) on the rest, and connect them with a p-random graph. [CG ‘89] Gave two proofs of (2). One using a counting argument, and another algebraic proof based on the rank of certain intersection matrices. [Janson ‘09] Gave another proofs of (2), using graph limits.

  6. A Touch on Hypergraphs Definition: A hypergraph G=(V,E) has a set of vertices V, and a set of hyper-edges E. A hyper-edge eE is a subset of V. G is k-uniform (k-graph for short) if all edges are of size k. Definition:Let =(1,…,k) satisfy 0<i<1 and i=1. Let P be the following property of k-graphs: Any (1n,…,kn)-cut is crossed by the “right” number of edges.

  7. Quasi-Random Hypergraphs What is a quasi-random k-graph? Answer 1: “Obvious” extension of quasi-random graphs. All vertex-sets have the “correct” edge density. Def: This is “weak” quasi-randomness [Chung-Graham ‘90]. Why? Because it does not imply certain (useful) things that are implied by quasi-randomness in graphs. Answer 2: “Strong” quasi-randomness [Frankl-Rodl ‘02]. Fact: Strong Quasi-Randomness  Weak Quasi-Randomness Notation: “P is Quasi-Random” means “P  Weak Quasi-Rand”

  8. Our Main Result Theorem [Chung-Graham ’89]:For =(c,1-c) let P be the graph property that every (c,1-c) cut is crossed by the correct number of edges. Then P is quasi-random iff c ½ . Def:For =(1,…,k), we say that a k-graph has property P if any (1n,…,kn)-cut is crossed by the “right” number of edges. Chung-Graham ‘91:Can one characterize the cut properties of k-graphs which are quasi-random? Theorem 1 [S-Yuster]P is quasi-random iff   (1/k,…,1/k).

  9. Our Main Result p 2p 0 C(n,p) Theorem [Chung-Graham ’89]:For =(c,1-c) let P be the graph property that every (c,1-c) cut is crossed by the correct number of edges. Then P is quasi-random iff c ½ . To see that c=½ is not quasi-random consider the random graph: Theorem 2 [S-Yuster]:C(n,p) is (essentially) the only non-quasi random graph satisfying P where =(½,½) . Theorem 3 [S-Yuster]:A similar holds for k-graphs.

  10. Proof Overview p 2p 0 Definition:Let =(1,…,k) satisfy 0<i<1 and i=1. We let P be the following property of k-graphs: Any (1n,…,kn)-cut has the correct number of edges crossing it. Theorem:If  = (1/k,…,1/k) then P is not quasi-random. [CG‘89] For k=2: i vertices k-i vertices 2ip/k For arbitrary k2:

  11. Proof Overview p2 =p p1 p3 V-A |V-A|=(1-)n A |A|=n =p Theorem:If   (1/k,…,1/k) then P is quasi-random. Proof (of k=2): It is enough to show that every set of vertices of size n has the correct edge density. Let A be such a set. =p Let 0c. Define a new partition (B,V-B). B gets a random subset of cn vertices from A and (-c)n vertices from V-A. 1. We know the expected number of edges in the new cut. 2. This expectation is a linear function in p1 , p2 , p3. 3. Using c{0,,/2} we get 3 linear equations. 4. When 1/2 they have a unique solution p1=p2=p3=p.

  12. Proof Overview p p p p p The random graph G(n,p) p p p p p The counter example C(n,p) 0 2p 2p 0 p p Theorem 2: When  = (½,½) the only way a non-quasi random graph can satisfy P is the “trivial” one. Let’s consider the fractional problem P*of assigning weights to the edges of the complete graph, s.t. the total (normalized) weight crossing any (n/2,n/2)-cut is p. What is “trivial”? Two ways a graph can satisfy P* There are many such solutions

  13. Proof Overview p 2p p 0 (1) The random graph (2) C(n,p) p p p p p p p p p/2 3p/2 0 2p p p p p p p What is “trivial”? Two ways a graph can satisfy P* • Satisfying P*  satisfying a set of linear equations: • Unknowns are the weights of the edges. • We have a linear equation for any (n/2,n/2)-cut Definition: A trivial solution of P* is any affine combination of solution (1) and (a collection of) solution (2). ½ · + ½ · =

  14. Proof Overview p p 2p 0 (1) G(n,p) (2) C(n,p) Definition: A trivial solution of P* is any affine combination of solution (1) and (a collection of) solution (2). Theorem 2:When  = (½,½) the only solutions satisfying P* are the trivial ones.

  15. Proof Overview p 2p 0 p (2) C(n,p) (1) G(n,p) Note:A is an matrix. Step 1:rank(A)  Definition: A trivial solution is any affine combination of solutions (1) and (2). Theorem 2:When  = (½,½) the only solutions satisfying P are the trivial ones. Recall Satisfying P* is equivalent to satisfying a set of linear equations: 1. Unknowns are the weights of the edges. 2. We have one equation for any (n/2,n/2)-cut Definition:Let us write this as Ax=p. Proof:Any solution is a solution of the linear system Ax=p. Step 2:dim(span[(2) – (1)]) n-1.

  16. Proof Overview p 2p 0 p (2) C(n,p) (1) G(n,p) p 0 2p • This collection spans Rn-1 Definition: A trivial solution is any affine combination of solutions (1) and (2). Step 2:Trivial solutions have dimension n-1. Proof: For every solution (2) consider the vector of pairs (v1,vi). n/2-1 of the entries are 2p, the other are p. After subtracting (1) from these vectors, we get, for every subset S [n-1] of size n/2, a vector vS satisfying: • vS(i) = 0 if iS. • vS(i) = p if i  S.

  17. Proof Overview Step 1:rank(A)  Note:A is an matrix 1 -1 -c C (vS – vS,t) vs-  t SV-S n/2n/2 n/2n/2-1 We first prove that matrix of (n/2,n/2-1)-cuts has full rank. Proof: Take the vector vS corresponding to some cut. vS,t = vector of cut obtained by moving t from S to V-S. (vS – vS,t) t

  18. Proof Overview Step 1:rank(A)  [Gottlieb ‘66]:rank(I(2,n/2,n-1)) =. Conclusion:A spans the rows of the matrix Inc(2,n/2,n-1) 2-element subsets of [n-1] S IS,T = 1 iif ST n/2-element subsets of [n-1] T

  19. Back to Graphs “Theorem”: When  = (½,½) the only way a non-quasi random graph can satisfy P is the “trivial” one. Instead of graphs, we consider the fractional problem of assigning weights to the complete graph, s.t. total weight crossing any (n/2,n/2)-cut is p. Theorem:When  = (½,½), only solutions of P* are affine combinations of G(n,p) and C(n,p). “Theorem”:If G is close to satisfying P then G is close to an affine combination of G(n,p) and C(n,p).

  20. Back to Graphs Theorem 1:When  = (½,½), only solutions to P* are affine combinations of G(n,p) and C(n,p). “Theo 2”:IfG is close to PG is close to an affine comb. of G(n,p) and C(n,p). We measure distance between graphs using the Cut-Norm Theorem 2:, ,t such that if G is -close to satisfying P then G is -close (in the cut-norm) to an affine combination of G(n,p) and t copies C(n,p). Proof:Application of Theorem 1 + Frieze-Kannan reg. lemma.

  21. The General Case Definition:Let =(1,…,k) satisfy 0<i<1 and i=1. We let P be the following property of k-uniform hypergraphs: Any (1n,…,kn)-cut has the correct number of edges crossing it. Theorem:If  = (1/k,…,1/k) then P is not quasi-random. 2ip/k i vertices Ck(n,p) : Theorem 2:When  = (1/k,…,1/k) every k-graph satisfying P is close to an affine combinations of Ck(n,p) and the randomk-graph.

  22. The General Case Note:A has columns and a row for any balanced k-cut. Instead of thinking about hyperraphs, let’s consider the fractional problem P* of assigning weights to the edges of the complete k-graph, s.t. total weight crossing any (n/k,…,n/k)-cut is p. Definition:Let us write this as Ax=p. Ae,c=1 edge e intersects each of the k parts of C. Proof:We need to show that any solution to Ax=p is an affine combination of Ck(n,p) and the random k-graphGk(n,p). Step 1: Show that dim({Ck(n,p)-Gk(n,p)}) n-1. Much harder than the graph problem. Similar to the graph problem. Key Step:rank(A)

  23. The General Case A has columns and a row for any balancedk-cut. Ae,c=1 edge e intersects each of the k parts of C. We need to show:rank(A) Can we reduce to Gottlieb Theorem? Lemma:If the rows of A correspond to non balanced cuts, then rank[A]= Tight for k=2 Only gives rank(A)

  24. The General Case A has columns and a row for any balanced k-cut. Ae,c=1 edge e intersects each of the k parts of C. Theorem:rank(A) Johnson Scheme:0tk, let Bt be matrix: Bt(S,T)=1  |ST|=t. Known facts: 1. Bt has k+1 eigenvalues 0t,…,kt 2. rt has multiplicity 3. If C = ct Btthen C has k+1 eigenvalues 0,…,k. Also, r = ct rt and has multiplicity .

  25. The General Case A has columns and a row for any balanced k-cut. Ae,c=1 edge e intersects each of the k parts of C. The Johnson Scheme: For 0tk, let Bt be the square matrix, where Bt(S,T)=1  |ST|=t. Claim:ATA= ct Bt Corollary:ATA has k+1 eigenvalues 0,…, k. r =  ct rt and has multiplicity . Conclusion: To show that rank[A]  , it is enough to show that for all r1, we have r 0.

  26. The General Case Conclusion: To finish the proof we “just” need to show that for all r1 we have r =  ctrt0. While ct have “civilized” expressions… rt = But…, note that ris a polynomial in n. So proving r(n)0, will prove the result for large n. Leading coefficient of r(n) is

  27. Concluding Remarks Meta problem: Is it true that any quasi-random property that involves edges, remains quasi-random if we replace the edge with a fixed graph H? [Simonovits-Sos ’97]: Let PH be the property that all vertex-sets span the correct number of copies of H. For any H, property PH is quasi-random. Definition:Let =(1,2,3) satisfy 0<i<1 and i=1. Let Q be the following graph property: Any (1n,2n,3n)-cut is crossed by the “correct” number of K3.

  28. Concluding Remarks Definition:Let =(1,2,3) satisfy 0<i<1 and i=1. Let Q be the following graph property: Any (1n,2n,3n)-cut is crossed by the “correct” number of K3. Corollary: If   (1/3,1/3,1/3) then Q is quasi-random Proof: Let H contain a 3-hyperedge for every K3 of G. H satisfies P and is thus quasi-random by Theorem 1. In G, any set of vertices has the “correct” number of K3. The Simonovits-Sos Theorem implies that G is quasi-random. Open Problem: What happens when  = (1/3,1/3,1/3)?

  29. Thank You

  30. Our Main Results p 2p 0 Definition:Let =(1,…,k) satisfy 0<i<1 and i=1. Let P be the following property of k-uniform hypergraphs: Any (1n,…,kn)-cut is crossed by the “right” number of edges. Theorem 1 [S-Yuster ’09]: 1. If  = (1/k,…,1/k) then P is not quasi-random. 2. If   (1/k,…,1/k) then P is quasi-random. Theorem 2 [S-Yuster ’09]: 1. When  = (½,½), the only way a non-quasi random graph can satisfy P is the “trivial” one. 2. Same result holds in hypergraphs.

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