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Testing the independence number of hypergraphs

This research focuses on testing the independence number of hypergraphs efficiently, distinguishing between cases where a hypergraph has a large independent set or is far from having one. By designing an efficient distinguishing algorithm, the goal is to prove the existence of large independent sets robustly. The research combines ideas from previous works and develops a proof paradigm for testing properties in hypergraphs, specifically the independence number. The findings provide insights into the efficiency and accuracy of property testing in hypergraphs.

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Testing the independence number of hypergraphs

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  1. Testing the independence number of hypergraphs Michael Langberg California Institute of Technology

  2. k-uniform hypergraph I • G=(V,E) • Each edge is of size k. • I  IS: no edges included in I. • (G) = size of max IS.

  3. Property testing of (G) • Input: G=(V,E). • Goal: distinguish between 2 cases • G has a largeIS. • G is far from having a largeIS. • Design efficient(,)-distinguishing algorithm: • Case I: (G)  n. • Case II: Must remove at least nkedges from G for it to have an IS of size n (-far). • Efficient = few samples of G (constant).

  4. Naïve PT algorithm • Let s be constant (will depend on  and ). • Sample s of vertices of G randomly = H. • Compute (H). • If (H)  s declare “case I” o.w. “case II”. • Like to prove: • (G)  n (H)  s. • G-far from having (G)  n (H) < s. Case I:(G)  n Case II: G-far from (G)  n

  5. Naïve PT algorithm • (G)  n (H)  s • Exp. |I H| = s. • G-far from (G)  n (H) < s. • (G) < n (H) < s. • Must use “-far” property of G. H I =½ =1

  6. Our result • Let G be -far from (G)  n. • H random subgraph of G. Thm: If |H| > exp(k)2k/3 then w.h.p. (H) < |H|. Repeating Naïve alg: • (G)  n Pr[Output = Case I] = large. • G-far from (G)  n Pr[Output = Case I] = small.

  7. Previous work Testing (G) in standard graphs (2-uniform): • [Goldreich,Goldwasser,Ron]: prove similar theorem for |H|~/4. • [Feige,L,Schechtman]: improve to |H|~4/3. PT of hypergraphs: • Chromatic number: Considered by [Czumaj,Sohler]. • Max-k-CNF: [Alon,Fernandez de la Vega,Kannan,Karpinski]. [Alon,Shapira],[Frieze,Kannan],[Andersson,Engebretsen] We combine ideas from [FLS] and [AS] to obtain: Thm: G -far, HrG, |H|~2k/3 then w.h.p. (H) < |H|.

  8. Remainder of talk • Present [FLS] proof paradigm for testing of (G) in standard graphs. • Present our proof.

  9. [FLS] proof Let G be -far from (G)  n. Let H be large random sample of G. • Thm: W.h.p. (H) <|H|. • Let R be random subgraph of G. • Analyze Pr[R is IS]. • Can be used to prove Thm. • Use union bound on all large R in H. • Pr[(H) |H|] ≤ #(RH, |R|=|H|)Pr[R is IS]. • Pr[R is IS] = ? H R G=(V,E)

  10. s s |H| |R| ( ) ( ) s |R| = [FLS] proof cont. R • Let G be -far from (G)  n. • Let R be random subset of G. • Pr[R is IS] = ? • G may have an IS of size ~ n. • Thus Pr[R is IS] > |R|. • [FLS] show that this is “tight”. • Union bound for |H|=s, |R|=s: • Pr[(H)  s] ≤ [FLS] fix this by considering Pr[R is a maximum IS in H]. IS G=(V,E) = large !!

  11. What about hypergraphs? R • Let G be -far from (G)  n. • Let R be random subset of G. • Pr[R is IS] = ? • We show that Pr[R is IS] ~ |R|. • Use ideas of [AS]: formalize “set of neighbors” • (R) = set system of all subsets “adjacent” to R. • Aadjacent to R:  edge that consists of A portion of R. • (R) = { { },{ },{ } }.

  12. Proof: Pr[R is IS]~|R| (R) R • Consider choosing the set R one by one. • Ris IS iff each inter. subset is IS. • If R is IS, then most steps: vertices of R must be chosen from subset of size n. • Initially, R0 = and (R0)= . • Consider new random vertex v. • Lemma:At each step, v must be chosenfrom a set of size n, otherwise “size” of (Riv)) >> “size” of (Ri). • Corollary: The size of (R) is bounded. For “large” R, most vertices v must be chosen from a set of size n. v

  13. Proof of lemma Ri • Consider step i: Ri = {r1,…,ri}. • Define “degree” of vertex v as: dv = size of (Riv) – size of (Ri). • Claim: only n vertices have low degree. • Prf: Look at n vertices of lowest degree, the induced subgraph has many edges  vertex of high degree. Lemma:At each step, v must be chosenfrom a set of size n otherwise size of (Riv) >> size of (Ri). v High deg. dense subgraph n of low deg.

  14. Concluding remarks • PT algorithm with sample size ~ 2k/3. • Lower bound of ~ 1/2. • Similar gap between 1/2 (upper) and 1/ (lower) exists in the case of testing chromatic number. • Thanks!

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