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hardness of testing 3-colorability in bounded degree graphs

hardness of testing 3-colorability in bounded degree graphs. Andrej Bogdanov Kenji Obata Luca Trevisan. testing sparse graph properties. A property tester is an algorithm A input: adjacency list of bounded deg graph G if G satisfies property P , accept w.p. ¾

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hardness of testing 3-colorability in bounded degree graphs

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  1. hardness of testing 3-colorability in bounded degree graphs Andrej Bogdanov Kenji Obata Luca Trevisan

  2. testing sparse graph properties A property tester is an algorithm A input: adjacency list of bounded deg graph G • if G satisfies property P, accept w.p. ¾ • if G is e-far from P, reject w.p. ¾ e-far: must modify e-fraction of adj. list What is the query complexity of A?

  3. examples of sparse testers [Goldreich, Goldwasser, Ron]

  4. examples of sparse testers have one-sided error: • if G satisfies property P, accept w.p. 1

  5. testing vs. approximation Approximating 3-colorability: • SDP finds 3-coloring good for 80% of edges • NP-hard to go above 98% Implies conditional lower bound on query complexity for smalle

  6. hardness of 3-colorability One-sided testers for 3-colorability: • For any e <⅓, A must make W(n) queries • Optimal: Every G is ⅓ close to 3-colorable Two sided testers: • There exists an e for which A must make W(n) queries

  7. other results With o(n) queries, it is impossible to • Approximate Max 3SAT within 7/8 +e • Approximate Max Cut within 16/17 + e • etc. Håstad showed these are inapproximable in poly time unless P = NP

  8. one-sided error lower bound Must see non 3-colorable subgraph to reject Claim. There exists a sparse G such that • G is ⅓ - δ far from 3-colorable • Every subgraph of size o(n) is 3-colorable Proof.G =O(1/δ2) random perfect matchings

  9. an explicit construction Efficiently construct sparse graph G such that • G is e far from 3-colorable • Every subgraph of size o(n) is 3-colorable

  10. an explicit construction Efficiently construct sparse CSP Asuch that • A is e far from satisfiable • Every subinstance of A with o(n)clauses is satisfiable There is a local, apx preserving reduction from CSP A to graph G

  11. an explicit construction CSP A:flow constraints on constant degree expander graph (Tseitin tautologies) C V-C 9 3 6 4 x34 + x36 + x39 = x43 + x63 + x93 + 1 small cuts are overloaded

  12. an explicit construction By expansion property, no cut (C, V-C) with |C|  n/2 is overloaded C V-C

  13. an explicit construction By expansion property, no cut (C, V-C) with |C|  n/2 is overloaded Flow on vertices in C = sat assignment for C C V-C

  14. two-sided error bound Construct two distributions for graph G: • If G ~ far, G is e far from 3-colorable whp • If G ~ col, G is 3-colorable • Restrictions on o(n) vertices look the same in far and col

  15. two-sided error bound Two distributions for E3LIN2 instance A: • If A ~ far, A is ½-δ far from satisfiable • If A ~ sat, A is satisfiable • Restrictions on o(n)equations look the same in far and sat Apply reduction from E3LIN2 to 3-coloring

  16. two-sided error bound Claim. Can choose left hand side of A: • Every xi appears in 3/δ2 equations • Every o(n) equations linearly independent Proof. Repeat 3/δ2 times: choose n/3 disjoint random triples xi + xj + xk

  17. two-sided error bound Distributions. Fix left hand side as in Claim x1 + x4 + x8 = x2 + x5 + x1 = x2 + x7 + x6 = x8 + x3 + x9 = x1 + x4 + x8 = x2 + x5 + x1 = x2 + x7 + x6 = x8 + x3 + x9 = A ~ far A ~ sat

  18. two-sided error bound Distributions. Fix left hand side as in Claim • A ~ far: Choose right hand side at random x1 + x4 + x8 = 0x2 + x5 + x1 = 1x2 + x7 + x6 = 1x8 + x3 + x9 = 1 x1 + x4 + x8 = x2 + x5 + x1 = x2 + x7 + x6 = x8 + x3 + x9 = A ~ far A ~ sat

  19. two-sided error bound Distributions. Fix left hand side as in Claim • A ~ far: Choose right hand side at random • A ~ sat: Choose random satisfiable rhs x1 + x4 + x8 = 0x2 + x5 + x1 = 1x2 + x7 + x6 = 1x8 + x3 + x9 = 1 x1 + x4 + x8 = x2 + x5 + x1 = x2 + x7 + x6 = x8 + x3 + x9 = A ~ far A ~ sat

  20. two-sided error bound Distributions. Fix left hand side as in Claim • A ~ far: Choose right hand side at random • A ~ sat: Choose random satisfiable rhs x1 + x4 + x8 = 0x2 + x5 + x1 = 1x2 + x7 + x6 = 1x8 + x3 + x9 = 1 0 + 1 + 1 = 01 + 0 + 0 = 11 + 0 + 0 = 11 + 1 + 1 = 1 A ~ far A ~ sat

  21. two-sided error bound Distributions. Fix left hand side as in Claim • A ~ far: Choose right hand side at random • A ~ sat: Choose random satisfiable rhs x1 + x4 + x8 = 0x2 + x5 + x1 = 1x2 + x7 + x6 = 1x8 + x3 + x9 = 1 x1 + x4 + x8 = 0x2 + x5 + x1 = 1x2 + x7 + x6 = 1x8 + x3 + x9 = 1 A ~ far A ~ sat

  22. two-sided error bound On any subset of o(n) equations • A ~ far: rhs uniform by construction • A ~ sat: rhs uniform by linear independence Instances look identical to any algorithm of query complexity o(n)

  23. two-sided error bound With o(n) queries, cannot distinguish satisfiable vs. ½ - δ far from satisfiable E3LIN instances By reduction, cannot distinguish 3-colorable vs. e far from 3-colorable graphs

  24. some open questions Conjecture. A two-sided tester for 3-colorability with error parameter ⅓ - δ must make W(n) queries Conjecture. Approximating Max CUT within ½ + δ requires W(n) queries • SDP approximates Max CUT within 87%

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