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

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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 -far from P, reject w.p. ¾ -far: must modify -fraction of adj. list What is the query complexity of A?

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examples of sparse testers [Goldreich, Goldwasser, Ron] propertyalgorithmlower bound connectivity Õ(1/ ) is a forest Õ(1/ ) bipartiteness Õ( n poly(1/ )) ( n)

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examples of sparse testers have one-sided error: if G satisfies property P, accept w.p. 1 propertyalgorithmlower bound connectivity Õ(1/ ) is a forest Õ(1/ ) bipartiteness Õ( n poly(1/ )) ( n)

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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 small

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

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

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one-sided error lower bound Must see non 3-colorable subgraph to reject Claim. There exists a sparse G such that G is δ far from 3-colorable Every subgraph of size o(n) is 3- colorable Proof. G = O(1/δ 2 ) random perfect matchings

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

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an explicit construction Efficiently construct sparse CSP A such that A is 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

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an explicit construction CSP A: flow constraints on constant degree expander graph (Tseitin tautologies) 3 64 9 x 34 + x 36 + x 39 = x 43 + x 63 + x 93 + 1 small cuts are overloaded C V C

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

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

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

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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

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two-sided error bound Claim. Can choose left hand side of A: Every x i appears in 3/δ 2 equations Every o(n) equations linearly independent Proof. Repeat 3/δ 2 times: choose n/3 disjoint random triples x i + x j + x k

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two-sided error bound Distributions. Fix left hand side as in Claim x 1 + x 4 + x 8 = x 2 + x 5 + x 1 = x 2 + x 7 + x 6 = x 8 + x 3 + x 9 = A farA sat

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two-sided error bound Distributions. Fix left hand side as in Claim A far: Choose right hand side at random x 1 + x 4 + x 8 = 0 x 2 + x 5 + x 1 = 1 x 2 + x 7 + x 6 = 1 x 8 + x 3 + x 9 = 1 x 1 + x 4 + x 8 = x 2 + x 5 + x 1 = x 2 + x 7 + x 6 = x 8 + x 3 + x 9 = A farA sat

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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 x 1 + x 4 + x 8 = 0 x 2 + x 5 + x 1 = 1 x 2 + x 7 + x 6 = 1 x 8 + x 3 + x 9 = 1 x 1 + x 4 + x 8 = x 2 + x 5 + x 1 = x 2 + x 7 + x 6 = x 8 + x 3 + x 9 = A farA sat

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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 x 1 + x 4 + x 8 = 0 x 2 + x 5 + x 1 = 1 x 2 + x 7 + x 6 = 1 x 8 + x 3 + x 9 = 1 0 + 1 + 1 = 0 1 + 0 + 0 = 1 1 + 0 + 0 = 1 1 + 1 + 1 = 1 A farA sat

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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 x 1 + x 4 + x 8 = 0 x 2 + x 5 + x 1 = 1 x 2 + x 7 + x 6 = 1 x 8 + x 3 + x 9 = 1 A farA sat

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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)

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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. far from 3-colorable graphs

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

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