hardness of testing 3 colorability in bounded degree

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 • 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?

examples of sparse testers property algorithm connectivity Õ(1/e) is a forest Õ(1/e 3) bipartiteness Õ( n poly(1/e)) [Goldreich, Goldwasser, Ron] lower bound W( n)

examples of sparse testers property algorithm connectivity Õ(1/e) is a forest Õ(1/e 3) bipartiteness Õ( n poly(1/e)) lower bound W( n) have one-sided error: • if G satisfies property P, accept w. p. 1

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 e

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

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

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 2

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

an explicit construction Efficiently construct sparse CSP A such 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

an explicit construction CSP A: flow constraints on constant degree expander graph (Tseitin tautologies) C V-C 9 3 6 4 x 34 + x 36 + x 39 = x 43 + x 63 + x 93 + 1 small cuts are overloaded

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

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

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

two-sided error bound Two distributions for E 3 LIN 2 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 E 3 LIN 2 to 3 -coloring

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

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 ~ far 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 ~ sat

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 A ~ far 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 ~ sat

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 1 + x 4 + x 8 = • A ~ sat: Choose random satisfiable rhs x 2 + x 5 + x 1 = 1 x 2 + x 7 + x 6 = 1 x 8 + x 3 + x 9 = 1 A ~ far x 2 + x 5 + x 1 = x 2 + x 7 + x 6 = x 8 + x 3 + x 9 = A ~ sat

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 0 + 1 +1 =0 • A ~ sat: Choose random satisfiable rhs x 2 + x 5 + x 1 = 1 x 2 + x 7 + x 6 = 1 x 8 + x 3 + x 9 = 1 A ~ far 1 + 0 +0 =1 1 +1 =1 A ~ sat

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 • A ~ sat: Choose random satisfiable rhs x 2 + x 5 + x 1 = 1 x 2 + x 7 + x 6 = 1 x 8 + x 3 + x 9 = 1 A ~ far x 2 + x 5 + x 1 = 1 x 2 + x 7 + x 6 = 1 x 8 + x 3 + x 9 = 1 A ~ sat

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)

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

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%