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hardness of testing 3-colorability in bounded degree graphs

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hardness of testing 3-colorability in bounded degree graphs

Andrej Bogdanov

Kenji Obata

Luca Trevisan

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?

[Goldreich, Goldwasser, Ron]

have one-sided error:

- if G satisfies property P, accept w.p. 1

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

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

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

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

Efficiently construct sparse graph G such that

- G is e far from 3-colorable
- Every subgraph of size o(n) is 3-colorable

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

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

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

C

V-C

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

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 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

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

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

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

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

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

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

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)

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

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%