hardness of testing 3-colorability in bounded degree graphs

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# hardness of testing 3-colorability in bounded degree graphs - PowerPoint PPT Presentation

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

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

[Goldreich, Goldwasser, Ron]

examples of sparse testers

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 smalle

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 3SAT 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

Proof.G =O(1/δ2) random perfect matchings

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 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

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

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

Flow on vertices in C = sat assignment for C

C

V-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 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

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

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

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

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

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

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

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 E3LIN 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%