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

hardness of testing 3-colorability in bounded

degree graphsAndrej Bogdanov

Kenji ObataLuca Trevisan

testing sparse graph properties

A property tester is an algorithm Ainput: 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. listWhat is the query complexity of A?

examples of sparse testers

[Goldreich, Goldwasser, Ron]

property algorithm lower bound

connectivity

Õ(1/)

is a forest Õ(1/)

bipartiteness

Õ(n poly(1/))

(n)

examples of sparse testers

have one-sided error:• if G satisfies property P, accept w.p. 1

property algorithm lower bound

connectivity

Õ(1/)

is a forest Õ(1/)

bipartiteness

Õ(n poly(1/))

(n)

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

hardness of 3-colorability

One-sided testers for 3-colorability:• For any < ⅓, A must make (n) queries• Optimal: Every G is ⅓ close to 3-

colorable

Two sided testers:• There exists an for which A must make

(n) queries

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

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 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 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)

3

6 4

9

x34 + x36 + x39 = x43 + x63 + x93 + 1small cuts are overloaded

C VC

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

an explicit construction

C VC

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

Flow on vertices in C = sat assignment for C

an explicit construction

C VC

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

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. ½δ far from satisfiable E3LIN instances

By reduction, cannot distinguish 3-colorable vs. far from 3-colorable graphs

some open questions

Conjecture. A two-sided tester for 3-colorability with error parameter ⅓ δ must make (n) queries

Conjecture. Approximating Max CUT within ½ + δ requires (n) queries

• SDP approximates Max CUT within 87%