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. 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

hardness of testing 3-colorability in bounded degree graphs

Andrej Bogdanov

Kenji Obata

Luca Trevisan


Testing sparse graph properties

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

examples of sparse testers

[Goldreich, Goldwasser, Ron]


Examples of sparse testers1

examples of sparse testers

have one-sided error:

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


Testing vs approximation

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

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

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

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

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 construction1

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 construction2

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

small cuts are overloaded


An explicit construction3

an explicit construction

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

C

V-C


An explicit construction4

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

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 bound1

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 bound2

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 bound3

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 bound4

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 bound5

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 bound6

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 bound7

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 bound8

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 bound9

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

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%


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