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%
