Testing Odd-Cycle Freeness of Boolean Functions

1. Testing Odd-Cycle Freeness of Boolean Functions AsafShapira (Georgia Tech) Joint work with: Arnab Bhattacharyya (MIT) Elena Grigorescu (Georgia Tech) Prasad Raghavendra (Georgia Tech)

2. Testing Boolean functions Input:Oracle access to a function • f is said to be -far from some property P if we need to modify f on at least 2ninputs to get a functionsatisfying P. Examples: • Linear functions [BLR93]: • Triangle-Freeness: What do tests for these families look like? • Check if defining pattern is satisfied on random sample • If f in P no violation exists (we accept with prob. 1) • If f is far from P there must exist a violation (to test, we need many violations) Triangle-freeness studied by [Green05]

3. Recap of Arnab’s Talk [BGS10] • Testing if a graph contains a certain (induced) sub-graph is fundamental to understanding the testability of graph properties • Testing if a Boolean function has an (induced) solution of a system of linear equations, is analogous to the notion of a graph having a certain (induced) sub-graph • Approach suggests a characterization of the linear-invariant properties of Boolean functions that are testable

4. Odd-Cycle Freeness (OCF) Definition: A function is odd-cycle-free (OCF) if for any odd tthere exist no x1,…,xtsatisfying • Note that we are forbidding solutions to an infinite set of equations. • In fact, OCF is the only monotone property defined by forbidding solutions to an infinite set of equations.

5. Our Main Result Theorem [BGRS11]: OCF is testable with O(1/ε2) queries. Comments: • Improvements from tower of exponential in generic results [BGS10] • First family defined by infinitely many constraints that is testable efficiently.

6. How to test OCF? Alg-1: A graph-test with queries: • Pick at random • Set • Accept ifff restricted to G is OCF. Proof technique: reduce to testing bipartiteness in graphs Alg-2: A subspace-sampling test with poly(1/) queries. • Pick at random • Set • Accept ifff restricted to S is OCF. Proof technique: Fourier analysis

7. The edge sampling test Definition: The Cayley graph of f , denoted C( f ), is defined as Example: Suppose f(1,0) = f(1,1) = 1 and f(0,0) = f(0,1) = 0. (0,1) (1,0) (1,1) (0,0) • If we change f(0,1)=1 • There is a 1-to-2n correspondence between “cycles” of f and cycles in C( f ).

8. The edge sampling test Definition: The Cayley graph of f , denoted C( f ), is defined as V {0,1}n and E  {(u,v) | f (u-v)  1} Observation: f is OCFC( f ) is bipartite Observation: If f is -far from OCF, then for any OCF function g, the Cayley graph C( f ) is -far from the Cayley graph C(g). Not enough to apply a graph test for checking bipartiteness.

9. The Key Lemma Lemma: If f is -far from OCF then C( f ) is /2-far from bipartite. Corollary:This proves correctness of the graph-test Proof: The bipartiteness test of [GGR96,AK02] works as follows: • Randomly pick 1/ε vertices in G • Query all edges between them • Check if the induced subgraph is bipartite • Our graph-test for OCF works on C( f ) as follows: • Pick random x1,…,x1/ (like picking vertices in C( f )) • Query f on all points in (like querying edges in C( f )) • Check if they contain an odd-cycle (corresponds to odd cycle in C( f ))

10. Additional Results Lemma: If f is ε-far from OCF then C( f ) is ε/2-far from bipartite. In fact: f isε-far from OCFC( f ) is ε/2-far from bipartite. [GGR96, AdlVKK03] One can estimate the distance of a graph to being bipartite up to an error of n2, using O(1/ε8) queries. Corollary 1 [BGRS11]: (Tolerant Test): One can estimate how far is f from OCF up to an error of 2n, using O(1/ε8) queries. Corollary 2 [BGRS11]: One can estimate the smallest Fourier coefficient of f up to an error of , using O(1/ε8) queries.

11. Proof of Key Lemma Lemma: If f is -far from OCF then C( f ) is /2-far from bipartite. Step 1: A geometric interpretation of OCF Step 2:Expressing OCF in terms of Fourier coefficients of f Step 3:A spectral characterization of OCF

12. Step 1: Geometric interpretation of OCF Claim: f is OCFiff there exists a half-space that contains no element of support( f ). Corollary: f is ε-far from OCF every half-space contains at least ε2n from support( f ). That is, support( f ){x : ax = 0}  2n H+a Supp( f ) H

13. Step 2: OCF and the Fourier Coefficients Step 1:f is ε-far from OCFiffa we have support( f ){x : ax = 0}  2n Suppose |support( f )|= 2n • If f is OCF then a such that support( f )  {x : ax = 1} Hence, • If f is ε-far, then a we have support( f ){x : ax = 0}  2n Hence, Corollary: The distance of f from OCF is

14. Step 3: A Spectral Characterization Fact: Theeigenvalues of the adjacency matrix of C( f ) are Corollary: If f is ε-far from OCF then

15. Smallest Eigenvalue and Bipartiteness Lemma:If G is d-regular and then G is /2-far from bipartite Proof: Enough to show that for any set of vertices U, we have Let u denote the characteristic vector of U. Then uTAu=2e(U) But using the fact the G is d-regular, we also have

16. A Canonical Test? [AFKS01,GT03] If a graph property is testable with q() queries, then it is also testable by a canonical test that samples a set S of q() vertices and inspects the graph induced by S. Note: This gives only a quadratic loss in query-complexity. Open Problem: Is there a canonical test for linear-invariant properties of Boolean functions?

17. A Canonical Test? Open Problem: Is there an efficient canonical test for linear-invariant properties of Boolean functions? That is, the test should work as follows: Test: Sample a set S of q() points Accept ifff restricted to span(S) satisfies P Note: We can assume that test operates as above [BGS10]. Problem is that k points “span” 2k points, so this gives an exponential loss in query complexity. Question is can we do it with only a poly loss, as in graphs [GT03].

18. A Canonical Test for OCF Reminder: The graph-testwith queries: • Pick at random • Set • Accept ifff restricted to G is OCF. Theorem [BGRS11]:OCF can be tested with acanonical test with only a poly loss in query complexity: The test: • Pick a set S of O(log 1/ε) points • Check if f restricted to span(S) is OCF

19. Open Problems • Is there an “efficient” canonical tester for linear-invariant and subspace-hereditary properties of Boolean functions: Suppose P is testable with q() queries. Is it then also testable with poly(q())by a canonical test? • Lower bounds for testing OCF (better than (1/))

20. Thanks!