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

Testing Boolean functions 2 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  2 n inputs to get a function satisfying 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]

Recap of Arnab’s Talk [BGS10] 3 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

Odd-Cycle Freeness (OCF) 4 Definition: A function is odd-cycle-free (OCF) if for any odd t there exist no x 1,…,x t satisfying 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.

Our Main Result 5 Theorem [BGRS11]: OCF is testable with O(1/ ε 2 ) queries. Comments: Improvements from tower of exponential in generic results [BGS10] First family defined by infinitely many constraints that is testable efficiently.

How to test OCF? 6 Alg-1: A graph-test with queries: 1. Pick at random 2. Set 3. Accept iff f restricted to G is OCF. Proof technique: reduce to testing bipartiteness in graphs Alg-2: A subspace-sampling test with poly(1/  ) queries. 1. Pick at random 2. Set 3. Accept iff f restricted to S is OCF. Proof technique: Fourier analysis

The edge sampling test 7 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. (0,0) (0,1) (1,1) (1,0) If we change f(0,1)=1 There is a 1-to-2 n correspondence between “cycles” of f and cycles in C( f ).

The edge sampling test 8 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.

The Key Lemma 9 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: 1. Randomly pick 1/ ε vertices in G 2. Query all edges between them 3. Check if the induced subgraph is bipartite Our graph-test for OCF works on C( f ) as follows: 1. Pick random x 1,…,x 1/  (like picking vertices in C( f )) 2. Query f on all points in (like querying edges in C( f )) 3. Check if they contain an odd-cycle (corresponds to odd cycle in C( f ))

Additional Results 10 Lemma: If f is ε -far from OCF then C( f ) is ε /2-far from bipartite. In fact: f is ε -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  n 2, using O(1/ ε 8 ) queries. Corollary 1 [BGRS11]: (Tolerant Test): One can estimate how far is f from OCF up to an error of  2 n, using O(1/ ε 8 ) queries. Corollary 2 [BGRS11]: One can estimate the smallest Fourier coefficient of f up to an error of , using O(1/ ε 8 ) queries.

Proof of Key Lemma 11 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

Step 1: Geometric interpretation of OCF 12 Claim: f is OCF iff 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 ε 2 n from support( f ). That is,  support( f )  {x : ax = 0}    2 n H+a H Supp( f )

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

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

Smallest Eigenvalue and Bipartiteness 15 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 u T Au=2e(U) But using the fact the G is d-regular, we also have

A Canonical Test? 16 [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?

A Canonical Test? 17 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 iff f restricted to span(S) satisfies P Note: We can assume that test operates as above [BGS10]. Problem is that k points “span” 2 k 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].

A Canonical Test for OCF 18 Reminder: The graph-test with queries: 1. Pick at random 2. Set 3. Accept iff f restricted to G is OCF. Theorem [BGRS11]: OCF can be tested with a canonical 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

Open Problems 19 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!