Given f : {−1, 1} n → {−1, 1}, define the spectral distribution of f to be the distribution on subsets of [n] in which the set S is sampled with probability f (S) 2 . Then the Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai [FK96] states that there is some absoluteHere, H[ f 2 ] denotes the Shannon entropy of f 's spectral distribution, and Inf [f ] is the total influence of f . This conjecture is one of the major open problems in the analysis of Boolean functions, and settling it would have several interesting consequences.Previous results on the FEI conjecture have been largely through direct calculation. In this paper we study a natural interpretation of the conjecture, which states that there exists a communication protocol which, given subset S of [n] distributed as f 2 , can communicate the value of S using at most C • Inf [f ] bits in expectation. Using this interpretation, we are able show the following results:• Next, if f has Inf [f ] ≥ 1 and is computable by a decision tree with expected depth d, then
Building on work of Cai, Fürer, and Immerman [CFI92], we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic n-vertex graphs G and H such that any sum-of-squares (SOS) proof of nonisomorphism requires degree Ω(n). In other words, we show an Ω(n)round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs G and H which are not even (1 − 10 −14 )-isomorphic. (Here we say that two n-vertex, m-edge graphs G and H are α-isomorphic if there is a bijection between their vertices which preserves at least αm edges.) Our second result is that under the R3XOR Hypothesis [Fei02] (and also any of a class of hypotheses which generalize the R3XOR Hypothesis), the robust Graph Isomorphism problem is hard. I.e. for every > 0, there is no efficient algorithm which can distinguish graph pairs which are (1 − )-isomorphic from pairs which are not even (1 − 0 )-isomorphic for some universal constant 0 . Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest. the heuristic is powerful enough to work correctly for all trees and for almost all n-vertex graphs in the Erdős-Rényi G(n, 1/2) model [BES80, BK79]. (We say the heuristic "works correctly" on a graph G if the stabilized coloring for G is distinct from the stabilized coloring of any graph not isomorphic to G.) To overcome WL's failure for regular graphs, several researchers independently introduced the "k-dimensional generalization" WL k (see [Wei76, CFI92] for discussion). Briefly, in the WL k heuristic, each k-tuple of vertices has a color, and color refinement involves looking at all "neighbors" of each k-tuple of vertices (v 1 , . . . , v k ) (where the neighbors are all tuples of the formThe WL k heuristic can be performed in time n k+O(1) and is thus a polynomial-time algorithm for any constant k.The WL k heuristic is very powerful. For example, it is known to work correctly in polynomial time for all graphs which exclude a fixed minor [Gro12], a class which includes all graphs of bounded treewidth or bounded genus. Spielman's 2 O(n 1/3 ) -time graph isomorphism algorithm [Spi96] for strongly regular graphs is achieved by WL k with k = O(n 1/3 ). The WL k algorithm with k = O( √ n) is also a key component in the 2 O( √ n log n) -time GISO algorithm [BL83]. Throughout the '80s there was some speculation that GISO might be solvable on all graphs by running the WL k algorithm with k = O(log n) or even k = O(1). However this was disproved in the notable work of Cai, Fürer, and Immerman [CFI92], which showed the existence of nonisomorphic n-vertex graphs G and H which are not distinguished by WL k unless k = Ω(n). 1 The GRAPHISOMORPHISM problem can be thought of as kind of constraint satisfaction problem (CSP), and readers familiar with LP/SDP hierarchies for CSPs might see an analogy between k-dimensional WL and level-k LP/SDP relaxations. A very interesting recent work of Atserias and Maneva [AM13] (see also [GO12]) shows tha...
Abstract. We consider the problem of testing whether a function f : {0, 1} n → {0, 1} is computable by a read-once, width-2 ordered binary decision diagram (OBDD), also known as a branching program. This problem has two variants: one where the variables must occur in a fixed, known order, and one where the variables are allowed to occur in an arbitrary order. We show that for both variants, any nonadaptive testing algorithm must make Ω(n) queries, and thus any adaptive testing algorithm must make Ω(log n) queries.We also consider the more general problem of testing computability by width-w OBDDs where the variables occur in a fixed order. We show that for any constant w ≥ 4, Ω(n) queries are required, resolving a conjecture of Goldreich [15].We prove all of our lower bounds using a new technique of Blais, Brody, and Matulef [6], giving simple reductions from known hard problems in communication complexity to the testing problems at hand. Our result for width-2 OBDDs provides the first example of the power of this technique for proving strong nonadaptive bounds.
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