An r-simple k-path is a path in the graph of length k that passes through each vertex at most r times. The r-SIMPLE k-PATH problem, given a graph G as input, asks whether there exists an r-simple k-path in G. We first show that this problem is NP-Complete. We then show that there is a graph G that contains an r-simple k-path and no simple path of length greater than 4 log k/ log r. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex.We then give a randomized algorithm that runs in time poly(n) · 2 O(k·log r/r) that solves the r-SIMPLE k-PATH on a graph with n vertices with one-sided error. We also show that a randomized algorithm with running time poly(n) · 2 (c/2)k/r with c < 1 gives a randomized algorithm with running time poly(n) · 2 cn for the Hamiltonian path problem in a directed graph -an outstanding open problem. So in a sense our algorithm is optimal up to an O(log r) factor.

Finding a maximal independent set (MIS) in a graph is a cornerstone task in distributed computing. The local nature of an MIS allows for fast solutions in a static distributed setting, which are logarithmic in the number of nodes or in their degrees [Luby 1986, Ghaffari 2015. By running a (static) distributed MIS algorithm after a topology change occurs, one can easily obtain a solution with the same complexity also for the dynamic distributed model, in which edges or nodes may be inserted or deleted.In this paper, we take a different approach which exploits locality to the extreme, and show how to update an MIS in a dynamic distributed setting, either synchronous or asynchronous, with only a single adjustment, meaning that a single node changes its output, and in a single round, in expectation. These strong guarantees hold for the complete fully dynamic setting: we handle all cases of insertions and deletions, of edges as well as nodes, gracefully and abruptly. This strongly separates the static and dynamic distributed models, as super-constant lower bounds exist for computing an MIS in the former.We prove that for any deterministic algorithm, there is a topology change that requires n adjustments, thus we also strongly separate deterministic and randomized solutions.Our results are obtained by a novel analysis of the surprisingly simple solution of carefully simulating the greedy sequential MIS algorithm with a random ordering of the nodes. As such, our algorithm has a direct application as a 3-approximation algorithm for correlation clustering. This adds to the important toolbox of distributed graph decompositions, which are widely used as crucial building blocks in distributed computing.Finally, our algorithm enjoys a useful history-independence property, which means that the distribution of the output structure depends only on the current graph, and does not depend on the history of topology changes that constructed that graph. This means that the output cannot be chosen, or even biased, by the adversary, in case its goal is to prevent us from optimizing some objective function. Moreover, history independent algorithms compose nicely, which allows us to obtain history independent coloring and matching algorithms, using standard reductions.

We consider the problem of testing if a given function f : F n q → F q is close to a n-variate degree d polynomial over the finite field F q of q elements. The natural, low-query, test for this property would be to pick the smallest dimension t = t q,d ≈ d/q such that every function of degree greater than d reveals this feature on some t-dimensional affine subspace of F n q and to test that f when restricted to a random t-dimensional affine subspace is a polynomial of degree at most d on this subspace. Such a test makes only q t queries, independent of n. hold only for fields of prime order). Thus to get a constant probability of detecting functions that were at constant distance from the space of degree d polynomials, the tests made q 2t queries. Kaufman and Ron also noted that when q is prime, then q t queries are necessary. Thus these tests were off by at least a quadratic factor from known lower bounds. It was unclear if the soundness analysis of these tests were tight and this question relates closely to the task of understanding the behavior of the Gowers Norm. This motivated the work of Bhattacharyya et al. [BKS + 10], who gave an optimal analysis for the case of the binary field and showed that the natural test actually rejects functions that were Ω(1)-far from degree d-polynomials with probability at least Ω(1).In this work we give an optimal analysis of this test for all fields showing that the natural test does indeed reject functions that are Ω(1)-far from degree d polynomials with Ω(1)-probability. Our analysis thus shows that this test is optimal (matches known lower bounds) when q is prime. (It is also potentially best possible for all fields.) Our approach extends the proof technique of Bhattacharyya et al., however it has to overcome many technical barriers in the process. The natural extension of their analysis leads to an O(q d ) query complexity, which is worse than that of Kaufman and Ron for all q except 2! The main technical ingredient in our work is a tight analysis of the number of "hyperplanes" (affine subspaces of co-dimension 1) on which the restriction of a degree d polynomial has degree less than d. We show that the number of such hyperplanes is at most O(q t q,d ) -which is tight to within constant factors.

In this paper we study the structure of polynomials of degree three and four that have high bias or high Gowers norm, over arbitrary prime fields. In particular we obtain the following results.1. Let f be a degree three polynomial with bias(f ) = δ then there exist r = O(log(1/δ)) quadratic polynomials. This result generalizes the corresponding result for quadratic polynomials. Let deg(f ) = 4 and biaswhere r = poly(1/δ), the i -s are linear, the q i -s are quadratics and the g i -s are cubic.3. Let deg(f ) = 4 and f U 4 = δ. Then there exists a partition of a subspaceItems 1,2 extend and improve previous results for degree three and four polynomials [KL08, GT07]. Item 3 gives a new result for the case of degree four polynomials with high U 4 norm. It is the first case where the inverse conjecture for the Gowers norm fails [LMS08,GT07], namely that such an f is not necessarily correlated with a cubic polynomial. Our result shows that instead f equals a cubic polynomial on a large subspace (in fact we show that a much stronger claim holds).Our techniques are based on finding a structure in the space of partial derivatives of f . For example, when deg(f ) = 4 and f has high U 4 norm we show that there exist quadratic polynomials {q i } i∈ [r] and linear functions { i } i∈ [R] such that (on a large enough subspace) every partial derivative of f can be written as ∆ y (f ) =

We consider the task of compression of information when the source of the information and the destination do not agree on the prior, i.e., the distribution from which the information is being generated. This setting was considered previously by Kalai et al. (ICS 2011) who suggested that this was a natural model for human communication, and efficient schemes for compression here could give insights into the behavior of natural languages. Kalai et al. gave a compression scheme with nearly optimal performance, assuming the source and destination share some uniform randomness. In this work we explore the need for this randomness, and give some non-trivial upper bounds on the deterministic communication complexity for this problem. In the process we introduce a new family of structured graphs of constant fractional chromatic number whose (integral) chromatic number turns out to be a key component in the analysis of the communication complexity. We provide some non-trivial upper bounds on the chromatic number of these graphs to get our upper bound, while using lower bounds on variants of these graphs to prove lower bounds for some natural approaches to solve the communication complexity question. Tight analysis of communication complexity of our problems and the chromatic number of the underlying graphs remains open.

We consider the problem of testing if a given function f : F n q → F q is close to a n-variate degree d polynomial over the finite field F q of q elements. The natural, low-query, test for this property would be to pick the smallest dimension t = t q,d ≈ d/q such that every function of degree greater than d reveals this feature on some t-dimensional affine subspace of F n q and to test that f when restricted to a random t-dimensional affine subspace is a polynomial of degree at most d on this subspace. Such a test makes only q t queries, independent of n. hold only for fields of prime order). Thus to get a constant probability of detecting functions that were at constant distance from the space of degree d polynomials, the tests made q 2t queries. Kaufman and Ron also noted that when q is prime, then q t queries are necessary. Thus these tests were off by at least a quadratic factor from known lower bounds. It was unclear if the soundness analysis of these tests were tight and this question relates closely to the task of understanding the behavior of the Gowers Norm. This motivated the work of Bhattacharyya et al. [BKS + 10], who gave an optimal analysis for the case of the binary field and showed that the natural test actually rejects functions that were Ω(1)-far from degree d-polynomials with probability at least Ω(1).In this work we give an optimal analysis of this test for all fields showing that the natural test does indeed reject functions that are Ω(1)-far from degree d polynomials with Ω(1)-probability. Our analysis thus shows that this test is optimal (matches known lower bounds) when q is prime. (It is also potentially best possible for all fields.) Our approach extends the proof technique of Bhattacharyya et al., however it has to overcome many technical barriers in the process. The natural extension of their analysis leads to an O(q d ) query complexity, which is worse than that of Kaufman and Ron for all q except 2! The main technical ingredient in our work is a tight analysis of the number of "hyperplanes" (affine subspaces of co-dimension 1) on which the restriction of a degree d polynomial has degree less than d. We show that the number of such hyperplanes is at most O(q t q,d ) -which is tight to within constant factors.

No abstract

Let D be a b-wise independent distribution over {0, 1} m. Let E be the "noise" distribution over {0, 1} m where the bits are independent and each bit is 1 with probability η/2. We study which tests f : {0, 1} m → [−1, 1] are ε-fooled by D + E, i.e., | E[f (D + E)] − E[f (U)]| ≤ ε where U is the uniform distribution. We show that D + E ε-fools product tests f : ({0, 1} n) k → [−1, 1] given by the product of k bounded functions on disjoint n-bit inputs with error ε = k(1 − η) Ω(b 2 /m) , where m = nk and b ≥ n. This bound is tight when b = Ω(m) and η ≥ (log k)/m. For b ≥ m 2/3 log m and any constant η the distribution D + E also 0.1-fools log-space algorithms. We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length m split among k parties. For Reed-Solomon codes of dimension m/k where k = O(1), communication Ω(ηm) − O(log m) is required to decode one message symbol from a codeword with ηm errors, and communication O(ηm log m) suffices. Second, we obtain pseudorandom generators. We can ε-fool product tests f : ({0, 1} n) k → [−1, 1] under any permutation of the bits with seed lengths 2n +Õ(k 2 log(1/ε)) and O(n) + O(nk log 1/ε). Previous generators have seed lengths ≥ nk/2 or ≥ n √ nk. For the special case where the k bounded functions have range {0, 1} the previous generators have seed length ≥ (n + log k) log(1/ε).