Elad Haramaty

On the structure of cubic and quartic polynomials

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. We give a canonical representation for degree three or four polynomials that have a significant bias (i.e. they are not equidistributed). This result generalizes the corresponding results from the theory of quadratic forms. This significantly improves previous results for such polynomials. 2. For the case of degree four polynomials with high Gowers norm we show that (a subspace of constant co-dimension of) Fn can be partitioned to subspaces of dimension Omega(n) such that on each of the subspaces the polynomial is equal to some degree three polynomial. It was previously shown that a quartic polynomial with a high Gowers norm is not necessarily correlated with any cubic polynomial. Our result shows that a slightly weaker statement does hold. The proof is based on finding a structure in the space of partial derivatives of the underlying polynomial.

All-Or-Nothing Generalized Assignment with Application to Scheduling Advertising Campaigns

We study a variant of the <i>generalized assignment problem</i> (<scp>gap</scp>), which we label <i>all-or-nothing</i> <scp>gap</scp> (<scp>agap</scp>). We are given a set of items, partitioned into <i>n</i> groups, and a set of <i>m</i> bins. Each item ℓ has size <i>s</i>_ℓ > 0, and utility <i>a</i>_ℓ<i>j</i> ⩾ 0 if packed in bin <i>j</i>. Each bin can accommodate at most one item from each group; the total size of the items in a bin cannot exceed its capacity. A group of items is <i>satisfied</i> if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from satisfied groups is maximized. We motivate the study of <scp>agap</scp> by pointing out a central application in scheduling advertising campaigns. Our main result is an <i>O</i>(1)-approximation algorithm for <scp>agap</scp> instances arising in practice, in which each group consists of at most <i>m</i>/2 items. Our algorithm uses a novel reduction of <scp>agap</scp> to maximizing submodular function subject to a matroid constraint. For <scp>agap</scp> instances with a fixed number of bins, we develop a randomized <i>polynomial time approximation scheme (PTAS)</i>, relying on a nontrivial LP relaxation of the problem. We present a (3 + ϵ)-approximation as well as PTASs for other special cases of <scp>agap</scp>, where the utility of any item does not depend on the bin in which it is packed. Finally, we derive hardness results for the different variants of <scp>agap</scp> studied in this paper.

On r-Simple k-Path

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 4logk/logr. 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.

Absolutely Sound Testing of Lifted Codes

In this work we present a strong analysis of the testability of a broad, and to date the most interesting known, class of “affine-invariant” codes. Affine-invariant codes are codes whose coordinates are associated with a vector space and are invariant under affine transformations of the coordinate space. Affine-invariant linear codes form a natural abstraction of algebraic properties such as linearity and low-degree, which have been of significant interest in theoretical computer science in the past. The study of affine-invariance is motivated in part by its relationship to property testing: Affine-invariant linear codes tend to be locally testable under fairly minimal and almost necessary conditions. Recent works by Ben-Sasson et al. (CCC 2011) and Guo et al. (ITCS 2013) have introduced a new class of affine-invariant linear codes based on an operation called “lifting”. Given a base code over a t-dimensional space, its m-dimensional lift consists of all words whose restriction to every t-dimensional affine subspace is a codeword of the base code. Lifting not only captures the most familiar codes, which can be expressed as lifts of low-degree polynomials, it also yields new codes when lifting “medium-degree” polynomials whose rate is better than that of corresponding polynomial codes, and all other combinatorial qualities are no worse. In this work we show that codes derived from lifting are also testable in an “absolutely sound” way. Specifically, we consider the natural test: Pick a random affine subspace of base dimension and verify that a given word is a codeword of the base code when restricted to the chosen subspace. We show that this test accepts codewords with probability one, while rejecting words at constant distance from the code with constant probability (depending only on the alphabet size). This work thus extends the results of Bhattacharyya et al. (FOCS 2010) and Haramaty et al. (FOCS 2011), while giving concrete new codes of higher rate that have absolutely sound testers.

Deterministic Compression with Uncertain Priors

Communication in “natural” settings, e.g., between humans, is distinctly different from that in classical designed settings, in that the former is invariably characterized by the sender and receiver not being in perfect agreement with each other. Solutions to classical communication problems thus have to overcome an extra layer of uncertainty introduced by this lack of prior agreement. One of the classical goals of communication is compression of information, and in this context lack of agreement implies that sender and receiver may not agree on the “prior” from which information is being generated. Most classical mechanisms for compressing turn out to be non-robust when sender and receiver do not agree on the prior. Juba et al. (Proc. ITCS 2011) showed that there do exists compression schemes with shared randomness between sender and receiver that do not share a prior that can compress information down roughly to its entropy. In this work, we explore the assumption of shared randomness between the sender and receiver and highlight why this assumption is problematic when dealing with natural communication. We initiate the study of deterministic compression schemes amid uncertain priors, and expose some of the mathematical facets of this problem. We show some non-trivial deterministic compression schemes, and some lower bounds on natural classes of compression schemes. We show that a full understanding of deterministic communication turns into challenging (open) questions in graph theory and communication complexity.

All-or-Nothing generalized assignment with application to scheduling advertising campaigns

We study a variant of the generalized assignment problem (gap) which we label all-or-nothing GAP (AGAP). We are given a set of items, partitioned into n groups, and a set of m bins. Each item l has size sl>0, and utility alj≥0 if packed in bin j. Each bin can accommodate at most one item from each group, and the total size of the items in a bin cannot exceed its capacity. A group of items is satisfied if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from satisfied groups is maximized. We motivate the study of agap by pointing out a central application in scheduling advertising campaigns. Our main result is an O(1)-approximation algorithm for agap instances arising in practice, where each group consists of at most m/2 items. Our algorithm uses a novel reduction of agap to maximizing submodular function subject to a matroid constraint. For agap instances with fixed number of bins, we develop a randomized polynomial time approximation scheme (PTAS), relying on a non-trivial LP relaxation of the problem. We present a (3+e)-approximation as well as PTASs for other special cases of agap, where the utility of any item does not depend on the bin in which it is packed. Finally, we derive hardness results for the different variants of agap studied in the paper.