We introduce the notion of covering complexity of a probabilistic
verifier. The covering complexity of a verifier on a given input is
the minimum number of proofs needed to ``satisfy'' the verifier on
every random string, i.e., on every random string, at least one of the
given proofs must be ... more >>>

In the {\sc $k$-center} problem, the input is a bound $k$
and $n$ points with the distance between every two of them,
such that the distances obey the triangle inequality.
The goal is to choose a set of $k$ points to serve as centers,
so that the maximum distance ... more >>>

We study the approximate-coloring(q,Q) problem: Given a graph G, decide
whether \chi(G) \le q or \chi(G)\ge Q. We derive conditional
hardness for this problem for any constant 3\le q < Q. For q \ge
4, our result is based on Khot's 2-to-1 conjecture [Khot'02].
For q=3, we base our hardness ... more >>>

A randomness extractor is an algorithm which extracts randomness from a low-quality random source, using some additional truly random bits. We construct new extractors which require only log n + O(1) additional random bits for sources with constant entropy rate. We further construct dispersers, which are similar to one-sided extractors, ... more >>>

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of $\GW + \eps$, for all $\eps > 0$; here $\GW \approx .878567$ denotes the approximation ratio achieved by the Goemans-Williamson algorithm~\cite{GW95}. This implies that if the Unique ... more >>>

For circuit classes R, the fundamental computational problem, Min-R,
asks for the minimum R-size of a boolean function presented as a truth
table. Prominent examples of this problem include Min-DNF, and
Min-Circuit (also called MCSP). We begin by presenting a new reduction
proving that Min-DNF is NP-complete. It is significantly ... more >>>

We consider the problem of finding a monomial (or a term) that maximizes the agreement rate with a given set of examples over the Boolean hypercube. The problem originates in learning and is referred to as {\em agnostic learning} of monomials. Finding a monomial with the highest agreement rate was ... more >>>

Given a set of monomials, the Minimum AND-Circuit problem asks for a
circuit that computes these monomials using AND-gates of fan-in two and
being of minimum size. We prove that the problem is not polynomial time
approximable within a factor of less than 1.0051 unless P = NP, even if
more >>>

We investigate non-parametric unit-demand pricing problems, in which the goal is to find revenue maximizing prices for a set of products based on consumer profiles obtained, e.g., from an e-Commerce website. A consumer profile consists of a number of non-zero budgets and a ranking of all the products the consumer ... more >>>

We show that, assuming the Unique Games Conjecture, it is NP-hard to approximate Max 2-Sat within $\alpha_{LLZ}^{-}+\epsilon$, where $0.9401 < \alpha_{LLZ}^{-} < 0.9402$ is the believed approximation ratio of the algorithm of Lewin, Livnat and Zwick.

This result is surprising considering the fact that balanced instances of Max 2-Sat, i.e. ... more >>>

A cycle cover of a graph is a set of cycles such that every vertex is
part of exactly one cycle. An L-cycle cover is a cycle cover in which
the length of every cycle is in the set L. The weight of a cycle cover
of an edge-weighted graph ... more >>>

We give improved inapproximability results for some minimization problems in the second level of the Polynomial-Time Hierarchy. Extending previous work by Umans [Uma99], we show that several variants of DNF minimization are $\Sigma_2^p$-hard to approximate to within factors of $n^{1/3-\epsilon}$ and $n^{1/2-\epsilon}$ (where the previous results achieved $n^{1/4 - \epsilon}$), ... more >>>

We study constraint satisfaction problems on the domain $\{-1,1\}$, where the given constraints are homogeneous linear threshold predicates. That is, predicates of the form $\mathrm{sgn}(w_1 x_1 + \cdots + w_n x_n)$ for some positive integer weights $w_1, \dots, w_n$. Despite their simplicity, current techniques fall short of providing a classification ... more >>>

The Minimum Distance Problem (MDP), i.e., the computational task of evaluating (exactly or approximately) the minimum distance of a linear code, is a well known NP-hard problem in coding theory. A key element in essentially all known proofs that MDP is NP-hard is the construction of a combinatorial object that ... more >>>

We study two natural extensions of Constraint Satisfaction Problems (CSPs). {\em Balance}-Max-CSP requires that in any feasible assignment each element in the domain is used an equal number of times. An instance of {\em Hard}-Max-CSP consists of {\em soft constraints} and {\em hard constraints}, and the goal is to maximize ... more >>>

We show that it is quasi-NP-hard to color $2$-colorable $12$-uniform hypergraphs with $2^{(\log n)^{\Omega(1) }}$ colors where $n$ is the number of vertices. Previously, Guruswami et al. [GHHSV14] showed that it is quasi-NP-hard to color $2$-colorable $8$-uniform hypergraphs with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors. Their result is obtained by composing a ... more >>>

We show that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{(\log N)^{1/4-o(1)}}$ colors, where $N$ is the number of vertices. There has been much focus on hardness of hypergraph coloring recently. Guruswami, H{\aa}stad, Harsha, Srinivasan and Varma showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with ... more >>>

We present in this paper some of the recent techniques and methods for proving best up to now explicit approximation hardness bounds for metric symmetric and asymmetric Traveling Salesman Problem (TSP) as well as related problems of Shortest Superstring and Maximum Compression. We attempt to shed some light on the ... more >>>

This paper studies how well the standard LP relaxation approximates a $k$-ary constraint satisfaction problem (CSP) on label set $[L]$. We show that, assuming the Unique Games Conjecture, it achieves an approximation within $O(k^3\cdot \log L)$ of the optimal approximation factor. In particular we prove the following hardness result: let ... more >>>

We give a family of dictatorship tests with perfect completeness and low-soundness for 2-to-2 constraints. The associated 2-to-2 conjecture has been the basis of some previous inapproximability results with perfect completeness. However, evidence towards the conjecture in the form of integrality gaps even against weak semidefinite programs has been elusive. ... more >>>

A hypergraph is $k$-rainbow colorable if there exists a vertex coloring using $k$ colors such that each hyperedge has all the $k$ colors. Unlike usual hypergraph coloring, rainbow coloring becomes harder as the number of colors increases. This work studies the rainbow colorability of hypergraphs which are guaranteed to be ... more >>>

We prove that pseudorandom sets in Grassmann graph have near-perfect expansion as hypothesized in [DKKMS-2]. This completes
the proof of the $2$-to-$2$ Games Conjecture (albeit with imperfect completeness) as proposed in [KMS, DKKMS-1], along with a
contribution from [BKT].

The Grassmann graph $Gr_{global}$ contains induced subgraphs $Gr_{local}$ that are themselves ... more >>>

In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets $A$ and $B$ of vectors, and the goal is to find $a \in A$ and $b \in B$ maximizing inner product $a \cdot b$. Max-IP is very basic and serves ... more >>>

The $k$-Even Set problem is a parameterized variant of the Minimum Distance Problem of linear codes over $\mathbb F_2$, which can be stated as follows: given a generator matrix $\mathbf A$ and an integer $k$, determine whether the code generated by $\mathbf A$ has distance at most $k$. Here, $k$ ... more >>>

The $d$-to-$1$ conjecture of Khot asserts that it is hard to satisfy an $\epsilon$ fraction of constraints of a satisfiable $d$-to-$1$ Label Cover instance, for arbitrarily small $\epsilon > 0$. We prove that the $d$-to-$1$ conjecture for any fixed $d$ implies the hardness of coloring a $4$-colorable graph with $C$ ... more >>>

The problem of learning $t$-term DNF formulas (for $t = O(1)$) has been studied extensively in the PAC model since its introduction by Valiant (STOC 1984). A $t$-term DNF can be efficiently learnt using a $t$-term DNF only if $t = 1$ i.e., when it is an AND, while even ... more >>>

Finite languages lie at the heart of literally every regular expression. Therefore, we investigate the approximation complexity of minimizing regular expressions without Kleene star, or, equivalently, regular expressions describing finite languages. On the side of approximation hardness, given such an expression of size~$s$, we prove that it is impossible to ... more >>>

A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal ... more >>>

A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints ${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from ${\cal F}$ to subsequences of the $n$ ... more >>>

An ordering constraint satisfaction problem (OCSP) is given by a positive integer $k$ and a constraint predicate $\Pi$ mapping permutations on $\{1,\ldots,k\}$ to $\{0,1\}$. Given an instance of OCSP$(\Pi)$ on $n$ variables and $m$ constraints, the goal is to find an ordering of the $n$ variables that maximizes the number ... more >>>

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on $n$ variables taking values in $\{0,\ldots,q-1\}$, we prove that improving over the trivial approximability by a factor of $q$ requires $\Omega(n)$ space even on instances with $O(n)$ constraints. We also identify ... more >>>

We study a natural and quite general model of branch-and-bound algorithms. In this model, an algorithm attempts to minimize (or maximize) a function $f : D \to \mathbb{R}_{\geq 0}$ by making oracle queries to a heuristic $h_f$ satisfying
\[
\min_{x \in S} f(x) \leq h_f(S) \leq \gamma \cdot ... more >>>

We initiate a study of the streaming complexity of constraint satisfaction problems (CSPs) when the constraints arrive in a random order. We show that there exists a CSP, namely Max-DICUT, for which random ordering makes a provable difference. Whereas a $4/9 \approx 0.445$ approximation of DICUT requires $\Omega(\sqrt{n})$ space with ... more >>>

We say that two given polynomials $f, g \in R[x_1, \ldots, x_n]$, over a ring $R$, are equivalent under shifts if there exists a vector $(a_1, \ldots, a_n)\in R^n$ such that $f(x_1+a_1, \ldots, x_n+a_n) = g(x_1, \ldots, x_n)$. This is a special variant of the polynomial projection problem in Algebraic ... more >>>

We establish strong inapproximability for finding the sparsest nonzero vector in a real subspace (where sparsity refers to the number of nonzero entries). Formally we show that it is NP-Hard (under randomized reductions) to approximate the sparsest vector in a subspace within any constant factor. By simple tensoring the inapproximability ... more >>>