# Subhash Khot's research while affiliated with New York University and other places

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## Publications (129)

We show improved monotonicity testers for the Boolean hypercube under the $p$-biased measure, as well as over the hypergrid $[m]^n$. Our results are: 1. For any $p\in (0,1)$, for the $p$-biased hypercube we show a non-adaptive tester that makes $\tilde{O}(\sqrt{n}/\varepsilon^2)$ queries, accepts monotone functions with probability $1$ and rejects...

The k-Clique problem is a canonical hard problem in parameterized complexity. In this paper, we study the parameterized complexity of approximating the k-Clique problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a clique of size at least k/F(k) whenever the graph G has a clique of size k. When such an...

Given an alphabet size $m\in\mathbb{N}$ thought of as a constant, and $\vec{k} = (k_1,\ldots,k_m)$ whose entries sum of up $n$, the $\vec{k}$-multi-slice is the set of vectors $x\in [m]^n$ in which each symbol $i\in [m]$ appears precisely $k_i$ times. We show an invariance principle for low-degree functions over the multi-slice, to functions over t...

We study the structure of non-expanding sets in the Grassmann graph. We put forth a hypothesis stating that every small set whose expansion is smaller than 1–δ must be correlated with one of a specified list of sets which are isomorphic to smaller Grassmann graphs. We develop a framework of Fourier analysis for analyzing functions over the Grassman...

Andoni, Krauthgamer, and Razenshteyn (AKR) proved (STOC 2015) that a finite-dimensional normed space (X, ∥⋅∥X) admits a O(1) sketching algorithm (namely, with O(1) sketch size and O(1) approximation) if and only if for every ε ∈ (0, 1), there exist α⩾1 and an embedding f : X → ℓ1−ε such that ∥x−y∥X⩽∥f(x)−f(y)∥1−ε⩽α∥x−y∥X for all x, y ∈ X. The “if p...

A seminal result of H\r{a}stad [J. ACM, 48(4):798--859, 2001] shows that it is NP-hard to find an assignment that satisfies $\frac{1}{|G|}+\varepsilon$ fraction of the constraints of a given $k$-LIN instance over an abelian group, even if there is an assignment that satisfies $(1-\varepsilon)$ fraction of the constraints, for any constant $\varepsi...

Andoni, Krauthgamer and Razenshteyn (AKR) proved (STOC 2015) that a finite-dimensional normed space $(X,\|\cdot\|_X)$ admits a $O(1)$ sketching algorithm (namely, with $O(1)$ sketch size and $O(1)$ approximation) if and only if for every $\varepsilon\in (0,1)$ there exist $\alpha\geqslant 1$ and an embedding $f:X\to \ell_{1-\varepsilon}$ such that...

We study the structure of non-expanding sets in the Grassmann graph. We put forth a hypothesis stating that every small set whose expansion is smaller than 1−δ must be correlated with one of a specified list of sets which are isomorphic to smaller Grassmann graphs. We develop a framework of Fourier analysis for analyzing functions over the Grassman...

We present a polynomial time reduction from gap-3LIN to label cover with 2-to-1 constraints. In the “yes” case the fraction of satisfied constraints is at least 1 −ε, and in the “no” case we show that this fraction is at most ε, assuming a certain (new) combinatorial hypothesis on the Grassmann graph. In other words, we describe a combinatorial hyp...

In the simultaneous Max-Cut problem, we are given $k$ weighted graphs on the same set of $n$ vertices, and the goal is to find a cut of the vertex set so that the minimum, over the $k$ graphs, of the cut value is as large as possible. Previous work [BKS15] gave a polynomial time algorithm which achieved an approximation factor of $1/2 - o(1)$ for t...

We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295-314]. As an application, we give a monotonicity testing algorithm that makes Õ(√n/ε²) nonadaptive queries to a function f : {0, 1}ⁿ 7→ {0, 1}, always accepts a monotone function, and rejects a function that is ε-far fro...

We present a candidate reduction from the 3-Lin problem to the 2-to-2 Games problem and present a combinatorial hypothesis about Grassmann graphs which, if correct, is sufficient to show the soundness of the reduction in a certain non-standard sense. A reduction that is sound in this non-standard sense implies that it is NP-hard to distinguish whet...

A Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is called a dictator if it depends on exactly one variable i.e $f(x_1, x_2, \ldots, x_n) = x_i$ for some $i\in [n]$. In this work, we study a $k$-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems. The dictatorship test is...

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 Harsha, H\aa stad, Srinivasan, and Varma showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors. Their re...

We present an adaptive tester for the unateness property of Boolean functions. Given a function $f:\{0,1\}^n \to \{0,1\}$ the tester makes $O(n \log(n)/\epsilon)$ adaptive queries to the function. The tester always accepts a unate function, and rejects with probability at least 0.9 if a function is $\epsilon$-far from being unate.

We propose a candidate reduction for ruling out polynomial-time algorithms for unique games, either under plausible complexity assumptions, or unconditionally for Lasserre semi-definite programs with a constant number of rounds. We analyze the completeness and Lasserre solution of our construction, and provide a soundness analysis in a certain sett...

In the Gap-clique (k, k/2) problem, the input is an n-vertex graph G, and the goal is to decide whether G contains a clique of size k or contains no clique of size k/2. It is an open question in the study of fixed parameterized tractability whether the Gap-clique (k, k/2) problem is fixed parameter tractable, i.e., whether it has an algorithm that...

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 \(\m...

We show that it is quasi-NP-hard to color 2-colorable 12-uniform hypergraphs with 2(logn}ω(1) colors where n is the number of vertices. Previously, Guruswami et al. [1] showed that it is quasi-NP-hard to color 2-colorable 8-uniform hypergraphs with 22ω &root;log log n) colors. Their result is obtained by composing a standard Outer PCP with an Inner...

A boolean predicate f:{0,1}
k
→ {0,1} is said to be somewhat approximation resistant if for some constant \(\tau > \frac{|f^{-1}(1)|}{2^k}\), given a τ-satisfiable instance of the MAX k-CSP(f) problem, it is NP-hard to find an assignment that strictly beats the naive algorithm that outputs a uniformly random assignment. Let τ(f) denote the supremum...

We prove that for an arbitrarily small constant ϵ>0, the preprocessing versions of the closest vector problem and the nearest codeword problem are hard to approximate within a factor 2 log 1-ϵ n , under the assumption that NP ⊈ SIZE (2 log O(1/ϵ) n ).

This work studies the hardness of finding independent sets in hypergraphs
which are either 2-colorable or are almost 2-colorable, i.e. can be 2-colored
after removing a small fraction of vertices and the incident hyperedges. To be
precise, say that a hypergraph is (1-eps)-almost 2-colorable if removing an eps
fraction of its vertices and all hypere...

For a predicate f: {-1, 1}k ↦ {0, 1} with ρ(f) = |f⁻¹(1)|/2k, we call the predicate strongly approximation resistant if given a near-satisfiable instance of CSP(f), it is computationally hard to find an assignment such that the fraction of constraints satisfied is outside the range [ρ(f) - Ω(1), ρ(f) + Ω(1)].
We present a characterization of strong...

In this paper, we disprove a conjecture of Goemans and Linial; namely, that
every negative type metric embeds into $\ell_1$ with constant distortion. We
show that for an arbitrarily small constant $\delta> 0$, for all large enough
$n$, there is an $n$-point negative type metric which requires distortion at
least $(\log\log n)^{1/6-\delta}$ to embed...

In the kernel clustering problem we are given a (large) $n\times n$ symmetric positive semidefinite matrix $A=(a_{ij})$ with $\sum_{i=1}^n\sum_{j=1}^n a_{ij}=0$ and a (small) $k\times k$ symmetric positive semidefinite matrix $B=(b_{ij})$. The goal is to find a partition $\{S_1,...,S_k\}$ of $\{1,... n\}$ which maximizes $ \sum_{i=1}^k\sum_{j=1}^k...

A constraint satisfaction problem (CSP) is said to be \emph{approximation
resistant} if it is hard to approximate better than the trivial algorithm which
picks a uniformly random assignment. Assuming the Unique Games Conjecture, we
give a characterization of approximation resistance for $k$-partite CSPs
defined by an even predicate.

We construct a PCP based on the hyper-graph linearity test with 3 free queries. It has near-perfect completeness and soundness strictly less than 1/8. Such a PCP was known before only assuming the Unique Games Conjecture, albeit with soundness arbitrarily close to 1/16. At a technical level, our main contribution is constructing a new outer PCP whi...

We show that for any ε >; 0, and positive integers k and q such that q ≥ 2k + 1, given a graph on N vertices that has a q-colorable induced subgraph of (1 - ε)N vertices, it is NP-hard to find an independent set of N/qk+1 vertices. This substantially improves upon the work of Dinur et al. [1] who gave a corresponding bound of N/q2. Our result impli...

We survey connections of the Grothendieck inequality and its variants to
combinatorial optimization and computational complexity.

We prove that for an arbitrarily small constant ε>0, assuming NP⊈ DTIME (2logO 1-ε n), the preprocessing versions of the closest vector problem and the nearest codeword problem are hard to approximate within a factor better than 2log1-ε n. This improves upon the previous hardness factor of (log n)δ for some δ>0 due to [AKKV05].

We show that for any fixed prime q ≥ 5 and constant ζ >; 0, it is NP-hard to distinguish whether a two prover one round game with q<sup>6</sup> answers has value at least 1 - ζ or at most 4/q. The result is obtained by combining two techniques: (i) An Inner PCP based on the point versus subspace test for linear functions. The test is analyzed Fouri...

We prove that for an arbitrarily small constant $\eps>0,$ assuming NP$\not
\subseteq$DTIME$(2^{{\log^{O(1/\eps)} n}})$, the preprocessing versions of the
closest vector problem and the nearest codeword problem are hard to approximate
within a factor better than $2^{\log ^{1-\eps}n}.$ This improves upon the
previous hardness factor of $(\log n)^\del...

We consider the combinatorial problem of embedding the metric defined by an unweighted graph into the real line, so as to minimize the distortion of the embedding. This problem is inspired by connections to Banach space theory and to computer science. After establishing a framework in which to study line embeddings, we focus on metrics defined by t...

We show that unless NP=RP, it is hard to (even) weakly PAC-learn intersection of two halfspaces in Rn using a hypothesis which is a function of up to ℓ halfspaces (linear threshold functions) for any integer ℓ. Specifically, we show that for every integer ℓ and an arbitrarily small constant ε>0, unless NP=RP, no polynomial time algorithm can distin...

In this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables. Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be "non-trivial". Here is an informal statement of our result: it is NP-ha...

For every ∈ > 0, and integer q ≥ 3, we show that given an N-vertex graph that has an induced q-colorable subgraph of size (1 - ∈)N, it is NP-hard to find an independent set of size N/q<sup>2</sup>.

We present a simple deterministic gap-preserving reduction from SAT to the Minimum Distance of Code Problem over \(\mathbb{F}_2\). We also show how to extend the reduction to work over any finite field (of constant size). Previously a randomized reduction was known due to Dumer, Micciancio, and Sudan [9], which was recently derandomized by Cheng an...

In this paper we present semidefinite programming (SDP) gap instances for the following variants of the Label-Cover problem, closely related to the Unique Games Conjecture: (i) 2-to-1 Label-Cover; (ii) 2-to-2 Label-Cover; (iii) α-constraint Label-Cover. All of our gap instances have perfect SDP solutions. For alphabet size K, the integral optimal s...

Assuming the Unique Games Conjecture (UGC), we show optimal inapproximability results for two classic scheduling problems.
We obtain a hardness of 2 − ε for the problem of minimizing the total weighted completion time in concurrent open shops. We also obtain a hardness of 2 − ε for minimizing the makespan in the assembly line problem.
These result...

This article surveys recently discovered connections between the Unique Games Conjecture and computational complexity, algorithms, discrete Fourier analysis, and geometry.

These are the lecture notes for the DIMACS Tutorial "Limits of Approximation Algorithms: PCPs and Unique Games" held at the DIMACS Center, CoRE Building, Rutgers University on 20-21 July, 2009. This tutorial was jointly sponsored by the DIMACS Special Focus on Hardness of Approximation, the DIMACS Special Focus on Algorithmic Foundations of the Int...

This article gives a survey of recent results that connect three areas in com-puter science and mathematics: (1) (Hardness of) computing approximate solutions to NP-complete problems. (2) Fourier analysis of boolean functions on boolean hypercube. (3) Certain problems in geometry, especially related to isoperimetry and embeddings between metric spa...

In this paper, we consider the problem of approximately solving a system of homogeneous
linear equations over reals, where each equation contains at most three variables.
Since the all-zero assignment always satisfies all the equations exactly, we restrict the
assignments to be “non-trivial”. Here is an informal statement of our result: it is NP-ha...

We study the polynomial reconstruction problem for low-degree multivariate polynomials over \mathbb{F}\left[ 2 \right]. In this problem, we are given a set of points {\rm X} \in \left\{ {0,1} \right\}^n and target values f({\rm X}) \in \left\{ {0,1} \right\} for each of these points, with the promise that there is a polynomial over \mathbb{F}\left[...

In this paper, we investigate whether a constant round Lasserre Semi-definite Programming (SDP) relaxation might give a good approximation to the Unique Games problem. We show that the answer is negative if the relaxation is insensitive to a sufficiently small perturbation of the constraints. Specifically, we construct an instance of Unique Games w...

We construct integrality gap instances for SDP relaxation of the MAXIMUM CUT and the SPARSEST CUT problems. If the triangle inequality constraints are added to the SDP, then the SDP vectors naturally define an n-point negative type metric where n is the number of vertices in the problem instance. Our gap-instances satisfy a stronger constraint that...

For arbitrarily small constants epsilon, delta ¿.¿ > 0, we present a long code test with one free bit, completeness 1-epsilon and soundness delta. Using the test, we prove the following two inapproximability results:1. Assuming the Unique Games Conjecture of Khot, given an n-vertex graph that has two disjoint independent sets of size (1/2-¿)n each,...

We study the inapproximability of Vertex Cover and Independent Set on degree d graphs. We prove that: • Vertex Cover is Unique Games-hard to approximate to within a factor 2-(2+od(1))/log log d/log d . This exactly matches the algorithmic result of Halperin [1] up to the o d(1) term. • Independent Set is Unique Games-hard to approximate to within a...

In this article, we present a survey of known inapproximability results for computational problems on lattices, viz. the Shortest
Vector Problem (SVP), the Closest Vector Problem (CVP), the Closest Vector Problem with Preprocessing (CVPP), the Covering
Radius Problem (CRP), the Shortest Independent Vectors Problem (SIVP), and the Shortest Basis Pro...

We study the learnability of several fundamental concept classes in the agnostic learning framework of Haussler [Hau92] and Kearns et al. [KSS94]. We show that under the uniform distribution, agnostically learning parities reduces to learning pari-ties with random classification noise, commonly referred to as the noisy parity problem. Together with...

Given a graph with maximum cut of (fractional) size c, the semidefinite programming (SDP)-based algorithm of Goemans and Williamson is guaranteed to find a cut of size at least ·878·c. However, this guarantee becomes trivial when c is near 1 2, since making random cuts guarantees a cut of size 1 2 (i.e., half of all edges). A few years ago, Charika...

We study the fundamental classification problems 0-Extension and Metric Labeling. 0-Extension is closely related to partitioning problems in graph theory and to Lip-schitz extensions in Banach spaces; its generalization Metric Labeling is motivated by applications in computer vision. Researchers had proposed using earthmover metrics to get polynomi...

We study the problem of finding the minimum size DNF formula for a function f : {0, 1}<sup>d</sup> rarr {0,1} given its truth table. We show that unless NP sube DTIME(n<sup>poly(log</sup> <sup>n)</sup>), there is no polynomial time algorithm that approximates this problem to within factor d<sup>1-epsiv</sup> where epsiv > 0 is an arbitrarily small...

We consider the following allocation problem arising in the setting of combinatorial auctions: a set of goods is to be allocated
to a set of players so as to maximize the sum of the utilities of the players (i.e., the social welfare). In the case when
the utility of each player is a monotone submodular function, we prove that there is no polynomial...

In the kernel clustering problem we are given a large $n\times n$ positive semi-definite matrix $A=(a_{ij})$ with $\sum_{i,j=1}^na_{ij}=0$ and a small $k\times k$ positive semi-definite matrix $B=(b_{ij})$. The goal is to find a partition $S_1,...,S_k$ of $\{1,... n\}$ which maximizes the quantity $$ \sum_{i,j=1}^k (\sum_{(i,j)\in S_i\times S_j}a_{...

We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander.
We introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique games when the graph is an expander.

We show that unless NP = RP, it is hard to (even) weakly PAC-learn intersection of two halfspaces in Rn using a hy- pothesis which is a function of up to ' linear threshold func- tions for any integer '. Specically, we show that for every integer ' and an arbitrarily small constant " > 0, unless NP = RP, no polynomial time algorithm can distinguish...

Based on a conjecture regarding the power of unique 2-prover-1-round games presented in [S. Khot, On the power of unique 2-Prover 1-Round games, in: Proc. 34th ACM Symp. on Theory of Computing, STOC, May 2002, pp. 767-775], we show that vertex cover is hard to approximate within any constant factor better than 2 we actually show a stronger result,...

We present an efficient algorithm to find a good solution to the Unique Games problem when the constraint graph is an expander. We introduce a new analysis of the standard SDP in this case that involves correlations among distant vertices. It also leads to a parallel repetition theorem for unique games when the graph is an expander.

We consider the problem of minimizing regret with respect to a given set S of pairs of time selection functions and modifications rules. We give an on- line algorithm that has O( p T log|S|) regret with respect to S when the algorithm is run for T time steps and there are N actions allowed. This im- proves the upper bound of O( p TN log(|I||F|)) gi...

We study the polynomial reconstruction problem, for low-degree multivariate polynomials over F[2]. In this problem, we are given a set of points x epsi {0, 1}<sup>n</sup> and target values f(x) epsi {0, 1} for each of these points, with the promise that there is a polynomial over F[2] of degree at most d that agrees with f at 1 - epsiv fraction of...

We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A={aijk}ij,k=1n such that for all i,j,kisin{1,...,n} we have aijk=aikj=akji=ajik=akij=akji and aiik=aijj=aiji=0, computes a number Alg(A) which satisfies with probability at least 1/2, Omega(radic(logn/n))ldrmaxxisin{-1,1}n Sigmai,j,k=1naijkxixjxklesAlg(A)lesma...

We prove a lower bound of Ω(n4/3 log 1/3n) on the randomized decision tree complexity of any nontrivial monotone n-vertex graph property, and of any nontrivial monotone bipartite graph property with bipartitions of size n. This improves the previous best bound of Ω(n4/3) due to Hajnal (Combinatorica 11 (1991) 131–143). Our proof works by improving...

We study the problem embedding an n-point metric space into another n-point metric space while minimizing distortion. We show that there is no polynomial time algorithm to approximate the minimum distortion within a factor of Ω((logn)1/4 − δ
) for any constant δ> 0, unless \(\textnormal{NP} \subseteq \textnormal{DTIME}(n^{\textnormal{poly}(\log n))...

The Max-Min allocation problem is to distribute indivisible goods to people so as to maximize the minimum utility of the people.
We show a (2k − 1)-approximation algorithm for Max-Min when there are k people with subadditive utility functions. We also give a k/α-approximation algorithm (for α ≤ k/2) if the utility functions are additive and the uti...

We address well-studied problems concerning the learnability of parities and halfspaces in the presence of classification noise. Learning of parities under the uniform distribution with random classification noise, also called the noisy parity problem is a famous open problem in computational learning. We reduce a number of basic problems regarding...

Given a graph with maximum cut of (fractional) size c, the Goemans-Williamson (GW95) semideflnite programming (SDP) algorithm is guaranteed to flnd a cut of size :878 ¢ c. However this guarantee becomes trivial when c is near 1=2, since making random cuts always guarantees a cut of size 1=2. Recently, Charikar and Worth (CW04) (analyzing an algorit...

We prove an improved hardness of approximation result for two problems, namely, the problem of finding the size of the largest clique in a graph and the problem of finding the chromatic number of a graph. We show that for any constant γ> 0, there is no polynomial time algorithm that approximates these problems within factor \(n/2^{(\log n)^{3/4+\ga...

We show that for every ffl? 0, there is a constant p(ffl) such that for all integers p * p(ffl), it isNP-hard to approximate the Shortest Vector Problem in Lp norm within factor p1\Gamma ffl under randomizedreductions. For large values of p, this improves the factor 21=p \Gamma ffi hardness shown by Micciancio [27]. 1 1 Introduction An n-dimensiona...

Assuming that NP ⊈∩ ϵ>0 BPTIME(2 n ϵ ), we show that graph min-bisection, dense k-subgraph, and bipartite clique have no Polynomial Time Approximation Scheme (PTAS). We give a reduction from the Minimum Distance of Code (MDC) problem. Starting with an instance of MDC, we build a quasi-random Probabilistically Checkable Proof (PCP) that suffices to...

Arora, Rao and Vazirani (2) showed that the standard semi-denite programming ( SDP) relaxation of the Sparsest Cut problem with the triangle inequality constraints has an inte- grality gap of O( p logn). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture (referred to as the ARV-Conjecture)...

Abstract We study the classi cation problem Metric Labeling and its special case 0 - Extension in the context of earthmover metrics Researchers recently proposed using earthmover metrics to get a polynomial time - solvable relaxation of Metric Labeling; until now, however, no one knew if the integrality ratio was constant or not, for either Metric...

We study a very basic open problem regarding the PCP characterization of NP, namely, the power of PCPs with 3 non-adaptive queries and perfect completeness. The lowest soundness known till now for such a PCP is 6/8 + epsi given by a construction of Hastad (1997). However, Zwick (1998) shows that a 3-query non-adaptive PCP with perfect completeness...

We show that the Minimum Linear Arrangement problem is hard to approximate to within any constant factor. The result is conditional and relies on a strengthening of the Unique Games Conjecture.

We consider the following allocation problem arising in the setting of combinatorial auctions: a set of goods is to be allocated
to a set of players so as to maximize the sum of the utilities of the players (i.e., the social welfare). In the case when
the utility of each player is a monotone submodular function, we prove that there is no polynomial...

We show that, unless NP⊆DTIME(2<sup>poly log(n)</sup>) the closest vector problem with pre-processing, for ℓ<sub>p</sub> norm for any p ≥ 1, is hard to approximate within a factor of (log n)<sup>1</sup>p - ε/' /P for any ε > 0. This improves the previous best factor of 3<sup>1</sup>p/ - ε due to Regev (2004). Our results also imply t...

Summary form only given. The discovery of the PCP theorem in 1992 led to an avalanche of hardness of approximation results, i.e. results showing that for certain NP hard optimization problems, computing even approximate solutions is hard. However, for many fundamental problems, obtaining satisfactory hardness results seems out of reach of current t...

Various new nonembeddability results (mainly into L
1) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on {0,1}
d
has L
1 distortion
We also give new lower bounds on the L
1 distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation cost (Earthmover) metric.

Various new nonembeddability results (mainly into $L_1$) are proved via Fourier analysis. In particular, it is shown that the Edit Distance on $\{0,1\}^d$ has $L_1$ distortion $(\log d)^{\frac12-o(1)}$. We also give new lower bounds on the $L_1$ distortion of flat tori, quotients of the discrete hypercube under group actions, and the transportation...

Let p > 1b ea ny fixed real. We show that assuming NP �⊆ RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in � p norm within a constant factor. Under the stronger assumption NP �⊆ RTIME(2,/2−� . Categories and Subject Descriptors: F.2 [Theory of Computation]: Analysis of Algorithms and Problem Complexity...

We study the complexity of approximating Max NM-E3SAT, a variant of Max 3SAT when the instances are guaranteed to not have any mixed clauses, i.e., every clause has either all its literals unnegated or all of them negated. This is a natural special case of Max 3SAT introduced Guruswami (2004), where the question of whether this variant can be appro...

Summer has come again. And what better way is there to spend a summer than to relax on a sandy beach, on a mountain top, or at a park's picnic tables, and... think theory! Summer is a particularly good time to attack the big questions whose openness just plain annoys you. In light of Reingold's L = SL result, does L-vs.-RL tempt you? If so, take it...

We study the fundamental classification problems O-EXTENSION and METRIC LABELING. MINIMUM WEIGHT TRIANGULATION is closely related to partitioning problems in graph theory and to Lipschitz extensions in Banach spaces; its generalization METRIC LABELING is motivated by applications in computer vision. Researchers had proposed using earthmover metrics...

In this paper we give evidence suggesting that MAX- CUT is NP-hard to approximate to within a factor of GW+ , for all > 0, where GW denotes the approximation ra- tio achieved by the Goemans-Williamson algorithm (14), GW :878567. This result is conditional, relying on two conjectures: a) the Unique Games conjecture of Khot (24); and, b) a very belie...

In this paper, we give evidence suggesting that MAX-CUT is NP-hard to approximate to within a factor of α<sub>cw</sub>+ ε, for all ε > 0, where α<sub>cw</sub> denotes the approximation ratio achieved by the Goemans-Williamson algorithm (1995). α<sub>cw</sub> ≈ .878567. This result is conditional, relying on two conjectures: a) the unique...

Let p > 1 be any fixed real. We show that assuming NP
⊄
‾ RP, it is hard to approximate the shortest vector problem (SVP) in l<sub>p</sub> norm within an arbitrarily large constant factor. Under the stronger assumption NP
⊄
‾ RTIME(2<sup>poly(log n)</sup>), we show that the problem is hard to approximate within factor 2<sup>(log n)<sup>1/2 - ε</s...

Assuming that NP
⊄
‾ ∩<sub>ε > 0</sub> BPTIME(2<sup>n<sup>ε</sup></sup>), we show that graph min-bisection, densest subgraph and bipartite clique have no PTAS. We give a reduction from the minimum distance of code problem (MDC). Starting with an instance of MDC, we build a quasi-random PCP that suffices to prove the desired inapproximability resul...

## Citations

... By letting A = B in the definition of a PCSP, one obtains the standard (non-promise) constraint satisfaction problem (CSP) [39]. PCSPs were introduced by Austrin, Guruswami, and Håstad [5] and Brakensiek and Guruswami [16] as a general framework for studying approximability of perfectly satisfiable CSPs and have emerged as a new exciting direction in constraint satisfaction that requires different techniques than CSPs. 2 Recent works on PCSPs include those using analytical methods [12,13,17,22] and those building on algebraic methods [3,7,10,15,18,19,26,31,45,63] developed in [8]. However, most basic questions are still left open, including complexity classifications and applicability of different types of algorithms. ...