We give simple deterministic reductions demonstrating the NP-hardness of approximating the nearest codeword problem and minimum distance problem within arbitrary constant factors (and almost-polynomial factors assuming NP cannot be solved in quasipolynomial time). The starting point is a simple NP-hardness result without a gap, and is thus "PCP-free." Our approach is inspired by that of Bhattiprolu and Lee [BL24] who give a PCP-free randomized reduction for similar problems over the integers and the reals. We leverage the existence of ε-balanced codes to derandomize and further simplify their reduction for the case of finite fields.

Finding sparse vectors is a fundamental problem that arises in several contexts including codes, subspaces, and lattices. In this work, we prove strong inapproximability results for all these variants using a novel approach that even bypasses the PCP theorem. Our main result is that it is NP-hard (under randomized reductions) to approximate the sparsest vector in a real subspace within any constant factor; the gap can be further amplified using tensoring. Our reduction has the property that there is a Boolean solution in the completeness case. As a corollary, this immediately recovers the state-of-the-art inapproximability factors for the shortest vector problem (SVP) on lattices. Our proof extends the range of l_p (quasi) norms for which hardness was previously known, from 'p at least one' to 'p at least zero', answering a question raised by (Khot, JACM 2005). Previous hardness results for SVP, and the related minimum distance problem (MDP) for error-correcting codes, all use lattice/coding gadgets that have an abundance of codewords in a ball of radius smaller than the minimum distance. In contrast, our reduction only needs many codewords in a ball of radius slightly larger than the minimum distance. This enables an easy derandomization of our reduction for finite fields, giving a new elementary proof of deterministic hardness for MDP. We believe this weaker density requirement might offer a promising approach to showing deterministic hardness of SVP, a long elusive goal. The key technical ingredient underlying our result for real subspaces is a proof that in the kernel of a random Rademacher matrix, the support of any two linearly independent vectors have very little overlap. A broader motivation behind this work is the development of inapproximability techniques for problems over the reals. Analytic variants of sparsest vector have connections to small set expansion, quantum separability and polynomial maximization over convex sets, all of which appear to be out of reach of current PCP techniques. We hope that the approach we develop could enable progress on some of these problems.

We study the problem of computing the p \rightarrow q norm of a matrix A \in Rm\times n, defined as \| A\| p\rightarrow q = maxx\in Rn\setminus \{ 0\} \| x\| p. This problem generalizes the spectral norm of a matrix (p = q = 2) and the Grothendieck problem (p = \infty , q = 1) and has been widely studied in various regimes. When p \geq q, the problem exhibits a dichotomy: constant factor approximation algorithms are known if 2 \in [q, p], and the problem is hard to approximate within almost polynomial factors when 2 \in/ [q, p]. The regime when p < q, known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with p = 2 and q > 2 was studied by 307--326], who gave subexponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the exponential time hypothesis. However, no NP-hardness of approximation is known for these problems for any p < q. We prove the first NP-hardness result (under randomized reductions) for approximating hypercontractive norms. We show that for any 1 < p < q < \infty with 2 \in/ [p, q], \| A\| p\rightarrow q is hard to approximate within 2O((log n)1 - \epsilon ) assuming NP \subseteq \not BPTIME(2(logn)O(1) ). En route to the above result, we also prove almost tight results for the case when p \geq q with 2 \in [q, p].

Abstract. We study the problem of computing the [Formula: see text] norm of a matrix [Formula: see text], defined as [Formula: see text]. This problem generalizes the spectral norm of a matrix ([Formula: see text]) and the Grothendieck problem ([Formula: see text], [Formula: see text]) and has been widely studied in various regimes. When [Formula: see text], the problem exhibits a dichotomy: constant factor approximation algorithms are known if [Formula: see text], and the problem is hard to approximate within almost polynomial factors when [Formula: see text]. The regime when [Formula: see text], known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with [Formula: see text] and [Formula: see text] was studied by Barak et al. [ Proceedings of the 44 th Annual ACM Symposium on Theory of Computing, 2012, pp. 307–326], who gave subexponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the exponential time hypothesis. However, no NP-hardness of approximation is known for these problems for any [Formula: see text]. We prove the first NP-hardness result (under randomized reductions) for approximating hypercontractive norms. We show that for any [Formula: see text] with [Formula: see text], [Formula: see text] is hard to approximate within [Formula: see text] assuming [Formula: see text]. En route to the above result, we also prove almost tight results for the case when [Formula: see text] with [Formula: see text].

Grothendieck’s inequality [Gro53] states that there is an absolute constant K > 1 such that for any n × n matrix A

We investigate the approximability of the following optimization problem. The input is an n × n matrix A =( A ij ) with real entries and an origin-symmetric convex body K ⊂ ℝ n that is given by a membership oracle. The task is to compute (or approximate) the maximum of the quadratic form ∑ i =1 n ∑ j =1 n A ij x i x j =⟨ x , Ax ⟩ as x ranges over K . This is a rich and expressive family of optimization problems; for different choices of matrices A and convex bodies K it includes a diverse range of optimization problems like max-cut, Grothendieck/non-commutative Grothendieck inequalities, small set expansion and more. While the literature studied these special cases using case-specific reasoning, here we develop a general methodology for treatment of the approximability and inapproximability aspects of these questions. The underlying geometry of K plays a critical role; we show under commonly used complexity assumptions that polytime constant-approximability necessitates that K has type-2 constant that grows slowly with n . However, we show that even when the type-2 constant is bounded, this problem sometimes exhibits strong hardness of approximation. Thus, even within the realm of type-2 bodies, the approximability landscape is nuanced and subtle. However, the link that we establish between optimization and geometry of Banach spaces allows us to devise a generic algorithmic approach to the above problem. We associate to each convex body a new (higher dimensional) auxiliary set that is not convex, but is approximately convex when K has a bounded type-2 constant. If our auxiliary set has an approximate separation oracle, then we design an approximation algorithm for the original quadratic optimization problem, using an approximate version of the ellipsoid method. Even though our hardness result implies that such an oracle does not exist in general, this new question can be solved in specific cases of interest by implementing a range of classical tools from functional analysis, most notably the deep factorization theory of linear operators. Beyond encompassing the scenarios in the literature for which constant-factor approximation algorithms were found, our generic framework implies that that for convex sets with bounded type-2 constant, constant factor approximability is preserved under the following basic operations: (a) Subspaces, (b) Quotients, (c) Minkowski Sums, (d) Complex Interpolation. This yields a rich family of new examples where constant factor approximations are possible, which were beyond the reach of previous methods. We also show (under commonly used complexity assumptions) that for symmetric norms and unitarily invariant matrix norms the type-2 constant nearly characterizes the approximability of quadratic maximization.

A number of recent works have studied algorithms for entrywise l(p)-low rank approximation, namely algorithms which given an n x d matrix A (with n >= d), output a rank-k matrix B minimizing parallel to A - B parallel to(p)(p) = Sigma(i,j) vertical bar Ai,j - B-i,B-j vertical bar(p) when p > 0; and parallel to A - B parallel to(0) = Sigma(i,j) [A(i,j) not equal B-i,B-j] for p = 0, where [.] is the Iverson bracket, that is, parallel to A - B parallel to(0) denotes the number of entries (i; j) for which Ai,j not equal B-i,B-j. For p = 1, this is often considered more robust than the SVD, while for p = 0 this corresponds to minimizing the number of disagreements, or robust PCA. This problem is known to be NP-hard for p is an element of {0, 1}, already for k = 1, and while there are polynomial time approximation algorithms, their approximation factor is at best poly(k). It was left open if there was a polynomial-time approximation scheme (PTAS) for l(p)-approximation for any p >= 0. We show the following: 1. On the algorithmic side, for p is an element of (0, 2), we give the first n(poly(k/epsilon)) time (1 + epsilon)-approximation algorithm. For p = 0, there are various problem formulations, a common one being the binary setting in which A is an element of {0, 1}(nxd) and B = U . V, where U is an element of {0, 1}(nxk) and V is an element of {0; 1)(kxd). There are also various notions of multiplication (U) over dot . V, such as a matrix product over the reals, over a finite field, or over a Boolean semiring. We give the first almost-linear time approximation scheme for what we call the Generalized Binary l(0)-Rank-k problem, for which these variants are special cases. Our algorithm computes (1 + epsilon)-approximation in time (1/epsilon)(2O(k)/epsilon 2) .nd(1+o(1)), where o(1) hides a factor (log log d)(1.1)/log d. In addition, for the case of finite fields of constant size, we obtain an alternate PTAS running in time n.d(poly(k/epsilon)). 2. On the hardness front, for p is an element of (1, 2), we show under the Small Set Expansion Hypothesis and Exponential Time Hypothesis (ETH), there is no constant factor approximation algorithm running in time 2(k delta) for a constant delta > 0, showing an exponential dependence on k is necessary. For p = 0, we observe that there is no approximation algorithm for the Generalized Binary l(0)-Rank-k problem running in time 2(2 delta k) for a constant delta > 0. We also show for finite fields of constant size, under the ETH, that any fixed constant factor approximation algorithm requires 2(k delta) time for a constant delta > 0.

We study the problem of computing the p -> q operator norm of a matrix A in R-mxn, defined as parallel to A parallel to(p -> q) := sup x is an element of R-n\{0} parallel to A parallel to(q)/parallel to A parallel to x parallel to A parallel to(p): This problem generalizes the spectral norm of a matrix (p = q = 2) and the Grothendieck problem (p = 1; q = 1), and has been widely studied in various regimes. When p >= q, the problem exhibits a dichotomy: constant factor approximation algorithms are known if 2 is in [q; p], and the problem is hard to approximate within almost polynomial factors when 2 is not in [q,p]. For the case when 2 is in [q; p] we prove almost matching approximation and NP-hardness results. The regime when p < q, known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with p = 2 and q > 2 was studied by [Barak et. al., STOC'12] who gave sub-exponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the Exponential Time Hypothesis. However, no NP-hardness of approximation is known for these problems for any p < q. We prove the first NP-hardness result for approximating hypercontractive norms. We show that for any 1 < p < q < infinity with 2 not in [p; q], parallel to A parallel to(p -> q) is hard to approximate within 2(O(log epsilon) n) assuming NP is not contained in BPTIME(2 log(O(1)n)).

I am primarily interested in analytic techniques applied to approximation algorithms and inapproximability. Across works with various collaborators, I have employed complex analytic techniques in the design and analysis of approximation algorithms for problems such as coloring hypergraphs with some promised structure [BGL15], optimizing polynomials over the sphere [BGG17], and approximating operator norms [BGG18a], and techniques and intuition from functional analysis have informed my inapproximability results for operator norms [BGG18b] (which I believe may shed some light on the pursuit of inapproximability for important combinatorial optimization problems such as small-set expansion and densest-k-subgraph). Very broadly, my goals can be summarized as understanding via the lens of continuous optimization problems, rounding algorithms for convex programming and convex programming hierarchies, as well as the class of problems for which such algorithms achieve optimal approximations. In other words, I strive to extend the beautiful theory surrounding constraint satisfaction problems, convex programming and the unique games conjecture, to continuous optimization.

We consider the $(\ell_p,\ell_r)$-Grothendieck problem, which seeks to maximize the bilinear form $y^T A x$ for an input matrix $A$ over vectors $x,y$ with $\|x\|_p=\|y\|_r=1$. The problem is equivalent to computing the $p \to r^*$ operator norm of $A$. The case $p=r=\infty$ corresponds to the classical Grothendieck problem. Our main result is an algorithm for arbitrary $p,r \ge 2$ with approximation ratio $(1+\epsilon_0)/(\sinh^{-1}(1)\cdot \gamma_{p^*} \,\gamma_{r^*})$ for some fixed $\epsilon_0 \le 0.00863$. Comparing this with Krivine's approximation ratio of $(\pi/2)/\sinh^{-1}(1)$ for the original Grothendieck problem, our guarantee is off from the best known hardness factor of $(\gamma_{p^*} \gamma_{r^*})^{-1}$ for the problem by a factor similar to Krivine's defect. Our approximation follows by bounding the value of the natural vector relaxation for the problem which is convex when $p,r \ge 2$. We give a generalization of random hyperplane rounding and relate the performance of this rounding to certain hypergeometric functions, which prescribe necessary transformations to the vector solution before the rounding is applied. Unlike Krivine's Rounding where the relevant hypergeometric function was $\arcsin$, we have to study a family of hypergeometric functions. The bulk of our technical work then involves methods from complex analysis to gain detailed information about the Taylor series coefficients of the inverses of these hypergeometric functions, which then dictate our approximation factor. Our result also implies improved bounds for factorization through $\ell_{2}^{\,n}$ of operators from $\ell_{p}^{\,n}$ to $\ell_{q}^{\,m}$ (when $p\geq 2 \geq q$)--- such bounds are of significant interest in functional analysis and our work provides modest supplementary evidence for an intriguing parallel between factorizability, and constant-factor approximability.

For an $n$-variate order-$d$ tensor $A$, define $ A_{\max} := \sup_{\| x \|_2 = 1} \langle A , x^{\otimes d} \rangle$ to be the maximum value taken by the tensor on the unit sphere. It is known that for a random tensor with i.i.d $\pm 1$ entries, $A_{\max} \lesssim \sqrt{n\cdot d\cdot\log d}$ w.h.p. We study the problem of efficiently certifying upper bounds on $A_{\max}$ via the natural relaxation from the Sum of Squares (SoS) hierarchy. Our results include: - When $A$ is a random order-$q$ tensor, we prove that $q$ levels of SoS certifies an upper bound $B$ on $A_{\max}$ that satisfies \[ B ~~~~\leq~~ A_{\max} \cdot \biggl(\frac{n}{q^{\,1-o(1)}}\biggr)^{q/4-1/2} \quad \text{w.h.p.} \] Our upper bound improves a result of Montanari and Richard (NIPS 2014) when $q$ is large. - We show the above bound is the best possible up to lower order terms, namely the optimum of the level-$q$ SoS relaxation is at least \[ A_{\max} \cdot \biggl(\frac{n}{q^{\,1+o(1)}}\biggr)^{q/4-1/2} \ . \] - When $A$ is a random order-$d$ tensor, we prove that $q$ levels of SoS certifies an upper bound $B$ on $A_{\max}$ that satisfies \[ B ~~\leq ~~ A_{\max} \cdot \biggl(\frac{\widetilde{O}(n)}{q}\biggr)^{d/4 - 1/2} \quad \text{w.h.p.} \] For growing $q$, this improves upon the bound certified by constant levels of SoS. This answers in part, a question posed by Hopkins, Shi, and Steurer (COLT 2015), who established the tight characterization for constant levels of SoS.

We prove an analog of Parikh’s theorem for weighted context-free grammars over commutative, idempotent semirings, and exhibit a stochastic context-free grammar with behavior that cannot be realized by any stochastic right-linear context-free grammar. Finally, we show that every unary stochastic context-free grammar with polynomially-bounded ambiguity has an equivalent stochastic right-linear context-free grammar.

We consider the following basic problem: given an n-variate degree-d homogeneous polynomial f with real coefficients, compute a unit vector x in R̂n that maximizes abs(f(x)). Besides its fundamental nature, this problem arises in diverse contexts ranging from tensor and operator norms to graph expansion to quantum information theory. The homogeneous degree-2 case is efficiently solvable as it corresponds to computing the spectral norm of an associated matrix, but the higher degree case is NP-hard. We give approximation algorithms for this problem that offer a trade-off between the approximation ratio and running time: in n̂O(q) time, we get an approximation within factor (O(n)/q)̂(d/2-1) for arbitrary polynomials, (O(n)/q)̂(d/4-1/2) for polynomials with non-negative coefficients, and (m /q)̂(1/2) for sparse polynomials with m monomials. The approximation guarantees are with respect to the optimum of the level-q sum-of-squares (SoS) SDP relaxation of the problem (though our algorithms do not rely on actually solving the SDP). Known polynomial time algorithms for this problem rely on “decoupling lemmas.” Such tools are not capable of offering a trade-off like our results as they blow up the number of variables by a factor equal to the degree. We develop new decoupling tools that are more efficient in the number of variables at the expense of less structure in the output polynomials. This enables us to harness the benefits of higher level SoS relaxations. Our decoupling methods also work with “folded polynomials,” which are polynomials with polynomials as coefficients. This allows us to exploit easy substructures (such as quadratics) by considering them as coefficients in our algorithms. We complement our algorithmic results with some polynomially large integrality gaps for d-levels of the SoS relaxation. For general polynomials this follows from known results for random polynomials, which yield a gap of Omega(n)̂(d/4-1/2). For polynomials with non-negative coefficients, we prove an Omega(n̂(1/6) /polylogs) gap for the degree-4 case, based on a novel distribution of 4-uniform hypergraphs. We establish an n̂Omega(d) gap for general degree-d, albeit for a slightly weaker (but still very natural) relaxation. Toward this, we give a method to lift a level-4 solution matrix M to a higher level solution, under a mild technical condition on M. From a structural perspective, our work yields worst-case convergence results on the performance of the sum-of-squareshierarchy for polynomial optimization. Despite the popularity of SoS in this context, such results were previously only known for the case of q = Omega(n).

We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem arises in diverse contexts ranging from tensor and operator norms to graph expansion to quantum information theory. The homogeneous degree $2$ case is efficiently solvable as it corresponds to computing the spectral norm of an associated matrix, but the higher degree case is NP-hard. We give approximation algorithms for this problem that offer a trade-off between the approximation ratio and running time: in $n^{O(q)}$ time, we get an approximation within factor $O_d((n/q)^{d/2-1})$ for arbitrary polynomials, $O_d((n/q)^{d/4-1/2})$ for polynomials with non-negative coefficients, and $O_d(\sqrt{m/q})$ for sparse polynomials with $m$ monomials. The approximation guarantees are with respect to the optimum of the level-$q$ sum-of-squares (SoS) SDP relaxation of the problem. Known polynomial time algorithms for this problem rely on lemmas. Such tools are not capable of offering a trade-off like our results as they blow up the number of variables by a factor equal to the degree. We develop new decoupling tools that are more efficient in the number of variables at the expense of less structure in the output polynomials. This enables us to harness the benefits of higher level SoS relaxations. We complement our algorithmic results with some polynomially large integrality gaps, albeit for a slightly weaker (but still very natural) relaxation. Toward this, we give a method to lift a level-$4$ solution matrix $M$ to a higher level solution, under a mild technical condition on $M$.

Given a set $\mathsf{P}$ of $n$ points in $\mathbb{R}^d$, we show how to insert a set $\mathsf{X}$ of $O( n^{1-1/d} )$ additional points, such that $\mathsf{P}$ can be broken into two sets $\mathsf{P}_1$ and $\mathsf{P}_2$, of roughly equal size, such that in the Voronoi diagram $\mathcal{V}( \mathsf{P} \cup \mathsf{X} )$, the cells of $\mathsf{P}_1$ do not touch the cells of $\mathsf{P}_2$; that is, $\mathsf{X}$ separates $\mathsf{P}_1$ from $\mathsf{P}_2$ in the Voronoi diagram. Given such a partition $(\mathsf{P}_1,\mathsf{P}_2)$ of $\mathsf{P}$, we present approximation algorithms to compute the minimum size separator realizing this partition.

A hypergraph is said to be $\chi$-colorable if its vertices can be colored with $\chi$ colors so that no hyperedge is monochromatic. $2$-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a $2$-colorable $k$-uniform hypergraph, it is NP-hard to find a $2$-coloring miscoloring fewer than a fraction $2^{-k+1}$ of hyperedges (which is achieved by a random $2$-coloring), and the best algorithms to color the hypergraph properly require $\approx n^{1-1/k}$ colors, approaching the trivial bound of $n$ as $k$ increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a $2$-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than $2$-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy $\ell \ll \sqrt{k}$, we give an algorithm to color the it with $\approx n^{O(\ell^2/k)}$ colors. However, for the maximization version, we prove NP-hardness of finding a $2$-coloring miscoloring a smaller than $2^{-O(k)}$ (resp. $k^{-O(k)}$) fraction of the hyperedges when $\ell = O(\log k)$ (resp. $\ell=2$). Assuming the UGC, we improve the latter hardness factor to $2^{-O(k)}$ for almost discrepancy-$1$ hypergraphs. (B) Rainbow colorability: If the hypergraph has a $(k-\ell)$-coloring such that each hyperedge is polychromatic with all these colors, we give a $2$-coloring algorithm that miscolors at most $k^{-\Omega(k)}$ of the hyperedges when $\ell \ll \sqrt{k}$, and complement this with a matching UG hardness result showing that when $\ell =\sqrt{k}$, it is hard to even beat the $2^{-k+1}$ bound achieved by a random coloring.

Given a set $\mathsf{P}$ of $n$ points in $\mathbb{R}^d$, we show how to insert a set $\mathsf{X}$ of $O( n^{1-1/d} )$ additional points, such that $\mathsf{P}$ can be broken into two sets $\mathsf{P}_1$ and $\mathsf{P}_2$, of roughly equal size, such that in the Voronoi diagram $\mathcal{V}( \mathsf{P} \cup \mathsf{X} )$, the cells of $\mathsf{P}_1$ do not touch the cells of $\mathsf{P}_2$; that is, $\mathsf{X}$ separates $\mathsf{P}_1$ from $\mathsf{P}_2$ in the Voronoi diagram. Given such a partition $(\mathsf{P}_1,\mathsf{P}_2)$ of $\mathsf{P}$, we present approximation algorithms to compute the minimum size separator realizing this partition.

We show that every unary stochastic context-free grammar with polynomially bounded ambiguity, has an equivalent probabilistic automaton.