We investigate the polynomial-time approximability of the multistage version of Min-Sum Set Cover ($mathrm{DSSC}$), a natural and intriguing generalization of the classical List Update problem. In $mathrm{DSSC}$, we maintain a sequence of permutation
s $(pi^0, pi^1, ldots, pi^T)$ on $n$ elements, based on a sequence of requests $(R^1, ldots, R^T)$. We aim to minimize the total cost of updating $pi^{t-1}$ to $pi^{t}$, quantified by the Kendall tau distance $mathrm{D}_{mathrm{KT}}(pi^{t-1}, pi^t)$, plus the total cost of covering each request $R^t$ with the current permutation $pi^t$, quantified by the position of the first element of $R^t$ in $pi^t$. Using a reduction from Set Cover, we show that $mathrm{DSSC}$ does not admit an $O(1)$-approximation, unless $mathrm{P} = mathrm{NP}$, and that any $o(log n)$ (resp. $o(r)$) approximation to $mathrm{DSSC}$ implies a sublogarithmic (resp. $o(r)$) approximation to Set Cover (resp. where each element appears at most $r$ times). Our main technical contribution is to show that $mathrm{DSSC}$ can be approximated in polynomial-time within a factor of $O(log^2 n)$ in general instances, by randomized rounding, and within a factor of $O(r^2)$, if all requests have cardinality at most $r$, by deterministic rounding.
In the classical Subset Sum problem we are given a set $X$ and a target $t$, and the task is to decide whether there exists a subset of $X$ which sums to $t$. A recent line of research has resulted in $tilde{O}(t)$-time algorithms, which are (near-)o
ptimal under popular complexity-theoretic assumptions. On the other hand, the standard dynamic programming algorithm runs in time $O(n cdot |mathcal{S}(X,t)|)$, where $mathcal{S}(X,t)$ is the set of all subset sums of $X$ that are smaller than $t$. Furthermore, all known pseudopolynomial algorithms actually solve a stronger task, since they actually compute the whole set $mathcal{S}(X,t)$. As the aforementioned two running times are incomparable, in this paper we ask whether one can achieve the best of both worlds: running time $tilde{O}(|mathcal{S}(X,t)|)$. In particular, we ask whether $mathcal{S}(X,t)$ can be computed in near-linear time in the output-size. Using a diverse toolkit containing techniques such as color coding, sparse recovery, and sumset estimates, we make considerable progress towards this question and design an algorithm running in time $tilde{O}(|mathcal{S}(X,t)|^{4/3})$. Central to our approach is the study of top-$k$-convolution, a natural problem of independent interest: given sparse polynomials with non-negative coefficients, compute the lowest $k$ non-zero monomials of their product. We design an algorithm running in time $tilde{O}(k^{4/3})$, by a combination of sparse convolution and sumset estimates considered in Additive Combinatorics. Moreover, we provide evidence that going beyond some of the barriers we have faced requires either an algorithmic breakthrough or possibly new techniques from Additive Combinatorics on how to pass from information on restricted sumsets to information on unrestricted sumsets.
Computing the convolution $Astar B$ of two length-$n$ integer vectors $A,B$ is a core problem in several disciplines. It frequently comes up in algorithms for Knapsack, $k$-SUM, All-Pairs Shortest Paths, and string pattern matching problems. For thes
e applications it typically suffices to compute convolutions of nonnegative vectors. This problem can be classically solved in time $O(nlog n)$ using the Fast Fourier Transform. However, often the involved vectors are sparse and hence one could hope for output-sensitive algorithms to compute nonnegative convolutions. This question was raised by Muthukrishnan and solved by Cole and Hariharan (STOC 02) by a randomized algorithm running in near-linear time in the (unknown) output-size $t$. Chan and Lewenstein (STOC 15) presented a deterministic algorithm with a $2^{O(sqrt{log tcdotloglog n})}$ overhead in running time and the additional assumption that a small superset of the output is given; this assumption was later removed by Bringmann and Nakos (ICALP 21). In this paper we present the first deterministic near-linear-time algorithm for computing sparse nonnegative convolutions. This immediately gives improved deterministic algorithms for the state-of-the-art of output-sensitive Subset Sum, block-mass pattern matching, $N$-fold Boolean convolution, and others, matching up to log-factors the fastest known randomized algorithms for these problems. Our algorithm is a blend of algebraic and combinatorial ideas and techniques. Additionally, we provide two fast Las Vegas algorithms for computing sparse nonnegative convolutions. In particular, we present a simple $O(tlog^2t)$ time algorithm, which is an accessible alternative to Cole and Hariharans algorithm. We further refine this new algorithm to run in Las Vegas time $O(tlog tcdotloglog t)$, matching the running time of the dense case apart from the $loglog t$ factor.
Computing the convolution $Astar B$ of two length-$n$ vectors $A,B$ is an ubiquitous computational primitive. Applications range from string problems to Knapsack-type problems, and from 3SUM to All-Pairs Shortest Paths. These applications often come
in the form of nonnegative convolution, where the entries of $A,B$ are nonnegative integers. The classical algorithm to compute $Astar B$ uses the Fast Fourier Transform and runs in time $O(nlog n)$. However, often $A$ and $B$ satisfy sparsity conditions, and hence one could hope for significant improvements. The ideal goal is an $O(klog k)$-time algorithm, where $k$ is the number of non-zero elements in the output, i.e., the size of the support of $Astar B$. This problem is referred to as sparse nonnegative convolution, and has received considerable attention in the literature; the fastest algorithms to date run in time $O(klog^2 n)$. The main result of this paper is the first $O(klog k)$-time algorithm for sparse nonnegative convolution. Our algorithm is randomized and assumes that the length $n$ and the largest entry of $A$ and $B$ are subexponential in $k$. Surprisingly, we can phrase our algorithm as a reduction from the sparse case to the dense case of nonnegative convolution, showing that, under some mild assumptions, sparse nonnegative convolution is equivalent to dense nonnegative convolution for constant-error randomized algorithms. Specifically, if $D(n)$ is the time to convolve two nonnegative length-$n$ vectors with success probability $2/3$, and $S(k)$ is the time to convolve two nonnegative vectors with output size $k$ with success probability $2/3$, then $S(k)=O(D(k)+k(loglog k)^2)$. Our approach uses a variety of new techniques in combination with some old machinery from linear sketching and structured linear algebra, as well as new insights on linear hashing, the most classical hash function.
We consider the problem of computing the Boolean convolution (with wraparound) of $n$~vectors of dimension $m$, or, equivalently, the problem of computing the sumset $A_1+A_2+ldots+A_n$ for $A_1,ldots,A_n subseteq mathbb{Z}_m$. Boolean convolution fo
rmalizes the frequent task of combining two subproblems, where the whole problem has a solution of size $k$ if for some $i$ the first subproblem has a solution of size~$i$ and the second subproblem has a solution of size $k-i$. Our problem formalizes a natural generalization, namely combining solutions of $n$ subproblems subject to a modular constraint. This simultaneously generalises Modular Subset Sum and Boolean Convolution (Sumset Computation). Although nearly optimal algorithms are known for special cases of this problem, not even tiny improvements are known for the general case. We almost resolve the computational complexity of this problem, shaving essentially a factor of $n$ from the running time of previous algorithms. Specifically, we present a emph{deterministic} algorithm running in emph{almost} linear time with respect to the input plus output size $k$. We also present a emph{Las Vegas} algorithm running in emph{nearly} linear expected time with respect to the input plus output size $k$. Previously, no deterministic or randomized $o(nk)$ algorithm was known. At the heart of our approach lies a careful usage of Knesers theorem from Additive Combinatorics, and a new deterministic almost linear output-sensitive algorithm for non-negative sparse convolution. In total, our work builds a solid toolbox that could be of independent interest.
In the long-studied problem of combinatorial group testing, one is asked to detect a set of $k$ defective items out of a population of size $n$, using $m ll n$ disjunctive measurements. In the non-adaptive setting, the most widely used combinatorial
objects are disjunct and list-disjunct matrices, which define incidence matrices of test schemes. Disjunct matrices allow the identification of the exact set of defectives, whereas list disjunct matrices identify a small superset of the defectives. Apart from the combinatorial guarantees, it is often of key interest to equip measurement designs with efficient decoding algorithms. The most efficient decoders should run in sublinear time in $n$, and ideally near-linear in the number of measurements $m$. In this work, we give several constructions with an optimal number of measurements and near-optimal decoding time for the most fundamental group testing tasks, as well as for central tasks in the compressed sensing and heavy hitters literature. For many of those tasks, the previous measurement-optimal constructions needed time either quadratic in the number of measurements or linear in the universe size. Most of our results are obtained via a clean and novel approach which avoids list-recoverable codes or related complex techniques which were present in almost every state-of-the-art work on efficiently decodable constructions of such objects.
In this paper, we consider the extensively studied problem of computing a $k$-sparse approximation to the $d$-dimensional Fourier transform of a length $n$ signal. Our algorithm uses $O(k log k log n)$ samples, is dimension-free, operates for any uni
verse size, and achieves the strongest $ell_infty/ell_2$ guarantee, while running in a time comparable to the Fast Fourier Transform. In contrast to previous algorithms which proceed either via the Restricted Isometry Property or via filter functions, our approach offers a fresh perspective to the sparse Fourier Transform problem.
In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of $x in mathbb{C}^n$ and design a recovery algorithm such that the output of the algorithm approximates $hat x$, the Di
screte Fourier Transform (DFT) of $x$. The randomized case has been well-understood, while the main work in the deterministic case is that of Merhi et al.@ (J Fourier Anal Appl 2018), which obtains $O(k^2 log^{-1}k cdot log^{5.5}n)$ samples and a similar runtime with the $ell_2/ell_1$ guarantee. We focus on the stronger $ell_{infty}/ell_1$ guarantee and the closely related problem of incoherent matrices. We list our contributions as follows. 1. We find a deterministic collection of $O(k^2 log n)$ samples for the $ell_infty/ell_1$ recovery in time $O(nk log^2 n)$, and a deterministic collection of $O(k^2 log^2 n)$ samples for the $ell_infty/ell_1$ sparse recovery in time $O(k^2 log^3n)$. 2. We give new deterministic constructions of incoherent matrices that are row-sampled submatrices of the DFT matrix, via a derandomization of Bernsteins inequality and bounds on exponential sums considered in analytic number theory. Our first construction matches a previous randomized construction of Nelson, Nguyen and Woodruff (RANDOM12), where there was no constraint on the form of the incoherent matrix. Our algorithms are nearly sample-optimal, since a lower bound of $Omega(k^2 + k log n)$ is known, even for the case where the sensing matrix can be arbitrarily designed. A similar lower bound of $Omega(k^2 log n/ log k)$ is known for incoherent matrices.
In the sparse polynomial multiplication problem, one is asked to multiply two sparse polynomials f and g in time that is proportional to the size of the input plus the size of the output. The polynomials are given via lists of their coefficients F an
d G, respectively. Cole and Hariharan (STOC 02) have given a nearly optimal algorithm when the coefficients are positive, and Arnold and Roche (ISSAC 15) devised an algorithm running in time proportional to the structural sparsity of the product, i.e. the set supp(F)+supp(G). The latter algorithm is particularly efficient when there not too many cancellations of coefficients in the product. In this work we give a clean, nearly optimal algorithm for the sparse polynomial multiplication problem.
In the problem of adaptive compressed sensing, one wants to estimate an approximately $k$-sparse vector $xinmathbb{R}^n$ from $m$ linear measurements $A_1 x, A_2 x,ldots, A_m x$, where $A_i$ can be chosen based on the outcomes $A_1 x,ldots, A_{i-1} x
$ of previous measurements. The goal is to output a vector $hat{x}$ for which $$|x-hat{x}|_p le C cdot min_{ktext{-sparse } x} |x-x|_q,$$ with probability at least $2/3$, where $C > 0$ is an approximation factor. Indyk, Price and Woodruff (FOCS11) gave an algorithm for $p=q=2$ for $C = 1+epsilon$ with $Oh((k/epsilon) loglog (n/k))$ measurements and $Oh(log^*(k) loglog (n))$ rounds of adaptivity. We first improve their bounds, obtaining a scheme with $Oh(k cdot loglog (n/k) +(k/epsilon) cdot loglog(1/epsilon))$ measurements and $Oh(log^*(k) loglog (n))$ rounds, as well as a scheme with $Oh((k/epsilon) cdot loglog (nlog (n/k)))$ measurements and an optimal $Oh(loglog (n))$ rounds. We then provide novel adaptive compressed sensing schemes with improved bounds for $(p,p)$ for every $0 < p < 2$. We show that the improvement from $O(k log(n/k))$ measurements to $O(k log log (n/k))$ measurements in the adaptive setting can persist with a better $epsilon$-dependence for other values of $p$ and $q$. For example, when $(p,q) = (1,1)$, we obtain $O(frac{k}{sqrt{epsilon}} cdot log log n log^3 (frac{1}{epsilon}))$ measurements.
