Factorization Norms and Hereditary Discrepancy

Let A be the incidence matrix of a set system with m sets and n points, m ≤ n , and let t= \mathop \rm tr M , where M= A T A . Finally, let σ = \mathop \rm tr M 2 be the sum of squares of the elements of M . We prove that the hereditary discrepancy of the set system is at least ${\frac14}\, c^{n\sigma/ t^2} \sqrt{t/n}$, with $c= \frac{1}{324}$ . This general trace bound allows us to resolve discrepancy-type questions for which spectral methods had previously failed. Also, by using this result in conjunction with the spectral lemma for linear circuits, we derive new complexity bounds for range searching. We show that the (red—blue) discrepancy of the set system formed by n points and n lines in the plane is Ω(n 1/6 ) in the worst case and always 1 \tilde O(n 1/6 ) . We give a simple explicit construction of n points and n halfplanes with hereditary discrepancy \tildeΩ(n 1/4 ) . We show that in any dimension d= Ω( log n/\kern -1ptlog log n ) , there is a set system of n points and n axis-parallel boxes in \bf R d with discrepancy n Ω(1/\kern -1pt log log n) . Applying these discrepancy results together with a new variation of the spectral lemma, we derive a lower bound of Ω(nlog n) on the arithmetic complexity of off-line range searching for points and lines (for nonmonotone circuits). We also prove a lower bound of Ω(nlog n/\kern -1ptlog log n) on the complexity of orthogonal range searching in any dimension Ω(log n/\kern -1ptlog log n) . 1 We use the notation \tilde O(m) and \tilde Ω(m) as shorthand for O(mlog c m) and Ω(m/\kern -1ptlog c m) , respectively, for some constant c>0 .

Random polygon Generation from a given set of points is very important for different applications. A heuristic termed as “GRP_AS” has been proposed here to generate a random simple polygon from a given set of `n' points in 2-Dimensional plane. The “2-Opt Move” heuristic with time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) is the best known among the existing heuristics to generate a simple polygon. We recently designed another heuristic “GRP_CH” for the same problem with time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) which was shown to be better than that of “2-Opt Move” heuristic in terms of random behaviour and time complexity. In this work, our main objective is to design an algorithm with reduced time complexity. The proposed heuristic, “GRP_AS” is based on the principle of random selection of a line segment and angular sorting of the point set. This “GRP_AS” heuristic takes O(nlogn) time which is less than that of “2-Opt Move” heuristic as well as “GRP_CH” heuristic. The lower bound and upper bound on the number of polygons generated in “GRP_AS” heuristic has been computed.

Given a Boolean formula in conjunctive normal form with \(n\) variables and \(m=rn\) clauses, if there exists a truth assignment satisfying \((1-2^{-k}-q(1-2^{-k}))m\) clauses, call the formula \(q\)-satisfiable. The Minimum Satisfiability Problem (MinSAT) is a special case of \(q\)-satisfiable, which asks for an assignment to minimize the number of satisfied clauses. When each clause contains \(k\) literals, it is called Min\(k\)SAT. If each clause is independently and randomly selected from all possible clauses over the \(n\) variables, it is called random MinSAT. In this paper, we give upper and lower bounds of \(r\) (the ratio of clauses to variables) for random \(k\)-CNF formula with \(q\)-satisfiable. The upper bound is proved by the first moment argument, while the proof of lower bound is the second moment with weighted scheme. Interestingly, our experimental results about MinSAT demonstrate that the lower and upper bounds are very tight. Moreover, these results give a partial explanation for the excellent performance of MinSatz, the state-of-the-art MinSAT solver, from the perspective of pruning effects. Finally, we give a conjecture about the relationship between the minimum number and the maximum number of satisfied clauses on random SAT instances.KeywordsMinSATUpper boundsLower bounds

The convex hull of a planer set of points can be defined as the set of vertices of the smallest convex polygon containing all the points. If S is a planer set of points then convex layers of S can be derived by iteratively computing the convex hull of S and then removing it from S. Computation of the convex layers has widely been studied in the static environment where the set of points is fixed. The theoretical lower bound of computing the convex layers of a fixed set of points is O (n log n). The static convex layers algorithms with the optimal time complexity have already been proposed in the literature. A set of points where the points can be inserted or deleted is called a dynamic set of points. The set of convex layers should be reconstructed from the scratch at each insertion or deletion if a static convex layers algorithm was used to handle a dynamic set of points. Therefore, it takes O (n log n) time to handle an insertion or a deletion even for an optimal static convex layers algorithm. A dynamic convex layers algorithm has been proposed recently that can perform an insertion or a deletion by doing a slight modification to the existing set of convex layers. It takes O (n) time for an insertion or a deletion. It assumes that the set of points does not contain collinear points and that assumption is not valid in practical applications. Furthermore, the notion of tangent used in that approach restricts the extension of the algorithm into higher dimensions. This paper proposes a novel dynamic convex layers algorithm to eliminate the drawbacks in the existing algorithm. Salient feature of this algorithm is that it represents each layer as a set of line segments. The layers are modified upon an insertion or a deletion of a point by adding some new line segments and deleting some existing line segments. The proposed algorithm takes O (n 3 /k 2 ) time for an insertion or a deletion where k is the number of convex layers. A computer implementation is also presented. The proposed algorithm can work with set of points with collinear points and coincident points. The notion used in the algorithm can easily be extended to higher dimensions. Suppose the set of convex layers have already been found for a given set of points. Further, suppose that the layers are close to each other and new points are expected to fall within the region bounded by the outermost layer. This kind of situations widely occurs in practice and then the proposed algorithm takes O (n) time for an inclusion or a deletion.

Herding is an algorithm of recent interest in the machine learning community, motivated by inference in Markov random fields. It solves the following sampling problem: given a setX R d with mean , construct an infinite sequence of points fromX such that, for every t 1, the mean of the first t points in that sequence lies within Euclidean distance O(1=t) of . The classic Perceptron boundedness theorem implies that such a result actually holds for a wide class of algorithms, although the factors suppressed by the O(1=t) notation are exponential in d. Thus, to establish a non-trivial result for the sampling problem, one must carefully analyze the factors suppressed by theO(1=t) error bound. This paper studies the best error that can be achieved for the sampling problem. Known analysis of the Herding algorithm give an error bound that depends on geometric properties ofX but, even under favorable conditions, this bound depends linearly on d. We present a new polynomialtime algorithm that solves the sampling problem with error O p d log 2:5 jXj=t assuming thatX is finite. Our algorithm is based on recent algorithmic results in discrepancy theory. We also show that any algorithm for the sampling problem must have error ( p d=t). This implies that our

Determination of minimum number of sensors and their locations for an automated facility: An algorithmic approach

The star discrepancy $D_N^*(\mathcal{P})$ is a quantitative measure for the irregularity of distribution of a finite point set $\mathcal{P}$ in the multi-dimensional unit cube which is intimately related to the integration error of quasi-Monte Carlo algorithms. It is known that for every integer $N \ge 2$ there are point sets $\mathcal{P}$ in $[0,1)^d$ with $|\mathcal{P}|=N$ and $D_N^*(\mathcal{P}) =O((\log N)^{d-1}/N)$. However, for small $N$ compared to the dimension $d$ this asymptotically excellent bound is useless (e.g. for $N \le {\rm e}^{d-1}$). In 2001 it has been shown by Heinrich, Novak, Wasilkowski and Wo\'{z}niakowski that for every integer $N \ge 2$ there exist point sets $\mathcal{P}$ in $[0,1)^d$ with $|\mathcal{P}|=N$ and $D_N^*(\mathcal{P}) \le C \sqrt{d/N}$. Although not optimal in an asymptotic sense in $N$, this upper bound has a much better (and even optimal) dependence on the dimension $d$. Unfortunately the result by Heinrich et al. and also later variants thereof by other authors are pure existence results and until now no explicit construction of point sets with the above properties are known. Quite recently L\"obbe studied lacunary subsequences of Kronecker's $(n \boldsymbol{\alpha})$-sequence and showed a metrical discrepancy bound of the form $C \sqrt{d (\log d)/N}$ with implied absolute constant $C>0$ independent of $N$ and $d$. In this paper we show a corresponding result for digital Kronecker sequences, which are a non-archimedean analog of classical Kronecker sequences.

We revisit the primitive relay channel, introduced by Cover in 1987. Recent work derived upper bounds on the capacity of this channel that are tighter than the classical cutset bound using the concentration of measure. In this paper, we recover, generalize, and improve upon some of these upper bounds with simpler proofs using reverse hypercontractivity. To our knowledge, this is the first application of reverse hypercontractivity in proving first-order converses in network information theory.

A novel technique for obtaining lower bounds for the time versus space complexity of certain functions in a general input oblivious sequential model of computation is developed. This is demonstrated by studying the intrinsic complexity of the following set equality problem SE(n,m): Given a sequence x1,x2,....,xn, y1,....,yn of 2n numbers of m bits each, decide whether the sets [x1,....,xn] and [y1,...,yn] coincide. We show that for any log log n ≤ m ≤1/2log n and any 1 ≤ s ≤ log n, any input oblivious sequential computation that solves SE(n,m) using 2m/s space, takes Ω(n ? s) time. This result is sharp for all admissible values of n,m,s and is the first known nontrivial time space tradeoff lower bound (for space = ω (log n) of a set recognition problem on such a general model of computation. Our method also supplies lower bounds on the length of arbitrary (not necessarily input oblivious) branching programs for several natural symmetric functions, improving results of Chandra, Furst and Lipton, of Pudlák and of Ajtai et. al. For example we show that for the majority - function any branching program of width w(n) has length ω(n · log w/n (n) · log w (n)), in particular for bounded width we get length ω (n log n) (independently of our work Babai et. al. [BPRS] have simultaneously proved this last result). Our lower bounds for branching programs imply lower bounds on the number of steps that are needed to pebble arbitrary computation graphs for the same computational problems. To establish our lower bounds we introduce the new concept of a meander that captures superconcentrator-type properties of sequences. We prove lower bounds on the length of meanders via a new Ramsey theoretic lemma that is of interest in its own right. This lemma has other applications, including a tight lower bound on the size of weak superconcentrators of depth 2 that strengthens the known lower bound of Pippenger [Pi]. A surprising new feature of these applications of Ramsey theory in lower bound arguments is the fact that no numbers are required to be unusually large and that several of the resulting superlinear lower bounds are in fact optimal.

An obstacle representation of a graph G is a set of points in the plane representing the vertices of G, together with a set of polygonal obstacles such that two vertices of G are connected by an edge in G if and only if the line segment between the corresponding points avoids all the obstacles. The obstacle number $${{\mathrm{obs}}}(G)$$ of G is the minimum number of obstacles in an obstacle representation of G. We provide the first non-trivial general upper bound on the obstacle number of graphs by showing that every n-vertex graph G satisfies $${{\mathrm{obs}}}(G) \le n\lceil \log {n}\rceil -n+1$$ . This refutes a conjecture of Mukkamala, Pach, and Pálvölgyi. For n-vertex graphs with bounded chromatic number, we improve this bound to O(n). Both bounds apply even when the obstacles are required to be convex. We also prove a lower bound $$2^{\Omega (hn)}$$ on the number of n-vertex graphs with obstacle number at most h for $$h<n$$ and a lower bound $$\Omega (n^{4/3}M^{2/3})$$ for the complexity of a collection of $$M \ge \Omega (n\log ^{3/2}{n})$$ faces in an arrangement of line segments with n endpoints. The latter bound is tight up to a multiplicative constant.

In this article we survey recent results on the explicit construction of finite point sets and infinite sequences with optimal order of $\mathcal{L}_q$ discrepancy. In 1954 Roth proved a lower bound for the $\mathcal{L}_2$ discrepancy of finite point sets in the unit cube of arbitrary dimension. Later various authors extended Roth's result to lower bounds also for the $\mathcal{L}_q$ discrepancy and for infinite sequences. While it was known already from the early 1980s on that Roth's lower bound is best possible in the order of magnitude, it was a longstanding open question to find explicit constructions of point sets and sequences with optimal order of $\mathcal{L}_2$ discrepancy. This problem was solved by Chen and Skriganov in 2002 for finite point sets and recently by the authors of this article for infinite sequences. These constructions can also be extended to give optimal order of the $\mathcal{L}_q$ discrepancy of finite point sets for $q \in (1,\infty)$. The main aim of this article is to give an overview of these constructions and related results.

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an n-point set $P$, and a collection $\mathcal{F} = \{F_1, ..., F_m\}$ of subsets of $P$, and our goal is color $P$ with two colors, red and blue, so that the maximum over the $F_i$ of the absolute difference between the number of red elements and the number of blue elements (the discrepancy) is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis algorithms. We investigate the combinatorial discrepancy of geometrically defined systems, in which $P$ is an n-point set in $d$-dimensional space ,and $\mathcal{F}$ is the collection of subsets of $P$ induced by dilations and translations of a fixed convex polytope $B$. Such set systems include systems of sets induced by axis-aligned boxes, whose discrepancy is the subject of the well known Tusnady problem. We prove new discrepancy upper and lower bounds for such set systems by extending the approach based on factorization norms previously used by the author and Matousek. We improve the best known upper bound for the Tusnady problem by a logarithmic factor, using a result of Banaszczyk on signed series of vectors. We extend this improvement to any arbitrary convex polytope $B$ by using a decomposition due to Matousek. Using Fourier analytic techniques, we also prove a nearly matching discrepancy lower bound for sets induced by any fixed bounded polytope $B$ satisfying a certain technical condition. We also outline applications of our results to geometric discrepancy, data structure lower bounds, and differential privacy.

We consider the problem of testing graph cluster structure: given access to a graph G = (V, E), can we quickly determine whether the graph can be partitioned into a few clusters with good inner conductance, or is far from any such graph? This is a generalization of the well-studied problem of testing graph expansion, where one wants to distinguish between the graph having good expansion (i.e. being a good single cluster) and the graph having a sparse cut (i.e. being a union of at least two clusters). A recent work of Czumaj, Peng, and Sohler (STOC'15) gave an ingenious sublinear time algorithm for testing k-clusterability in time O(n^1/2 poly(k)). Their algorithm implicitly embeds a random sample of vertices of the graph into Euclidean space, and then clusters the samples based on estimates of Euclidean distances between the points. This yields a very efficient testing algorithm, but only works if the cluster structure is very strong: it is necessary to assume that the gap between conductances of accepted and rejected graphs is at least logarithmic in the size of the graph G. In this paper we show how one can leverage more refined geometric information, namely angles as opposed to distances, to obtain a sublinear time tester that works even when the gap is a sufficiently large constant. Our tester is based on the singular value decomposition of a natural matrix derived from random walk transition probabilities from a small sample of seed nodes. We complement our algorithm with a matching lower bound on the query complexity of testing clusterability. Our lower bound is based on a novel property testing problem, which we analyze using Fourier analytic tools. As a byproduct of our techniques, we also achieve new lower bounds for the problem of approximating MAX-CUT value in sublinear time.

Approximation and online algorithms for multidimensional bin packing: A survey

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Polygon Placement Revisited: (Degree of Freedom + 1)-SUM Hardness and an Improvement via Offline Dynamic Rectangle UnionMarvin Künnemann and André NusserMarvin Künnemann and André Nusserpp.3181 - 3201Chapter DOI:https://doi.org/10.1137/1.9781611977073.124PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We revisit a classical problem in computational geometry: Determine the largest copy of a simple polygon P that can be placed into the simple polygon Q. Despite significant effort studying a number of settings, known algorithms require high polynomial running times, even for the interesting case when either P or Q have constant size. (Barequet and Har-Peled, 2001) give a conditional lower bound of n2–o(1) under the 3SUM conjecture when P and Q are (convex) polygons with Θ(n) vertices each. This leaves open whether we can establish (1) hardness beyond quadratic time and (2) any superlinear bound for constant-sized P or Q. In this paper, we affirmatively answer these questions under the higher-order kSUM conjecture, proving natural hardness results that increase with each degree of freedom (scaling, x-translation, y-translation, rotation): (scaling, x-translation:) Finding the largest copy of P that can be x-translated into Q requires time n2–o(1) under the 3SUM conjecture, even for orthogonal (rectilinear) polygons P, Q with O(1) and n vertices, respectively. (scaling, x-translation, y-translation:) Finding the largest copy of P that can be arbitrarily translatedinto Q requires time n2–o(1) under the 4SUM conjecture, even for orthogonal polygons P, Q with O(1) and n vertices, respectively. This establishes the same lower bound under the assumption that Subset Sum cannot be solved in time O((2–∊)n/2) for any ∊ > 0. The above lower bounds are almost tight when one of the polygons is of constant size: Using an offline dynamic algorithm for maintaining the area of a union of rectangles due to Overmars and Yap, we obtain an -time algorithm for orthogonal polygons P, Q with p and q vertices, respectively. This matches the lower bounds up to an n1/2+o(1)-factor when P, Q have O(1) and n vertices. (scaling, x-translation, y-translation, rotation:) Finally, finding the largest copy of P that can be arbitrarily rotated and translated into Q requires time n3–o(1) under the 5SUM conjecture. As in our reductions, each degree of freedom determines one summand in a kSUM instance, these lower bounds appear likely to be best possible under kSUM. We are not aware of any other such natural (degree of freedom + 1)-SUM hardness for a geometric optimization problem. Finally, we prove an additional tight OV hardness of the translations-only case. Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771