We show that for @math and for measurable subsets of the unit square with Lebesgue measure @math there exist bi-Lipschitz maps with bounded Lipschitz constant (uniformly over all such sets) which are identity on the boundary and increases the Lebesgue measure of the set to at least @math .
Shortly before uploading this paper we came across a very recent paper @cite_5 on arXiv which investigates related questions and proves a variant of Theorem for sets @math of sufficiently small measure (Proposition 11, @cite_5 ), as well as much more. Their proof uses a covering lemma based on a result of @cite_2 , which is no longer true in higher dimensions. Our proof is different and we believe can be adapted to work in higher dimensions as well. Moreover, the stretching result only for sufficiently small sets is not sufficient for our purposes in proving rough isometry. Our work was independent of @cite_5 .
@cite_5 @cite_2
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We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . The full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary @math of a set @math , as a function of @math . A weaker (but more widely-used) lower bound is @math , where equality holds iff @math is a subcube. In 2011, the first author obtained a sharp stability' version of the latter result, proving that if @math , then there exists a subcube @math such that @math . The weak' version of the edge isoperimetric inequality has the following well-known generalization for the @math -biased' measure @math on the discrete cube: if @math , or if @math and @math is monotone increasing, then @math . In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if @math , then there exists a subcube @math such that @math , where @math . This result is a central component in recent work of the authors proving sharp stability versions of a number of Erd o s-Ko-Rado type theorems in extremal combinatorics, including the seminal complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem. In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of ( )-numbers. These ( )-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones’ result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones’ ( )-numbers. This variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set. In this paper we give a version of P. Jones’ theorem for sets of arbitrary (integer) dimension lying in Euclidean space. Our main result is a lower bound for the d-dimensional Hausdorff measure of a set in terms of an analogous sum of ( )-type numbers. We also show an upper bound of this type. The combination of these results gives a Jones theorem for higher dimensional sets. While there is no assumption of Ahlfors regularity, or of a measure on the underlying set, there is an assumption of a lower bound on the Hausdorff content. We adapt David and Semmes’ version of Jones’ ( )-numbers by redefining them using a Choquet integral, allowing them to be defined for arbitrary sets (and not just sets of locally finite measure). A key tool in the proof is G. David and T. Toro’s parametrization of Reifenberg flat sets (with holes). We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ≥ 4 alternatives and n voters will be manipulable with probability at least 10−4∈2 n −3 q −30, where ∈ is the minimal statistical distance between f and the family of dictator functions. Our results extend those of [11], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [4,6,9,15,7]) cannot hide manipulations completely. Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies. A recent theorem of [arXiv:0910.1824v2, 2010], proved using results about the cluster expansion in statistical mechanics, extends the Lovasz local lemma by weakening the conditions under which its conclusion holds. In this note, we prove an algorithmic analogue of this result, extending Moser and Tardos's recent algorithmic local lemma [J. ACM, 57 (2010), 11], and providing an alternative proof of the theorem of applicable in the Moser--Tardos algorithmic framework. In this article, we show several results obtained by combining the use of stable distributions with pseudorandom generators for bounded space. In particular:---We show that, for any p ∈ (0, 2], one can maintain (using only O(log n e2) words of storage) a sketch C(q) of a point q ∈ lnp under dynamic updates of its coordinates. The sketch has the property that, given C(q) and C(s), one can estimate Vq − sVp up to a factor of (1 p e) with large probability. This solves the main open problem of [1999].---We show that the aforementioned sketching approach directly translates into an approximate algorithm that, for a fixed linear mapping A, and given x ∈ ℜn and y ∈ ℜm, estimates VAx − yVp in O(n p m) time, for any p ∈ (0, 2]. This generalizes an earlier algorithm of Wasserman and Blum [1997] which worked for the case p e 2.---We obtain another sketch function C′ which probabilistically embeds ln1 into a normed space lm1. The embedding guarantees that, if we set m e log(1 Δ)O(1 e), then for any pair of points q, s ∈ ln1, the distance between q and s does not increase by more than (1 p e) with constant probability, and it does not decrease by more than (1 − e) with probability 1 − Δ. This is the only known dimensionality reduction theorem for the l1 norm. In fact, stronger theorems of this type (i.e., that guarantee very low probability of expansion as well as of contraction) cannot exist [Brinkman and Charikar 2003].---We give an explicit embedding of ln2 into lnO(log n)1 with distortion (1 p 1 nΘ(1)). We study the metric properties of finite subsets of L1. The analysis of such metrics is central to a number of important algorithmic problems involving the cut structure of weighted graphs, including the Sparsest Cut Problem, one of the most compelling open problems in the field of approximation algorithms. Additionally, many open questions in geometric non-linear functional analysis involve the properties of finite subsets of L1.We present some new observations concerning the relation of L1 to dimension, topology, and Euclidean distortion. We show that every n-point subset of L1 embeds into L2 with average distortion O(√log n), yielding the first evidence that the conjectured worst-case bound of O(√log n) is valid. We also address the issue of dimension reduction in Lp for p ∈ (1, 2). We resolve a question left open by M. Charikar and A. Sahai [Dimension reduction in the l1 norm, in: Proceedings of the 43rd Annual IEEE Conference on Foundations of Computer Science, ACM, 2002, pp. 251-260] concerning the impossibility of dimension reduction with a linear map in the above cases, and we show that a natural variant of the recent example of Brinkman and Charikar [On the impossibility of dimension reduction in l1, in: Proceedings of the 44th Annual IEEE Conference on Foundations of Computer Science, ACM, 2003, pp. 514-523], cannot be used to prove a lower bound for the non-linear case. This is acomplished by exhibiting constant-distortion embeddings of snowflaked planar metrics into Euclidean space. We study the recovery of sparse vectors from subsampled random convolutions via @math -minimization. We consider the setup in which both the subsampling locations as well as the generating vector are chosen at random. For a subgaussian generator with independent entries, we improve previously known estimates: if the sparsity @math is small enough, i.e., @math , we show that @math measurements are sufficient to recover @math -sparse vectors in dimension @math with high probability, matching the well-known condition for recovery from standard Gaussian measurements. If @math is larger, then essentially @math measurements are sufficient, again improving over previous estimates. Our results are shown via the so-called robust null space property which is weaker than the standard restricted isometry property. Our method of proof involves a novel combination of small ball estimates with chaining techniques which should be of independent interest. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 For any integers @math and @math , we show the existence of a set of @math vectors @math such that any embedding @math satisfying @math must have @math This lower bound matches the upper bound given by the Johnson-Lindenstrauss lemma [JL84]. Furthermore, our lower bound holds for nearly the full range of @math of interest, since there is always an isometric embedding into dimension @math (either the identity map, or projection onto @math ). Previously such a lower bound was only known to hold against linear maps @math , and not for such a wide range of parameters @math [LN16]. The best previously known lower bound for general @math was @math [Wel74, Lev83, Alo03], which is suboptimal for any @math . We give a sharp lower bound on the number of matchings of a given size in a bipartite graph. When specialized to regular bipartite graphs, our results imply Friedland's Lower Matching Conjecture and Schrijver's theorem proven by Gurvits and Csikvari. Indeed, our work extends the recent work of Csikvari done for regular and bi-regular bipartite graphs. Moreover, our lower bounds are order optimal as they are attained for a sequence of @math -lifts of the original graph as well as for random @math -lifts of the original graph when @math tends to infinity. We then extend our results to permanents and subpermanents sums. For permanents, we are able to recover the lower bound of Schrijver recently proved by Gurvits using stable polynomials. Our proof is algorithmic and borrows ideas from the theory of local weak convergence of graphs, statistical physics and covers of graphs. We provide new lower bounds for subpermanents sums and obtain new results on the number of matching in random @math -lifts with some implications for the matching measure and the spectral measure of random @math -lifts as well as for the spectral measure of infinite trees. We study the asymptotic behavior of the persistent homology of i.i.d. samples from a @math -Ahlfors regular measure --- one that satisfies uniform bounds of the form for some @math all @math in the support of @math and all sufficiently small @math Our main result is that if @math are sampled from a @math -Ahlfors regular measure on @math and @math denotes the @math -weight of the minimal spanning tree on @math [E_ (x_1, ,x_n )= e T (x_1, ,x_n ) |e|^ ] then [E_ (x_1, ,x_n ) n^ d- d ] with high probability as @math We also prove theorems about the asymptotic behavior of weighted sums defined in terms of higher-dimensional persistent homology. As an application, we exhibit hypotheses under which the fractal dimension of a measure can be computed from the persistent homology of i.i.d. samples from that space, in a manner similar to that proposed in the experimental work of (2018). The Lova´sz local lemma due to Erdods and Lova´sz (Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai 11, 1975, pp. 609-627) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP-hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson (Combinatorica, 7 (1987), pp. 365-374) to derive good approximation algorithms for such problems. We use our extension of the local lemma to prove that randomized rounding produces, with nonzero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column-sparse (e.g., routing using short paths, problems on hypergraphs with small dimension degree). This complements certain well-known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan (J. Comput. System Sci., 37 (1988), pp. 130-143), to obtain constructive (algorithmic) versions of our results for covering integer programs. also with probability 1. This result is proved in ?5. The difficulty in proving lower bounds like (1.2) is that one has to consider all possible coverings of the path by small convex sets in the definition of 4-measure. In the past the only successful method has been to use the connection between Hausdorff measures and generalized capacities. The first result of this kind appears in [16] where it is proved that for k >2, the Hausdorff measure with respect to ta is infinite for all a 3 we obtain a "law of the iterated logarithm" for the total time Tk(a, co) spent by the path X in a sphere of radius a as a-* 0+. We also determine the local behavior of the first passage time Pk(a, co) out of a sphere of radius a for k > 1. In order to obtain the required asymptotic results we required good estimates of the distribution functions for the random variables Tk(a, co), Pk(a, c). We use the method developed by Mark Kac to compute these dis- Spanners, emulators, and approximate distance oracles can be viewed as lossy compression schemes that represent an unweighted graph metric in small space, say O(n1+Δ) bits. There is an inherent tradeoff between the sparsity parameter Δ and the stretch function f of the compression scheme, but the qualitative nature of this tradeoff has remained a persistent open problem. It has been known for some time that when Δ ≥ 1 3 there are schemes with constant additive stretch (distance d is stretched to at most f(d) = d + O(1)), and recent results of Abboud and Bodwin show that when Δ In this paper we show that the lower bound of Abboud and Bodwin is just the first step in a hierarchy of lower bounds that characterize the asymptotic behavior of the optimal stretch function f for sparsity parameter Δ ∈ (0,1 3). Specifically, for any integer k ≥ 2, any compression scheme with size [EQUATION] has a sublinear additive stretch function f: f(d) = d + Ω(d1−1 k). This lower bound matches Thorup and Zwick's (2006) construction of sublinear additive emulators. It also shows that Elkin and Peleg's (1 + ϵ, β)-spanners have an essentially optimal tradeoff between Δ, ϵ, and β, and that the sublinear additive spanners of Pettie (2009) and Chechik (2013) are not too far from optimal. To complement these lower bounds we present a new construction of (1 + ϵ, O(k ϵ)k−1)-spanners with size [EQUATION], where hk Our lower bound technique exhibits several interesting degrees of freedom in the framework of Abboud and Bodwin. By carefully exploiting these freedoms, we are able to obtain lower bounds for several related combinatorial objects. We get lower bounds on the size of (β, ϵ)-hopsets, matching Elkin and Neiman's construction (2016), and lower bounds on shortcutting sets for digraphs that preserve the transitive closure. Our lower bound simplifies Hesse's (2003) refutation of Thorup's conjecture (1992), which stated that adding a linear number of shortcuts suffices to reduce the diameter to polylogarithmic. Finally, we show matching upper and lower bounds for graph compression schemes that work for graph metrics with girth at least 2γ + 1. One consequence is that 's (2010) additive O(γ)-spanners with size [EQUATION] cannot be improved in the exponent. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error @math in trace distance required @math copies for a @math -dimensional density matrix of rank @math . Here, we give a theoretical measurement scheme (POVM) that requires @math copies to estimate @math to error @math in infidelity, and a matching lower bound up to logarithmic factors. This implies @math copies suffice to achieve error @math in trace distance. We also prove that for independent (product) measurements, @math copies are necessary in order to achieve error @math in infidelity. For fixed @math , our measurement can be implemented on a quantum computer in time polynomial in @math . This paper addresses two problems lying at the intersection of geometric analysis and theoretical computer science: The non-linear isomorphic Dvoretzky theorem and the design of good approximate distance oracles for large distortion. We introduce the notion of Ramsey partitions of a finite metric space, and show that the existence of good Ramsey partitions implies a solution to the metric Ramsey problem for large distortion (a.k.a. the non-linear version of the isomorphic Dvoretzky theorem, as introduced by Bourgain, Figiel, and Milman in BFM86 ). We then proceed to construct optimal Ramsey partitions, and use them to show that for every @math , every @math -point metric space has a subset of size @math which embeds into Hilbert space with distortion @math . This result is best possible and improves part of the metric Ramsey theorem of Bartal, Linial, Mendel and Naor BLMN05 , in addition to considerably simplifying its proof. We use our new Ramsey partitions to design approximate distance oracles with a universal constant query time, closing a gap left open by Thorup and Zwick in TZ05 . Namely, we show that for every @math point metric space @math , and @math , there exists an @math -approximate distance oracle whose storage requirement is @math , and whose query time is a universal constant. We also discuss applications of Ramsey partitions to various other geometric data structure problems, such as the design of efficient data structures for approximate ranking. We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis. We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. (Random Str Algorithm 37(1):85–99, 2010) asked whether there exists an increasing (M )-Lipschitz embedding from one i.i.d. Bernoulli sequence into an independent copy with positive probability. We give a positive answer for large enough (M ). A closely related problem is to show that two independent Poisson processes on ( R ) are roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled (Ann Appl Probab 20:462–494, 2010). Our theorem also gives a new proof to Winkler’s compatible sequences problem. Our approach does not explicitly depend on the particular geometry of the problems and we believe it will be applicable to a range of multi-scale and random embedding problems.