FOCS 2011
TechTalks from event: FOCS 2011
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8

On Range Searching in the Group Model and Combinatorial DiscrepancyIn this paper we establish an intimate connection between dynamic range searching in the group model and combinatorial discrepancy. Our result states that, for a broad class of range searching data structures (including all known upper bounds), it must hold that $t_u t_q = \Omega(\disc^2/\lg n)$ where $t_u$ is the worst case update time, $t_q$ the worst case query time and $\disc$ is the combinatorial discrepancy of the range searching problem in question. This relation immediately implies a whole range of exceptionally high and neartight lower bounds for all of the basic range searching problems. We list a few of them in the following: \begin{itemize} \item For halfspace range searching in $d$dimensional space, we get a lower bound of $t_u t_q = \Omega(n^{11/d}/\lg n)$. This comes within a $\lg n \lg \lg n$ factor of the best known upper bound. \item For orthogonal range searching in $d$dimensional space, we get a lower bound of $t_u t_q = \Omega(\lg^{d2+\mu(d)}n)$, where $\mu(d)>0$ is some small but strictly positive function of $d$. \item For ball range searching in $d$dimensional space, we get a lower bound of $t_u t_q = \Omega(n^{11/d}/\lg n)$. \end{itemize} We note that the previous highest lower bound for any explicit problem, due to Patrascu [STOC'07], states that $t_q =\Omega((\lg n/\lg(\lg n+t_u))^2)$, which does however hold for a less restrictive class of data structures. Our result also has implications for the field of combinatorial discrepancy. Using textbook range searching solutions, we improve on the best known discrepancy upper bound for axisaligned rectangles in dimensions $d \geq 3$.

A Randomized Rounding Approach to the Traveling Salesman ProblemFor some positive constant $\epsilon_0$, we give a $(\frac{3}{2}\epsilon_0$)approximation algorithm for the following problem: given a graph $G_0=(V,E_0)$, find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in $G_0$. The result improves on the $\frac{3}{2}$approximation algorithm due to Christofides for this special case.Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or Tjoin) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation. Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the Tjoin polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest.

Approximating Graphic TSP by MatchingsWe present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graphTSP), the approach yields a 1.461approximation algorithm with respect to the HeldKarp lower bound. For graphTSP restricted to a class of graphs that contains degree three bounded and clawfree graphs, we show that the integrality gap of the HeldKarp relaxation matches the conjectured ratio 4/3. The framework allows for generalizations in a natural way and also leads to a 1.586approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.
9A

A Unified Continuous Greedy Algorithm for Submodular MaximizationThe study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic game theory and combinatorial optimization. Classical works on these problems are mostly combinatorial in nature. Recently, however, many results based on continuous algorithmic tools have emerged. The main bottleneck of such continuous techniques is how to approximately solve a nonconvex relaxation for the submodular problem at hand. Thus, the efficient computation of better fractional solutions immediately implies improved approximations for numerous applications. A simple and elegant method, called ``continuous greedy'', successfully tackles this issue for monotone submodular objective functions, however, only much more complex tools are known to work for general nonmonotone submodular objectives. In this work we present a new unified continuous greedy algorithm which finds approximate fractional solutions for both the nonmonotone and monotone cases, and improves on the approximation ratio for many applications.For general nonmonotone submodular objective functions, our algorithm achieves an improved approximation ratio of about 1/e. For monotone submodular objective functions, our algorithm achieves an approximation ratio that depends on the density of the polytope defined by the problem at hand, which is always at least as good as the previously known best approximation ratio of 1  1/e. Some notable immediate implications are an improved 1/eapproximation for maximizing a nonmonotone submodular function subject to a matroid or O(1)knapsack constraints, and informationtheoretic tight approximations for Submodular MaxSAT and Submodular Welfare with k players, for any number of players k. A framework for submodular optimization problems, called the contention resolution framework, was introduced recently by Chekuri et al. The improved approximation ratio of the unified continuous greedy algorithm implies improved approximation ratios for many problems through this framework. Moreover, via a parameter called stopping time, our algorithm merges the relaxation solving and renormalization steps of the framework, and achieves, for some applications, further improvements. We also describe new monotone balanced contention resolution schemes for various matching, scheduling and packing problems, thus, improving the approximations achieved for these problems via the framework.

Enumerative Lattice Algorithms in any Norm via Mellipsoid CoveringsWe give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming (IP). Our enumeration technique relies on a classical concept from asymptotic convex geometry known as the \emph{Mellipsoid}, and uses as a crucial subroutine the recent algorithm of Micciancio and Voulgaris (STOC 2010) for lattice problems in the $\ell_{2}$ norm. As a main technical contribution, which may be of independent interest, we build on the techniques of Klartag (Geometric and Functional Analysis, 2006) to give an expected $2^{O(n)}$time algorithm for computing an Mellipsoid for any $n$dimensional convex body. As applications, we give deterministic $2^{O(n)}$time and space algorithms for solving exact SVP, and exact CVP when the target point is sufficiently close to the lattice, on $n$dimensional lattices \emph{in any (semi)norm} given an Mellipsoid of the unit ball. In many norms of interest, including all $\ell_{p}$ norms, an Mellipsoid is computable in deterministic $\poly(n)$ time, in which case these algorithms are fully deterministic. Here our approach may be seen as a derandomization of the ``AKS sieve'' for exact SVP and CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). As a further application of our SVP algorithm, we derive an expected $O(f^*(n))^n$time algorithm for Integer Programming, where $f^*(n)$ denotes the optimal bound in the socalled ``flatness theorem,'' which satisfies $f^*(n) = O(n^{4/3} \polylog(n))$ and is conjectured to be $f^{*}(n)=\Theta(n)$. Our runtime improves upon the previous best of $O(n^{2})^{n}$ by Hildebrand and K{\"o}ppe (2010).

A nearly mlogn time solver for SDD linear systemsWe present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ nonzero entries and a vector $b$ such that $A\bar{x} = b$ for some (unknown) vector $\bar{x}$, our algorithm computers a vector $x$ such that ${x}\bar{x}_A < \epsilon \bar{x}_A $ \footnote{$\cdot_A$ denotes the Anorm} in time $${\tilde O}(m\log n \log (1/\epsilon)). The solver utilizes in a standard way a `preconditioning' chain of progressively sparser graphs. To claim the faster running time we make a twofold improvement in the algorithm for constructing the chain. The new chain exploits previously unknown properties of the graph sparsification algorithm given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning properties. We also present an algorithm of independent interest that constructs nearlytight lowstretch spanning trees in time $\tilde{O}(m\log{n})$, a factor of $O(\log{n})$ faster than the algorithm in [Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the construction time of the preconditioning chain.

Balls and Bins: Smaller Hash Families and Faster EvaluationA fundamental fact in the analysis of randomized algorithm is that when $n$ balls are hashed into $n$ bins independently and uniformly at random, with high probability each bin contains at most $O(\log n / \log \log n)$ balls. In various applications, however, the assumption that a truly random hash function is available is not always valid, and explicit functions are required. In this paper we study the size of families (or, equivalently, the description length of their functions) that guarantee a maximal load of $O(\log n / \log \log n)$ with high probability, as well as the evaluation time of their functions. Whereas such functions must be described using $\Omega(\log n)$ bits, the best upper bound was formerly $O(\log^2 n / \log \log n)$ bits, which is attained by $O(\log n / \log \log n)$wise independent functions. Traditional constructions of the latter offer an evaluation time of $O(\log n / \log \log n)$, which according to Siegel's lower bound [FOCS '89] can be reduced only at the cost of significantly increasing the description length. We construct two families that guarantee a maximal load of $O(\log n / \log \log n)$ with high probability. Our constructions are based on two different approaches, and exhibit different tradeoffs between the description length and the evaluation time. The first construction shows that $O(\log n / \log \log n)$wise independence can in fact be replaced by ``gradually increasing independence'', resulting in functions that are described using $O(\log n \log \log n)$ bits and evaluated in time $O(\log n \log \log n)$. The second construction is based on derandomization techniques for spacebounded computations combined with a tailored construction of a pseudorandom generator, resulting in functions that are described using $O(\log^{3/2} n)$ bits and evaluated in time $O(\sqrt{\log n})$. The latter can be compared to Siegel's lower bound stating that $O(\log n / \log \log n)$wise independent functions that are evaluated in time $O(\sqrt{\log n})$ must be described using $\Omega(2^{\sqrt{\log n}})$ bits.
9B

Lexicographic Products and the Power of NonLinear Network CodingWe introduce a technique for establishing and amplifying gaps between parameters of network coding and index coding. The technique uses linear programs to establish separations between combinatorial and codingtheoretic parameters and applies hypergraph lexicographic products to amplify these separations. This entails combining the dual solutions of the lexicographic multiplicands and proving that they are a valid dual of the product. Our result is general enough to apply to a large family of linear programs. This blend of linear programs and lexicographic products gives a recipe for constructing hard instances in which the gap between combinatorial or codingtheoretic parameters is polynomially large. We find polynomial gaps in cases in which the largest previously known gaps were only small constant factors or entirely unknown. Most notably, we show a polynomial separation between linear and nonlinear network coding rates. This involves exploiting a connection between matroids and index coding to establish a previously unknown separation between linear and nonlinear index coding rates. We also construct index coding problems with a polynomial gap between the broadcast rate and the trivial lower bound for which no gap was previously known.

Quadratic GoldreichLevin TheoremsDecomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The GoldreichLevin algorithm can be viewed as an algorithmic analogue of such a decomposition as it gives a way to efficiently find the linear phases associated with large Fourier coefficients. In the study of "quadratic Fourier analysis", higherdegree analogues of such decompositions have been developed in which the pseudorandomness property is stronger but the structured part correspondingly weaker. For example, it has previously been shown that it is possible to express a bounded function as a sum of a few quadratic phases plus a part that is small in the $U^3$ norm, defined by Gowers for the purpose of counting arithmetic progressions of length 4. We give a polynomial time algorithm for computing such a decomposition. A key part of the algorithm is a local selfcorrection procedure for ReedMuller codes of order 2 (over $\F_2^n$) for a function at distance $1/2\epsilon$ from a codeword. Given a function $f:\F_2^n \to \{1,1\}$ at fractional Hamming distance $1/2\epsilon$ from a quadratic phase (which is a codeword of ReedMuller code of order 2), we give an algorithm that runs in time polynomial in $n$ and finds a codeword at distance at most $1/2\eta$ for $\eta = \eta(\epsilon)$. This is an algorithmic analogue of Samorodnitsky's result, which gave a tester for the above problem. To our knowledge, it represents the first instance of a correction procedure for any class of codes, beyond the listdecoding radius. In the process, we give algorithmic versions of results from additive combinatorics used in Samorodnitsky's proof and a refined version of the inverse theorem for the Gowers $U^3$ norm over $\F_2^n$.

Optimal testing of multivariate polynomials over small prime fieldsWe consider the problem of testing if a given function f:F_q^n > F_q is close to a nvariate degree d polynomial over the finite field F_q of q elements. The natural, lowquery, test for this property would be to pick the smallest dimension t = t_{q,d}~ d/q such that every function of degree greater than d reveals this feature on some tdimensional affine subspace of F_q^n and to test that f when restricted to a random tdimensional affine subspace is a polynomial of degree at most d on this subspace. Such a test makes only q^t queries, independent of n. Previous works, by Alon et al. (AKKLR), and Kaufman & Ron and Jutla et al., showed that this natural test rejected functions that were \Omega(1)far from degree dpolynomials with probability at least \Omega(q^{t}) (the results of Kaufman & Ron hold for all fields F_q, while the results of Jutla et al. hold only for fields of prime order). Thus to get a constant probability of detecting functions that were at constant distance from the space of degree d polynomials, the tests made q^{2t} queries. Kaufman & Ron also noted that when q is prime, then q^t queries are necessary. Thus these tests were off by at least a quadratic factor from known lower bounds. It was unclear if the soundness analysis of these tests were tight and this question relates closely to the task of understanding the behavior of the Gowers Norm. This motivated the work of Bhattacharyya et al., who gave an optimal analysis for the case of the binary field and showed that the natural test actually rejects functions that were \Omega(1)far from degree dpolynomials with probability at least \Omega(1). In this work we give an optimal analysis of this test for all fields showing that the natural test does indeed reject functions that are \Omega(1)far from degree $d$ polynomials with \Omega(1)probability. Our analysis thus shows that this test is optimal (matches known lower bounds) when q is prime. (It is also potentially best possible for all fields.) Our approach extends the proof technique of Bhattacharyya et al., however it has to overcome many technical barriers in the process. The natural extension of their analysis leads to an O(q^d) query complexity, which is worse than that of Kaufman and Ron for all q except 2! The main technical ingredient in our work is a tight analysis of the number of ``hyperplanes'' (affine subspaces of codimension $1$) on which the restriction of a degree d polynomial has degree less than $d$. We show that the number of such hyperplanes is at most O(q^{t_{q,d}})  which is tight to within constant factors.

Tight lower bounds for 2query LCCs over finite fieldsA Locally Correctable Code (LCC) is an error correcting code that has a probabilistic selfcorrecting algorithm that, with high probability, can correct any coordinate of the codeword by looking at only a few other coordinates, even if a fraction $\delta$ of the coordinates are corrupted. LCC's are a stronger form of LDCs (Locally Decodable Codes) which have received a lot of attention recently due to their many applications and surprising constructions.In this work we show a separation between 2query LDCs and LCCs over finite fields of prime order. Specifically, we prove a lower bound of the form $p^{\Omega(\delta d)}$ on the length of linear $2$query LCCs over $\F_p$, that encode messages of length $d$. Our bound improves over the known bound of $2^{\Omega(\delta d)}$ \cite{GKST06,KdW04, DS07} which is tight for LDCs. Our proof makes use of tools from additive combinatorics which have played an important role in several recent results in Theoretical Computer Science. We also obtain, as corollaries of our main theorem, new results in incidence geometry over finite fields. The first is an improvement to the SylvesterGallai theorem over finite fields \cite{SS10} and the second is a new analog of Beck's theorem over finite fields.