An algebraic approach to approximate evaluation of a polynomial on a set of real points

The previous best algorithm for approximate evaluation of a polynomial on a real set was due to Rokhlin and required of the order ofmu+nu ³ infinite precision arithmetic operations to approximate [on a fixed bounded setX(m) ofm+1 real points] a degreen polynomial $p\left( z \right) = \sum\nolimits_{i = 0}^n {p_i x^i } $ within the error bound $2^{ - u} \sum\nolimits_{i = 0}^n {\left| {p_i } \right|} $ . We develop an approximation algorithm which exploits algebraic computational techniques and decreases Rokhlin's record estimate toO(mlog² u+nmin-u, logn}). For logu=o(logn), this result may also be favorably compared with the record boundO(m+n)log² n) on the complexity of the exact multipoint polynomial evaluation. The new algorithm can be performed in the fields (or rings) generated by the input values, which enables us to decrease the precision of the computations [by using modular (residue) arithmetic] and to simplify our computations further in the case whereu=O(logn). Our algorithm allows NC and simultaneously processor efficient parallel implementation. Because of the fundamental nature of the multipoint polynomial evaluation, our results have further applications to numerical and algebraic computational problems. In passing, we also show a substantial improvement in the Chinese remainder algorithm for integers based on incorporating Kaminski's fast residue computation.     

Extended Formulations for Polygons

The extension complexity of a polytope?P is the smallest integer?k such that?P is the projection of a polytope?Q with?k facets. We study the extension complexity of n-gons in the plane. First, we give a new proof that the extension complexity of regular n-gons is O(logn), a result originating from work by Ben-Tal and Nemirovski (Math. Oper. Res. 26(2), 193?C205, 2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), and Kaibel and Pashkovich (2011). Second, we prove a lower bound of $\sqrt{2n}$ on the extension complexity of generic n-gons. Finally, we prove that there exist n-gons whose vertices lie on an O(n)×O(n ²) integer grid with extension complexity $\varOmega (\sqrt{n}/\sqrt{\log n})$ .     

Lines Avoiding Balls in Three Dimensions Revisited

Let $\mathcal{B}$ be a collection of n arbitrary balls in ?³. We establish an almost-tight upper bound of O(n ^3+?? ), for any ??>0, on the complexity of the space $\mathcal{F}(\mathcal{B})$ of all the lines that avoid all the members of $\mathcal{B}$ . In particular, we prove that the balls of $\mathcal{B}$ admit O(n ^3+?? ) free isolated tangents, for any ??>0. This generalizes the result of Agarwal et al.?(Discrete Comput. Geom. 34:231?C250, 2005), who established this bound only for congruent balls, and solves an open problem posed in that paper. Our bound almost meets the recent lower bound of ??(n ³) of Glisse and Lazard (Proc. 26th Annu. Symp. Comput. Geom., pp. 48?C57, 2010). Our approach is constructive and yields an algorithm that computes the discrete representation of the boundary of $\mathcal{F}(B)$ in O(n ^3+?? ) time, for any ??>0.     

New lower bounds for Hopcroft's problem

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set ofn points andm hyperplanes in $\mathbb{R}^d $ , is any point contained in any hyperplane? We define a general class ofpartitioning algorithms, and show that in the worst case, for allm andn, any such algorithm requires time Ω(n logm + n ^2/3m^2/3 + m logn) in two dimensions, or Ω(n logm + n ^5/6m^1/2 + n^1/2m^5/6 + m logn) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2O(log^*(n+m)) of the best known upper bound, due to Matou?ek. Previously, the best known lower bound, in any dimension, was Ω(n logm + m logn). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called amonochromatic cover, and derive lower bounds on its size in the worst case. We then show that the running time of any partitioning algorithm is bounded below by the size of some monochromatic cover. As a related result, using a straightforward adversary argument, we derive aquadratic lower bound on the complexity of Hopcroft's problem in a surprisingly powerful decision tree model of computation.     

Approximation algorithms for TTP(2)

We consider the traveling tournament problem, which is a well-known benchmark problem in tournament timetabling. It consists of designing a schedule for a sports league of n teams such that the total traveling costs of the teams are minimized. The most important variant of the traveling tournament problem imposes restrictions on the number of consecutive home games or away games a team may have. We consider the case where at most two consecutive home games or away games are allowed. We show that the well-known independent lower bound for this case cannot be reached and present two approximation algorithms for the problem. The first algorithm has an approximation ratio of ${3/2+\frac{6}{n-4}}$ in the case that n/2 is odd, and of ${3/2+\frac{5}{n-1}}$ in the case that n/2 is even. Furthermore, we show that this algorithm is applicable to real world problems as it yields close to optimal tournaments for many standard benchmark instances. The second algorithm we propose is only suitable for the case that n/2 is even and n????12, and achieves an approximation ratio of 1?+?16/n in this case, which makes it the first ${1+\mathcal{O}(1/n)}$ -approximation for the problem.     

Inverse 1-median problem on trees under weighted Hamming distance

The inverse 1-median problem consists in modifying the weights of the customers at minimum cost such that a prespecified supplier becomes the 1-median of modified location problem. A linear time algorithm is first proposed for the inverse problem under weighted l _?? norm. Then two polynomial time algorithms with time complexities O(n log n) and O(n) are given for the problem under weighted bottleneck-Hamming distance, where n is the number of vertices. Finally, the problem under weighted sum-Hamming distance is shown to be equivalent to a 0-1 knapsack problem, and hence is ${\mathcal{NP}}$ -hard.     

Approximation algorithms for art gallery problems in polygons

In this paper, we present approximation algorithms for minimum vertex and edge guard problems for polygons with or without holes with a total of n vertices. For simple polygons, approximation algorithms for both problems run in O(n⁴) time and yield solutions that can be at most O(logn) times the optimal solution. For polygons with holes, approximation algorithms for both problems give the same approximation ratio of O(logn), but the running time of the algorithms increases by a factor of n to O(n⁵).     

Reeb Graphs: Approximation and Persistence

Given a continuous function f:X→? on a topological space X, its level set f ^?1(a) changes continuously as the real value a changes. Consequently, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f summarizes this information into a graph structure. Previous work on Reeb graph mainly focused on its efficient computation. In this paper, we initiate the study of two important aspects of the Reeb graph, which can facilitate its broader applications in shape and data analysis. The first one is the approximation of the Reeb graph of a function on a smooth compact manifold M without boundary. The approximation is computed from a set of points P sampled from M. By leveraging a relation between the Reeb graph and the so-called vertical homology group, as well as between cycles in M and in a Rips complex constructed from P, we compute the H ₁-homology of the Reeb graph from P. It takes O(nlogn) expected time, where n is the size of the 2-skeleton of the Rips complex. As a by-product, when M is an orientable 2-manifold, we also obtain an efficient near-linear time (expected) algorithm for computing the rank of H ₁(M) from point data. The best-known previous algorithm for this problem takes O(n ³) time for point data. The second aspect concerns the definition and computation of the persistent Reeb graph homology for a sequence of Reeb graphs defined on a filtered space. For a piecewise-linear function defined on a filtration of a simplicial complex K, our algorithm computes all persistent H ₁-homology for the Reeb graphs in $O(n n_{e}^{3})$ time, where n is the size of the 2-skeleton and n _e is the number of edges in K.     

Improved bounds on the max-flow min-cut ratio for multicommodity flows

In this paper we consider the worst case ratio between the capacity of min-cuts and the value of max-flow for multicommodity flow problems. We improve the best known bounds for the min-cut max-flow ratio for multicommodity flows in undirected graphs, by replacing theO(logD) in the bound byO(logk), whereD denotes the sum of all demands, andk demotes the number of commodities. In essence we prove that up to constant factors the worst min-cut max-flow ratios appear in problems where demands are integral and polynomial in the number of commodities.Klein, Rao, Agrawal, and Ravi have previously proved that if the demands and the capacities are integral, then the min-cut max-flow ratio in general undirected graphs is bounded byO(logClogD), whereC denotes the sum of all the capacities. Tragoudas has improved this bound toO(lognlogD), wheren is the number of nodes in the network. Garg, Vazirani and Yannakakis further improved this toO(logklogD). Klein, Plotkin and Rao have proved that for planar networks, the ratio isO(logD).Our result improves the bound for general networks toO(log² k) and the bound for planar networks toO(logk). In both cases our result implies the first non-trivial bound that is independent of the magnitude of the numbers involved. The method presented in this paper can be used to give polynomial time approximation algorithms to the minimum cuts in the network up to the above factors.Preliminary version appeared in Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, 1993, 691-697.Research supported by U.S. Army Research Office Grant DAAL-03-91-G-0102, and by a grant from Mitsubishi Electric Laboratories.Research supported in part by a Packard Fellowship, an NSF PYI award, a Sloan Fellowship, and by the National Science Foundation, the Air Force Office of Scientific Research, and the Office of Naval Research, through NSF grant DMS-8920550.     

A note on propositional proof complexity of some Ramsey-type statements

A Ramsey statement denoted ${n \longrightarrow (k)^2_2}$ says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(n ^k ) and with terms of size ${\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}$ . Let r _k be the minimal n for which the statement holds. We prove that RAM(r _k , k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák??s work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4 ^k , k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R.     

How Close is the Sample Covariance Matrix to?the?Actual Covariance Matrix?

Given a probability distribution in ? ⁿ with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N=N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose that the distribution is supported in a centered Euclidean ball of radius $O(\sqrt{n})$ . We conjecture that the optimal sample size is N=O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson (J. Funct. Anal. 164:60?C72, 1999), which states that N=O(nlog?n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535?C561, 2010), which guarantees that N=O(n) for subexponential distributions.     

Combinatorial constructions for the Zeckendorf sum of digits of polynomial values

Let $p(X)\in\mathbb{Z}[X]$ with $p(\mathbb{N})\subset\mathbb{N}$ be of degree h≥2 and denote by s _F (n) the sum of digits in the Zeckendorf representation of n. We study by combinatorial means three analogues of problems of Gelfond (Acta Arith. 13:259–265, 1967/1968), Stolarsky (Proc. Am. Math. Soc. 71:1–5, 1978) and Lindström (J. Number Theory 65:321–324, 1997) concerning the distribution of s _F on polynomial sequences. First, we show that for m≥2 we have #{n<N:s _F (p(n))≡amodm}? _p,m N ^4/(6h+1) (Gelfond). Secondly, we find the extremal minimal and maximal orders of magnitude of the ratio s _F (p(n))/s _F (n) (Stolarsky). Third, we prove that lim?sup _n→∞ s _F (p(n))/log _φ (p(n))=1/2, where φ denotes the golden ratio (Lindström).     

On lines missing polyhedral sets in 3-space

We show some combinatorial and algorithmic results concerning finite sets of lines and terrains in 3-space. Our main results include:

1. An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of the set of lines that can be translated to infinity without intersecting a given finite set ofn lines, wherec is a suitable constant. This bound is almost tight.
2. AnO(n ^1.5+ε) randomized expected time algorithm that tests whether a directionv exists along which a set ofn red lines can be translated away from a set ofn blue lines without collisions. ε>0 is an arbitrary small but fixed constant.
3. An $O(n^3 2^{c\sqrt {\log n} } )$ upper bound on the worst-case complexity of theenvelope of lines above a terrain withn edges, wherec is a suitable constant.
4. An algorithm for computing the intersection of two polyhedral terrains in 3-space withn total edges in timeO(n ^4/3+ε+k ^1/3 n ^1+ε+klog² n), wherek is the size of the output, and ε>0 is an arbitrary small but fixed constant. This algorithm improves on the best previous result of Chazelleet al. [5].

The tools used to obtain these results include Plücker coordinates of lines, random sampling, and polarity transformations in 3-space.     

Monotone loop models and rational resonance

Let ${T_{n,m}=\mathbb Z_n\times\mathbb Z_m}$ , and define a random mapping ${\phi\colon T_{n,m}\to T_{n,m}}$ by ${\phi(x,y)=(x+1,y)}$ or (x, y?+?1) independently over x and y and with equal probability. We study the orbit structure of such ??quenched random walks?? ${\phi}$ in the limit m, n ?? ??, and show how it depends sensitively on the ratio m/n. For m/n near a rational p/q, we show that there are likely to be on the order of ${\sqrt{n}}$ cycles, each of length O(n), whereas for m/n far from any rational with small denominator, there are a bounded number of cycles, and for typical m/n each cycle has length on the order of n ^4/3.     

The distribution of extreme points in best complex polynomial approximation

LetK be a compact point set in the complex plane having positive logarithmic capacity and connected complement. For anyf continuous onK and analytic in the interior ofK we investigate the distribution of the extreme points for the error in best uniform approximation tof onK by polynomials. More precisely, if $$A_n (f): = \{ z \in K:|f(z) - p_n^* (f;z)| = \parallel f - p_n^* (f)\parallel _K \} ,$$ wherep _n ^* (f) is the polynomial of degree≤n of best uniform approximation tof onK, we show that there is a subsequencen _k with the property that the sequence of (n _k +2)-point Fekete subsets of \(A_{n_k }\) has limiting distribution (ask→∞) equal to the equilibrium distribution forK. Analogues for weighted approximation are also given.     

Monotonicity testing and shortest-path routing on the cube

We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any set of source-sink pairs on the directed hypercube (with all sources and all sinks distinct) can be connected with edge-disjoint paths, then monotonicity of functions $f:\{ 0,1\} ^n \to \mathcal{R}$ can be tested with O(n/∈) queries, for any totally ordered range $\mathcal{R}$ . More generally, if at least a µ(n) fraction of the pairs can always be connected with edge-disjoint paths then the query complexity is O(n/(∈µ(n))). We construct a family of instances of Ω(2 ⁿ ) pairs in n-dimensional hypercubes such that no more than roughly a $\frac{1} {{\sqrt n }}$ fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [16], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≈ n ^3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanski’s conjecture about routing on the hypercube. In particular, we show that for any δ > 0, the n-dimensional hypercube is not $n^{\tfrac{1} {2} - \delta }$ -realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths. We also prove a lower bound of Ω(n/∈) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.     

Graded Identities of Some Simple Lie Superalgebras

We study ?₂-graded identities of Lie superalgebras of the type b(t), t?≥?2, over a field of characteristic zero. Our main result is that the n-th codimension is strictly less than \((\dim b(t))^n\) asymptotically. As a consequence we obtain an upper bound for ordinary (non-graded) PI-exponent for each simple Lie superalgebra b(t), t?≥?3.     

The Euclidean Bottleneck Steiner Path Problem and Other Applications of (α,β)-Pair Decomposition

We consider a geometric optimization problem that arises in network design. Given a set P of n points in the plane, source and destination points s, t∈P, and an integer k>0, one has to locate k Steiner points, such that the length of the longest edge of a bottleneck path between s and t is minimized. In this paper, we present an O(nlog² n)-time algorithm that computes an optimal solution, for any constant k. This problem was previously studied by Hou et al. (in Wireless Networks 16, 1033–1043, 2010), who gave an O(n ²logn)-time algorithm. We also study the dual version of the problem, where a value λ>0 is given (instead of k), and the goal is to locate as few Steiner points as possible, so that the length of the longest edge of a bottleneck path between s and t is at most λ. Our algorithms are based on two new geometric structures that we develop—an (α,β)-pair decomposition of P and a floor (1+ε)-spanner of P. For real numbers β>α>0, an (α,β)-pair decomposition of P is a collection $\mathcal{W}=\{(A_{1},B_{1}),\ldots,(A_{m},B_{m})\}$ of pairs of subsets of P, satisfying the following: (i) For each pair $(A_{i},B_{i}) \in\mathcal {W}$ , both minimum enclosing circles of A _i and B _i have a radius at most α, and (ii) for any p, q∈P, such that |pq|≤β, there exists a single pair $(A_{i},B_{i}) \in\mathcal{W}$ , such that p∈A _i and q∈B _i , or vice versa. We construct (a compact representation of) an (α,β)-pair decomposition of P in time O((β/α)³ nlogn). In some applications, a simpler (though weaker) grid-based version of an (α,β)-pair decomposition of P is sufficient. We call this version a weak (α,β)-pair decomposition of P. For ε>0, a floor (1+ε)-spanner of P is a (1+ε)-spanner of the complete graph over P with weight function w(p,q)=?|pq|?. We construct such a spanner with O(n/ε ²) edges in time O((1/ε ²)nlog² n), even though w is not a metric. Finally, we present two additional applications of an (α,β)-pair decomposition of P. In the first, we construct a strong spanner of the unit disk graph of P, with the additional property that the spanning paths also approximate the number of substantial hops, i.e., hops of length greater than a given threshold. In the second application, we present an O((1/ε ²)nlogn)-time algorithm for computing a one-sided approximation for distance selection (i.e., given k, $1 \le k \le{n \choose2}$ , find the k’th smallest Euclidean distance induced by P), significantly improving the running time of the algorithm of Bespamyatnikh and Segal.     

A computational study and survey of methods for the single-row facility layout problem

The single-row facility layout problem (SRFLP) is an NP-hard combinatorial optimization problem that is concerned with the arrangement of n departments of given lengths on a line so as to minimize the weighted sum of the distances between department pairs. (SRFLP) is the one-dimensional version of the facility layout problem that seeks to arrange rectangular departments so as to minimize the overall interaction cost. This paper compares the different modelling approaches for (SRFLP) and applies a recent SDP approach for general quadratic ordering problems from Hungerländer and Rendl to (SRFLP). In particular, we report optimal solutions for several (SRFLP) instances from the literature with up to 42 departments that remained unsolved so far. Secondly we significantly reduce the best known gaps and running times for large instances with up to 110 departments.     

On approximation of a function and its derivatives by a polynomial and its derivatives

Let g∈C~q[-1, 1] be such that g~((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomialof degree at most n, such that P_n~((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivativesP_n~((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi-mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x~2)~(1/2)~q‖This result is easily extended to an arbitrary f∈C~q[-1, 1], by subtracting from f the polynomial ofminnimal degree which interpolates f~((0))…,f~((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a givenPolynomial to a given function, our results further explain the similarities and differences betweenalgebraic polynomial approximation in C~q[-1, 1] and trigonometric polynomial approximation in thespace of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character,permitting the construction of actual error estimates for simultaneous approximation proedures for smallvalues of q.