pinecone/core-2020-05-10-deduplication huggingface dataset:
Quantum Lower Bounds by Quantum Arguments

AbstractWe propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with one input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Ω(N) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via a variety of different techniques
Finding Smooth Integers in Short Intervals Using CRT Decoding

AbstractWe present a new algorithm for CRT list decoding. An instance of the, CRT list decoding problem consists of integers B, 〈p1,…,pn〉 and 〈r1,…,rn〉, where p1<p2<ċ<pn is a sequence of relatively prime integers. The CRT list decoding problem is to find all positive integers x<B such that x=ri modpi for all but e values of i∈{1,…,n}. Suppose B=∏ri=1pi for some integer k. Goldreich, Ron, and Sudan (in “Proc. of STOC'99”, pp. 225–234, 1999) recently gave several applications for this problem and presented the first efficient algorithm that works whenever e (approximately) satisfies e<n−2knlogpnlogp1. Our new algorithm achieves the stronger bound e<n−knlogpnlogp1 (approximately). The improvement is significant when k is relatively close to n, e.g. k>n/3. The bounds we obtain are similar to the bounds obtained by Guruswami and Sudan for Reed–Solomon list decoding. Hence, our algorithm reduces the gap between CRT list decoding and list decoding of Reed–Solomon codes. In addition, we give a new application for CRT list decoding: finding smooth integers in short intervals. Problems of this type come up in several algorithms for factoring large integers. We define and solve a generalized CRT list decoding problem and discuss how it might be used within the quadratic sieve factoring method
Query Strategies for Priced Information

AbstractWe consider a class of problems in which an algorithm seeks to compute a function f over a set of n inputs, where each input has an associated price. The algorithm queries inputs sequentially, trying to learn the value of the function for the minimum cost. We apply the competitive analysis of algorithms to this framework, designing algorithms that incur large cost only when the cost of the cheapest “proof” for the value of f is also large. We provide algorithms that achieve the optimal competitive ratio for functions that include arbitrary Boolean AND/OR trees, and for the problem of searching in a sorted array. We also investigate a model for pricing in this framework and construct, for every AND/OR tree, a set of prices that satisfies a very strong type of equilibrium property
Connectivity and Inference Problems for Temporal Networks

AbstractMany network problems are based on fundamental relationships involving time. Consider, for example, the problems of modeling the flow of information through a distributed network, studying the spread of a disease through a population, or analyzing the reachability properties of an airline timetable. In such settings, a natural model is that of a graph in which each edge is annotated with a time label specifying the time at which its endpoints “communicated.” We will call such a graph a temporal network. To model the notion that information in such a network “flows” only on paths whose labels respect the ordering of time, we call a path time-respecting if the time labels on its edges are non-decreasing. The central motivation for our work is the following question: how do the basic combinatorial and algorithmic properties of graphs change when we impose this additional temporal condition? The notion of a path is intrinsic to many of the most fundamental algorithmic problems on graphs; spanning trees, connectivity, flows, and cuts are some examples. When we focus on time-respecting paths in place of arbitrary paths, many of these problems acquire a character that is different from the traditional setting, but very rich in its own right. We provide results on two types of problems for temporal networks. First, we consider connectivity problems, in which we seek disjoint time-respecting paths between pairs of nodes. The natural analogue of Menger's Theorem for node-disjoint paths fails in general for time-respecting paths; we give a non-trivial characterization of those graphs for which the theorem does hold in terms of an excluded subdivision theorem, and provide a polynomial-time algorithm for connectivity on this class of graphs. (The problem on general graphs is NP-complete.) We then define and study the class of inference problems, in which we seek to reconstruct a partially specified time labeling of a network in a manner consistent with an observed history of information flow
A New Proof of the Weak Pigeonhole Principle

AbstractThe exact complexity of the weak pigeonhole principle is an old and fundamental problem in proof complexity. Using a diagonalization argument, J. B. Paris et al. (J. Symbolic Logic53 (1988), 1235–1244) showed how to prove the weak pigeonhole principle with bounded-depth, quasipolynomial-size proofs. Their argument was further refined by J. Krajı́cek (J. Symbolic Logic59 (1994), 73–86). In this paper, we present a new proof: we show that the weak pigeonhole principle has quasipolynomial-size LK proofs where every formula consists of a single AND/OR of polylog fan-in. Our proof is conceptually simpler than previous arguments, and is optimal with respect to depth
Tighter Lower Bounds for Nearest Neighbor Search and Related Problems in the Cell Probe Model

AbstractWe prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n,d) cells, each containing poly(d, logn) bits. We get a lower bound of Ω(d/logn) on the search time, a significant improvement over the recent bound of Ω(logd) of Borodin et al. This should be contrasted with the upper bound of O(loglogd) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems
New Lowness Results for ZPPNP and Other Complexity Classes

AbstractWe show that the class AM∩coAM is low for ZPPNP. As a consequence, it follows that Graph Isomorphism and several group-theoretic problems are low for ZPPNP. We also show that the class IP[P/poly], consisting of sets that have interactive proof systems with honest provers in P/poly, is also low for ZPPNP. For the nonuniform function classes NPMV/poly, NPSV/poly, and NPMVt/poly, we show the following lowness results: Sets whose characteristic functions are in NPSV/poly and that have program checkers are low for AM and ZPPNP. Self-reducible sets with characteristic functions in NPMVt/poly are low for Σ2p. Sets whose characteristic functions are in NPMV/poly and that have program checkers are low for Σ2p. We also give applications of these lowness results
Some Decision Problems Concerning Semilinearity and Commutation

AbstractLet M be a class of automata (in a precise sense to be defined) and Mc the class obtained by augmenting each automaton in M with finitely many reversal-bounded counters. We show that if the languages defined by M are effectively semilinear, then so are the languages defined by Mc, and, hence, their emptiness problem is decidable. We give examples of how this result can be used to show the decidability of certain problems concerning the equivalence of morphisms on languages. We also prove a surprising undecidability result for commutation of languages: given a fixed two-element code K, it is undecidable whether a given context-free language L commutes with K, i.e., LK=KL
On the Complexity of Pattern Matching for Highly Compressed Two-Dimensional Texts

AbstractWe consider the complexity of problems related to two-dimensional texts (2D-texts) described succinctly. In a succinct description, larger rectangular subtexts are defined in terms of smaller parts in a way similar to that of Lempel–Ziv compression for 1D-texts, or in strings with shortened descriptions as in (Nordic J. Comput.4 (1997) 172–186), or in hierarchical graphs described by context-free graph grammars. A given 2D-text T with many internal repetitions can have a hierarchical description which may be exponentially smaller and which can be given as an input for a pattern-matching algorithm which gives information about T. Such a hierarchical description is given in terms of a straight-line program, or SLP (see Nordic J. Comput.4 (1997) 172–186) or, equivalently, of a 2D grammar. We consider compressed pattern matching, where the input consists of a 2D-pattern P and a hierarchical description of a 2D-text T, and fully compressed pattern matching, where the input consists of hierarchical descriptions of both the pattern P and the text T. For 1D strings, there exist polynomial-time deterministic algorithms for these problems for similar types of succinct text descriptions (J. Comput. System Sci.52 (2) (1996) 299–307; “Proceedings of the 27th Annual Symposium on the Theory of Computing,” pp. 703–712; “Proceedings of the 5th Scandinavian Workshop on Algorithm Theory,” Springer-Verlag, Berlin, 1996; Nordic J. Comput.4(2) (1997), 172–186). We show that the complexity dramatically increases in a 2D setting. For example, compressed 2D-matching is NP-complete, fully compressed 2D-matching is ΣP2-complete, and testing a given occurrence of a 2D compressed pattern is co-NP-complete. On the other hand, we give efficient algorithms for the related problems of randomized equality testing and testing for a given occurrence of an uncompressed pattern. We also show the surprising fact that the compressed size of a subrectangle of a compressed 2D array can grow exponentially, unlike the 1D case
Generalization Error of Combined Classifiers

AbstractWe derive an upper bound on the generalization error of classifiers which can be represented as thresholded convex combinations of thresholded convex combinations of functions. Such classifiers include single hidden-layer threshold networks and voted combinations of decision trees (such as those produced by boosting algorithms). The derived bound depends on the proportion of training examples with margin less than some threshold and the average complexity of the combined functions (where the average is over the weights assigned to each function in the convex combination). The complexity of the individual functions in the combination depends on their closeness to threshold. By representing a decision tree as a thresholded convex combination of weighted leaf functions, we apply this result to bound the generalization error of combinations of decision trees. Previous bounds depend on the margin of the combined classifier and the average complexity of the decision trees in the combination, where the complexity of each decision tree depends on the total number of leaves. Our bound also depends on the margin of the combined classifier and the average complexity of the decision trees, but our measure of complexity for an individual decision tree is based on the distribution of training examples over leaves and can be significantly smaller than the total number of leaves
Unary Context-Free Grammars and Pushdown Automata, Descriptional Complexity and Auxiliary Space Lower Bounds

AbstractIt is well known that a context-free language defined over a one-letter alphabet is regular. This implies that unary context-free grammars and unary pushdown automata can be transformed into equivalent finite automata. In this paper, we study these transformations from a descriptional complexity point of view. In particular, we give optimal upper bounds for the number of states of nondeterministic and deterministic finite automata equivalent to unary context-free grammars in Chomsky normal form. These bounds are functions of the number of variables of the given grammars. We also give upper bounds for the number of states of finite automata simulating unary pushdown automata. As a main consequence, we are able to prove a log log n lower bound for the workspace used by one-way auxiliary pushdown automata in order to accept nonregular unary languages. The notion of space we consider is the so-called weak space concept
A Constant-Factor Approximation Algorithm for the k-Median Problem

AbstractWe present the first constant-factor approximation algorithm for the metric k-median problem. The k-median problem is one of the most well-studied clustering problems, i.e., those problems in which the aim is to partition a given set of points into clusters so that the points within a cluster are relatively close with respect to some measure. For the metric k-median problem, we are given n points in a metric space. We select k of these to be cluster centers and then assign each point to its closest selected center. If point j is assigned to a center i, the cost incurred is proportional to the distance between i and j. The goal is to select the k centers that minimize the sum of the assignment costs. We give a 623-approximation algorithm for this problem. This improves upon the best previously known result of O(logk loglog k), which was obtained by refining and derandomizing a randomized O(log n loglog n)-approximation algorithm of Bartal
A Fully Dynamic Algorithm for Maintaining the Transitive Closure

AbstractThis paper presents an efficient fully dynamic graph algorithm for maintaining the transitive closure of a directed graph. The algorithm updates the adjacency matrix of the transitive closure with each update to the graph; hence, each reachability query of the form “Is there a directed path from i to j?” can be answered in O(1) time. The algorithm is randomized and has a one-sided error; it is correct when answering yes, but has O(1/nc) probability of error when answering no, for any constant c. In acyclic graphs, worst case update time is O(n2). In general graphs, the update time is O(n2.26). The space complexity of the algorithm is O(n2)
Asymptotically Good List-Colorings

AbstractThelist-chromatic index,χl′(H), of a hypergraph H is the leasttsuch that for any assignment oft-setsS(A) to the edgesAof H, there is a proper coloringσof H withσ(A)∈S(A) for allA∈H.Theorem.Let k be fixed and H a hypergraph having edges of size at most k and maximum degree D, and satisfying[formula]Then[formula]Thus if edge sizes are bounded and pairwise degrees are relatively small, we have asymptotic agreement ofχl′ with its trivial lower bound, the maximum degree. The corresponding result for ordinary chromatic index is a theorem of Pippenger and Spencer (J. Combin. Theory Ser. A51(1989), 24–42). On the other hand, even for simple graphs, earlier work falls far short of deciding the asymptotic behavior ofχl′.The “guided-random” method used in the proof is in the spirit of some earlier work and is thought to be of particular interest. One simple ingredient is a martingale inequality which ought to prove useful beyond the present context
A Formulation for the Genus Series for Regular Maps

AbstractEmbedding of vertex regular maps on orientable surfaces have been studied extensively, particularly in the case of low genus where several classes have proved to be tractable. However, the question of determining exact information about embeddings on surfaces of arbitrary genus has proved to be less so. In this paper we show that the genus series for vertex regular maps can be formulated elegantly as the constant term in a series in a way which fully captures information about the genus. It is hoped that the cosntant term formulation will lead to further study of the genus series for regular maps
Log-Concavity and Related Properties of the Cycle Index Polynomials

AbstractLetAndenote thenth-cycle index polynomial, in the variablesXj, for the symmetric group onnletters. We show that if the variablesXjare assigned nonnegative real values which are log-concave, then the resulting quantitiesAnsatisfy the two inequalitiesAn−1An+1⩽A2n⩽((n+1)/n)An−1An+1. This implies that the coefficients of the formal power series exp(g(u)) are log-concave whenever those ofg(u) satisfy a condition slightly weaker than log-concavity. The latter includes many familiar combinatorial sequences, only some of which were previously known to be log-concave. To prove the first inequality we show that in fact the differenceA2n−An−1An+1can be written as a polynomial with positive coefficients in the expressionsXjandXjXk−Xj−1Xk+1,j⩽k. The second inequality is proven combinatorially, by working with the notion of amarkedpermutation, which we introduce in this paper. The latter is a permutation each of whose cycles is assigned a subset of available markers {Mi,j}. Each marker has aweight, wt(Mi,j)=xj, and we relate the second inequality to properties of theweight enumerator polynomials. Finally, using asymptotic analysis, we show that the same inequalities hold fornsufficiently large when theXjare fixed with only finite many nonzero values, with no additional assumption on theXj
Disproofs of Generalized Gilbert–Pollak Conjecture on the Steiner Ratio in Three or More Dimensions

AbstractThe Gilbert–Pollak conjecture, posed in 1968, was the most important conjecture in the area of “Steiner trees.” The “Steiner minimal tree” (SMT) of a point setPis the shortest network of “wires” which will suffice to “electrically” interconnectP. The “minimum spanning tree” (MST) is the shortest such network when onlyintersite line segmentsare permitted. The generalized GP conjecture stated thatρd=infP⊂Rd(lSMT(P)/lMST(P)) was achieved whenPwas the vertices of a regulard-simplex. It was showed previously that the conjecture is true ford=2 and false for 3⩽d⩽9. We settle remaining cases completely in this paper. Indeed, we show that any point set achievingρdmust have cardinality growing at least exponentially withd. The real question now is: What are the true minimal-ρpoint sets? This paper introduces the “d-dimensional sausage” point sets, which may have a lit to do with the answer
Optimal Pooling Designs with Error Detection

AbstractConsider a collection of objects, some of which may be “bad,” and a test which determines whether or not a given subcollection contains no bad objects. The nonadaptive pooling (or group testing) problem involves identifying the bad objects using the least number of tests applied in parallel. The “hypergeometric” case occurs when an upper bound on the number of bad objects is knowna priori. Here, practical considerations lead us to impose the additional requirement ofa posterioriconfirmation that the bound is satisfied. A generalization of the problem in which occasional errors in the test outcomes can occur is also considered. Optimal solutions to the general problem are shown to be equivalent to maximum-size collections of subsets of a finite set satisfying a union condition which generalizes that considered by Erdős and co-workers. Lower bounds on the number of tests required are derived when the number of bad objects is believed to be either 1 or 2. Steiner systems are shown to be optimal solutions in some cases
Central Sets and Their Combinatorial Characterization

AbstractCentral sets in semigroups are known to have very rich combinatorial structure, described by the “Central Sets Theorem”. It has been unknown whether the Central Sets Theorem in fact characterizes central sets, and if not whether some other combinatorial characterization could be found. We derive here a combinatorial characterization of central sets and of the weaker notion of quasi-central sets. We show further that in (N, +) these notions are different and strictly stronger than the characterization provided by the Central Sets Theorem. In addition, we derive an algebraic characterization of sets satisfying the conclusion of the Central Sets Theorem and use this characterization to show that the conclusion of the Central Sets Theorem is a partition regular property in any commutative semigroup
Multicolored Subsets in Colored Hypergraphs

AbstractWe consider anti-Ramsey type problems fork-uniform hypergraphs. A subsetYof ann-element setXis totally multicolored, if the restriction of a coloring ofk-element subsets ofXto [Y]kis a one-to-one coloring. We study the maximum size of totally multicolored subsets for colorings which satisfy the following restriction: any two distinctk-element subsets with at least h elements in the intersection are colored differently. It is shown that the size of the largest totally multicolored subset can be bounded from below by[formula]whereckis a positive constant and this lower bound is sharp (up to a multiplicative constant). Extending earlier results of present and other authors, various related problems concerning edge-colorings of complete graphs are considered. We consider also some application to Sidon sets in arbitrary groups. We show for example the existence of a proper coloring of [X]2with no totally multicolored subset of size 2.21n1/3(lnn)1/3, providedn⩾n0. This sharpens an earlier result of L. Babai and is up to a multiplicative constant the best possible (this follows e.g. from the above mentioned result applied withk=2 andh=1). As a tool for proving the existence of such a coloring we derive an upper bound for permanents of rectangular (0, 1)-matrice
Arities of Permutation Groups: Wreath Products andk-Sets

AbstractWe introduce an invariant of finite permutation groups called the arity which is well known to model theorists but has not been examined from an algebraic point of view. There are few cases in which this invariant is known explicitly. We analyze the behavior of this invariant in power representations of wreath products. We compute it exactly for the action of the symmetric group onnletters on the set ofk-sets from ann-element set, and we estimate it rather closely for symmetric powers of these actions. In the casek=1 we formulate an explicit combinatorial conjecture which would pin down the values exactly in all cases
The Modular Homology of Inclusion Maps and Group Actions

AbstractLetΩbe a finite set ofnelements,Ra ring of characteristicp>0 and denote byMktheR-module withk-element subsets of 0 as basis. The set inclusion map ∂: Mk→Mk−1is the homomorphism which associates to ak-element subset 2 the sum ∂(Δ)=Γ1+Γ2+…+Γkof all its (k−1)-element subsetsΓi. In this paper we study the chain[formula]arising from ∂. We introduce the notion ofp-exactness for a sequence and show that any interval of (*) not includingMn/2orMn+1/2respectively, isp-exact for any primep>0. This result can be extended to various submodules and quotient modules, and we give general constructions for permutation groups onΩof order not divisible byp. If an interval of (*) , or an equivalent sequence arising from a permutation group onΩ, does include the middle term then proper homologies can occur. In these cases we have determined all corresponding Betti numbers. A further application arep-rank formulae for orbit inclusion matrices
Otter's Method and the Homology of Homeomorphically Irreduciblek-Trees

AbstractLet T(k)ndenote the collection of trees withnk+2 labelled leaves which have the property that every internal node has degreemk+2 for somem⩾1. Partially order the elements of T(k)nby definingT⩽SifTcan be obtained fromSby contracting some collection of internal edges. Then T(k)nis a Cohen–Macaulay poset with an action ofSnk+2. We show that the top homology of T(k)nas anSnk+2-module is[formula]whereLie(k)Nis the action ofSNon the 1N-homogeneous piece of the free Liek-algebra. This generalizes the result obtained by Sarah Whitehouse [W] in the casek=1