pinecone/core-2020-05-10-deduplication huggingface dataset:
Attribute-Efficient Learning in Query and Mistake-Bound Models

AbstractWe consider the problem ofattribute-efficientlearning in query and mistake-bound models. Attribute-efficient algorithms make a number of queries or mistakes that is polynomial in the number of relevant variables in the target function, but only sublinear in the number of irrelevant variables. We consider a variant of the membership query model in which the learning algorithm is given as input the number of relevant variables of the target function. We show that in this model, any projection and embedding closed class of functions (including parity) that can be learned in polynomial time can be learned attribute-efficiently in polynomial time. We show that this does not hold in the randomized membership query model. In the mistake-bound model, we consider the problem of learning attribute-efficiently using hypotheses that are formulas of small depth. Our results extend the work of A. Blum, L. Hellerstein, and N. Littlestone (J. Comput. System Sci.50(1995), 32–40) and N. Bshouty, R. Cleve, S. Kannan, and C. Tamon (in “Proceedings, 7th Annu. ACM Workshop on Comput. Learning Theory,” pp. 130–139, ACM Press, New York, 1994)
Optimal Circular Arc Representations: Properties, Recognition, and Construction

AbstractWe investigate some properties of minimal interval and circular arc representations and give several optimal sequential and parallel recognition and construction algorithms. We show that, among other things, given ans×tinterval or circular arc representation matrix, •deciding if the representation is minimal can be done inO(logs) time withO(st/logs) EREW PRAM processors, or inO(1) time withO(st) common CRCW PRAM processors; •constructing an equivalent minimum interval representation can be done inO(log(st)) time withO(st/log(st)) EREW PRAM processors, or inO(logt/loglogt) time withO(stloglogt/logt) common CRCW PRAM processors, or inO(1) time withO(st) BSR processors; •constructing an equivalent minimal circular arc representation can be done inO(st) time
The Equivalence of Bottom-Up and Top-Down Tree-to-Graph Transducers

AbstractWe introduce the bottom-up tree-to-graph transducer, which is very similar to the usual (total deterministic) bottom-up tree transducer except that it translates trees into hypergraphs rather than trees, using hypergraph substitution instead of tree substitution. If every output hypergraph of the transducer is a jungle, i.e., a hypergraph that can be unfolded into a tree, then the tree-to-graph transducer is said to be tree-generating and naturally defines a tree-to-tree translation. We prove that bottom-up tree-to-graph transducers define the same tree-to-tree translations as the previously introduced top-down tree-to-graph transducers. This is in contrast with the well-known incomparability of the usual bottom-up and top-down tree transducers
The Relative Complexity of NP Search Problems

AbstractPapadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, and a number of them are complete problems. We consider the question of the relative complexity of these search problem classes. We prove several separations which show that in a generic relativized world the search classes are distinct and there is a standard search problem in each of them that is not computationally equivalent to any decision problem. (Naturally, absolute separations would imply thatP≠NP.) Our separation proofs have interesting combinatorial content and go to the heart of the combinatorial principles on which the classes are based. We derive one result via new lower bounds on the degrees of poly- nomials asserted to exist by Hilbert's nullstellensatz over finite fields
On Data Structures and Asymmetric Communication Complexity

AbstractIn this paper we consider two-party communication complexity, the “asymmetric case”, when the input sizes of the two players differ significantly. Most of previous work on communication complexity only considers the total number of bits sent, but we study trade-offs between the number of bits the first player sends and the number of bits the second sends. These types of questions are closely related to the complexity of static data structure problems in the cell probe model. We derive two generally applicable methods of proving lower bounds and obtain several applications. These applications include new lower bounds for data structures in the cell probe model. Of particular interest is our “round elimination” lemma, which is interesting also for the usual symmetric communication case. This lemma generalizes and abstracts in a very clean form the “round reduction” techniques used in many previous lower bound proofs
Randomized Query Processing in Robot Path Planning

AbstractThe subject of this paper is the analysis of a randomized preprocessing scheme that has been used for query processing in robot path planning. The attractiveness of the scheme stems from its general applicability to virtually any path-planning problem, and its empirically observed success. In this paper we initiate a theoretical basis for explaining this empirical success. Under a simple assumption about the configuration space, we show that it is possible to perform preprocessing following which queries can be answered quickly. En route, we consider related problems on graph connectivity in the evasiveness model and art-gallery theorems
Approximations for the Disjoint Paths Problem in High-Diameter Planar Networks

AbstractWe consider the problem of connecting distinguished terminal pairs in a graph via edge-disjoint paths. This is a classical NP-complete problem for which no general approximation techniques are known; it has recently been brought into focus in papers discussing applications to admission control in high-speed networks and to routing in all- optical networks. In this paper we provideO(logn)-approximation algorithms for two natural optimization versions of this problem for the class of nearly Eulerian, uniformly high-diameter planar graphs, which includes two-dimensional meshes and other common planar interconnection networks. We give anO(logn)-approximation to the maximum number of terminal pairs that can be simultaneously connected via edge-disjoint paths, and anO(logn)-approximation to the minimum number of wavelengths needed to route a collection of terminal pairs in the “optical routing” model considered by Raghavan, Upfal, and others. The latter result improves on anO(log2n)-approximation for the special case of the mesh obtained independently by Aumann and Rabani. For both problems theO(logn)-approximation is a consequence of anO(1)-approximation for the special case when all terminal pairs are roughly the same distance apart. Our algorithms make use of a number of new techniques, including the construction of a “crossbar” structure in any nearly Eulerian planar graph, and develops some connections with classical matroid algorithms
Sorting in Linear Time?

AbstractWe show that a unit-cost RAM with a word length ofwbits can sortnintegers in the range 0…2w−1 inO(nloglogn) time for arbitraryw⩾logn, a significant improvement over the bound ofO(nlogn) achieved by the fusion trees of Fredman and Willard. Provided thatw⩾(logn)2+εfor some fixedε>0, the sorting can even be accomplished in linear expected time with a randomized algorithm. Both of our algorithms parallelize without loss on a unit-cost PRAM with a word length ofwbits. The first one yields an algorithm that usesO(logn) time andO(nloglogn) operations on a deterministic CRCW PRAM. The second one yields an algorithm that usesO(logn) expected time andO(n) expected operations on a randomized EREW PRAM, provided thatw⩾(logn)2+εfor some fixedε>0. Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting of multiple-precision integers represented in several words
What's Decidable about Hybrid Automata?

AbstractHybrid automata model systems with both digital and analog components, such as embedded control programs. Many verification tasks for such programs can be expressed as reachability problems for hybrid automata. By improving on previous decidability and undecidability results, we identify a boundary between decidability and undecidability for the reachability problem of hybrid automata. On the positive side, we give an (optimal) PSPACE reachability algorithm for the case of initialized rectangular automata, where all analog variables follow independent trajectories within piecewise-linear envelopes and are reinitialized whenever the envelope changes. Our algorithm is based on the construction of a timed automaton that contains all reachability information about a given initialized rectangular automaton. The translation has practical significance for verification, because it guarantees the termination of symbolic procedures for the reachability analysis of initialized rectangular automata. The translation also preserves theω-languages of initialized rectangular automata with bounded nondeterminism. On the negative side, we show that several slight generalizations of initialized rectangular automata lead to an undecidable reachability problem. In particular, we prove that the reachability problem is undecidable for timed automata augmented with a single stopwatch
Reductions in Circuit Complexity: An Isomorphism Theorem and a Gap Theorem

AbstractWe show that all sets that are complete for NP under nonuniformAC0reductions are isomorphic under nonuniformAC0-computable isomorphisms. Furthermore, these sets remain NP-complete even under nonuniformNC0reductions. More generally, we show two theorems that hold for any complexity class C closed under (uniform)NC1-computable many–one reductions.Gap: The sets that are complete for C underAC0andNC0reducibility coincide.Isomorphism: The sets complete for C underAC0reductions are all isomorphic under isomorphisms computable and invertible byAC0circuits of depth three. Our Gap Theorem does not hold for strongly uniform reductions; we show that there are Dlogtime-uniformAC0-complete sets forNC1that are not Dlogtime-uniformNC0-complete
Integer Programming as a Framework for Optimization and Approximability

AbstractStructural approximation theory seeks to provide a framework for expressing optimization problems and isolating structural or syntactic conditions that explain the apparent difference in the approximation properties of different NP-optimization problems. In this paper, we initiate a study of structural approximation using integer programming (an optimization problem in its own right) as a general framework for expressing optimization problems. We isolate three classes of constant-approximable maximization problems, based on restricting appropriately the syntactic form of the integer programs expressing them. The first of these classes subsumes MaxΣ1, which is the syntactic version of the well-studied class MaxNP. Moreover, by allowing variables to take on not just 0/1 values but rather values in a polylogarithmic or polynomial range, we obtain syntactic maximization classes that are polylog-approximable and poly-approximable, respectively. The other two classes contain problems, such as MaxMatching, for which no previous structural explanation of approximability has been found. We also investigate structurally defined classes of integer programs for minimization problems and show a difference between their maximization counterparts
DNA Models and Algorithms for NP-Complete Problems

AbstractA goal of research on DNA computing is to solve problems that are beyond the capabilities of the fastest silicon-based supercomputers. Adleman and Lipton present exhaustive search algorithms for 3Sat and 3-coloring, which can only be run on small instances and, hence, are not practical. In this paper, we show how improved algorithms can be developed for the 3-coloring and independent set problems. Our algorithms use only the DNA operations proposed by Adleman and Lipton, but combine them in more powerful ways and use polynomial preprocessing on a standard computer to tailor them to the specific instance to be solved. The main contribution of this paper is a more general model of DNA algorithms than that proposed by Lipton. We show that DNA computation for NP-complete problems can do more than just exhaustive search. Further research in this direction will help determine whether or not DNA computing is viable for NP-hard problems. A second contribution is the first analysis of errors that arise in generating the solution space for DNA computation
Zero Knowledge and the Chromatic Number

AbstractWe present a new technique, inspired by zero-knowledge proof systems, for proving lower bounds on approximating the chromatic number of a graph. To illustrate this technique we present simple reductions frommax-3-coloringandmax-3-sat, showing that it is hard to approximate the chromatic number withinΩ(Nδ) for someδ>0. We then apply our technique in conjunction with the probabilistically checkable proofs of Håstad and show that it is hard to approximate the chromatic number to withinΩ(N1−ε) for anyε>0, assuming NP⊈ZPP. Here, ZPP denotes the class of languages decidable by a random expected polynomial-time algorithm that makes no errors. Our result matches (up to low order terms) the known gap for approximating the size of the largest independent set. PreviousO(Nδ) gaps for approximating the chromatic number (such as those by Lund and Yannakakis, and by Furer) did not match the gap for independent set nor extend beyondΩ(N1/2−ε)
NondeterministicNC1Computation

AbstractWe define the counting classes #NC1,GapNC1,PNC1, andC=NC1. We prove that boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that #NC1⊆#L, thatPNC1⊆L, and thatC=NC1⊆L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separations ofACC0fromMOD-PHand that ofTC0from the counting hierarchy. Moreover, we obtain that if dlogtime-uniformity and logspace-uniformity forAC0coincide then the polynomial time hierarchy equalsPSPACE
Deterministic and Randomized Bounded Truth-Table Reductions of P, NL, and L to Sparse Sets

AbstractWe prove that there is no sparse hard set for P under logspace computable bounded truth-table reductions unless P=L. In case of reductions computable in NC1, the collapse goes down to P=NC1. We parameterize this result and obtain a generic theorem allowing us to vary the sparseness condition, the space bound and the number of queries of the truth-table reduction. Another instantiation yields that there is no quasipolynomially dense hard set for P under polylog-space computable truth-table reductions using polylogarithmically many queries unless P is in polylog-space. We also apply the proof technique to NL and L. We establish that there is no sparse hard set for NL under logspace computable bounded truth-table reductions unless NL=L and that there is no sparse hard set for L under NC1-computable bounded truth-table reductions unless L=NC1. We show that all these results carry over to the randomized setting: If we allow two-sided error randomized reductions with confidence at least inversely polynomial, we obtain collapses to the corresponding randomized classes in the multiple access model. In addition, we prove that there is no sparse hard set for NP under two-sided error randomized polynomial-time bounded truth-table reductions with confidence at least inversely polynomial unless NP=RP
Analysis of Practical Backoff Protocols for Contention Resolution with Multiple Servers

AbstractBackoff protocols are probably the most widely used protocols for contention resolution in multiple access channels. In this paper, we analyze the stochastic behavior of backoff protocols for contention resolution among a set of clients and servers. each server being a multiple access channel that deals with contention like an ethernet channel. We use the standard model in which each client generates requests for a given server according to a Bernoulli distribution with a specified mean. Theclient–server request rateof a system is the maximum over all client–server pairs (i,j) of the sum of all request rates associated with either clientior serverj. (Having a subunit client–server request rate is a necessary condition for stability for single-server systems.) Our main result is that any superlinear polynomial backoff protocol is stable for any multiple-server system with a subunit client–server request rate. Our result is the first proof of stability for any backoff protocol for contention resolution with multiple servers. (The multiple-server problem does not reduce to the single-server problem, because each client can only send a single message at any step.) Our result is also the first proof thatanyweakly acknowledgment based protocol is stable for contention resolution with multiple servers and such high request rates. Two special cases of our result are of interest. Hastad, Leighton, and Rogoff have shown that for a single-server system with a subunit client–server request rate anymodifiedsuperlinear polynomial backoff protocol is stable. These modified backoff protocols are similar to standard backoff protocols but require more random bits to implement. The special case of our result in which there is only one server extends the result of Hastad, Leighton, and Rogoff to standard (practical) backoff protocols. Finally, our result applies to dynamic routing in optical networks. Specifically, a special case of our result demonstrates that superlinear polynomial backoff protocols are stable for dynamic routing in optical networks
Characterizing Multiterminal Flow Networks and Computing Flows in Networks of Small Treewidth

AbstractWe show that if a flow network haskinput/output terminals (for the traditional maximum-flow problem,k=2), its external flow pattern (the possible values of flow into and out of the terminals) has two characterizations of size independent of the total number of vertices: a set of 2k+1 inequalities inkvariables representing flow values at the terminals, and a mimicking network with at most 22kvertices and the same external flow pattern as the original network. For the case in which the underlying graph has bounded treewidth, we present sequential and parallel algorithms that can compute these characterizations as well as a flow consistent with any desired feasible external flow (including a maximum flow between two given terminals). For constantk, the sequential algorithm runs inO(n) time onn-vertex networks, and the parallel algorithm runs inO(logn) time on an EREW PRAM withO(n/logn) processors if an explicit tree decomposition of the network of sizeO(n) is given; if not, known algorithms can compute such a tree decomposition inO((logn)2) time usingO(n/(logn)2) processors
Approximating Hyper-Rectangles: Learning and Pseudorandom Sets

AbstractThe PAC learning of rectangles has been studied because they have been found experimentally to yield excellent hypotheses for several applied learning problems. Also, pseudorandom sets for rectangles have been actively studied recently because (i) they are a subproblem common to the derandomization of depth-2 (DNF) circuits and derandomizing randomized logspace, and (ii) they approximate the distribution ofnindependent multivalued random variables. We present improved upper bounds for a class of such problems of “approximating” high-dimensional rectangles that arise in PAC learning and pseudo- randomness
Mining Optimized Association Rules for Numeric Attributes

AbstractGiven a huge database, we address the problem of finding association rules for numeric attributes, such as(Balance∈I)⇒(CardLoan=yes),which implies that bank customers whose balances fall in a rangeIare likely to use card loan with a probability greater thanp. The above rule is interesting only if the rangeIhas some special feature with respect to the interrelation betweenBalanceandCardLoan. It is required that the number of customers whose balances are contained inI(called thesupportofI) is sufficient and also that the probabilitypof the conditionCardLoan=yesbeing met (called theconfidence ratio) be much higher than the average probability of the condition over all the data. Our goal is to realize a system that finds such appropriate ranges automatically. We mainly focus on computing twooptimized ranges: one that maximizes the support on the condition that the confidence ratio is at least a given threshold value, and another that maximizes the confidence ratio on the condition that the support is at least a given threshold number. Using techniques from computational geometry, we present novel algorithms that compute the optimized ranges in linear time if the data are sorted. Since sorting data with respect to each numeric attribute is expensive in the case of huge databases that occupy much more space than the main memory, we instead apply randomized bucketing as the preprocessing method and thus obtain an efficient rule-finding system. Tests show that our implementation is fast not only in theory but also in practice. The efficiency of our algorithm enables us to compute optimized rules for all combinations of hundreds of numeric and Boolean attributes in a reasonable time
Managing Conflicts between Rules

AbstractRules are used as a programming paradigm in several application domains, including active databases, planning, expert systems, and billing. For example, active databases have rules that execute upon the occurrence of particular events if specified condition predicates are satisfied. It is often the case that multiple rules are fireable when a particular event occurs. We propose a declarative mechanism to control the interaction and execution of multiple rules. The mechanism is based upon logical meta-rules that can express various types of relationships between rules. The meta-rules allow us to reason statically about the rule behavior. We can determine, in polynomial time, whether a rule will never execute, whether two rules can ever be executed together, and whether a rule system is guaranteed to have a unique execution set for all possible rules that become fireable. In this paper, we illustrate our techniques using rules in an active database. A system based upon the meta-rules and the static analysis presented here has been found to be of value in a billing application at AT & T to control interactions between discount plans