On Randomization in On-Line Computation

This paper concerns two fundamental but somewhat neglected issues, both related to the design and analysis of randomized on-line algorithms. Motivated by early results in game theory we define several types of randomized on-line algorithms, discuss known conditions for their equivalence, and give a natural example distinguishing between two kinds of randomizations. In particular, we show thatmixedrandomized memoryless paging algorithms can achieve strictly better competitive performance thanbehavioralrandomized algorithms. Next we summarize known—and derive new—“Yao principle” theorems for lower bounding competitive ratios of randomized on-line algorithms. This leads to four different theorems for bounded/unbounded and minimization/maximization problems.     

Reductions among number theoretic problems

Many number theoretic problems such as integer factorization and the discrete logarithm problem have defied all attempts to classify their complexities. Thirteen such problems are considered, none of which is known either to have a deterministic polynomial time solution, or to be complete for any natural complexity class. Failing this, the next best goal is to determine which among these are the “easiest” and which are the “hardest” problems. Toward this end, this paper gives an overview of reductions among the problems. Two reductions are new: a deterministic polynomial time reduction from squarefreeness to Euler's function φ(n), and a probabilistic polynomial time reduction from order modulo a prime power to discrete logarithm modulo a prime power.     

Computational complexity of auditing finite attributes in statistical databases

We study the computational complexity of auditing finite attributes in databases allowing statistical queries. Given a database that supports statistical queries, the auditing problem is to check whether an attribute can be completely determined or not from a given set of statistical information. Some restricted cases of this problem have been investigated earlier, e.g. the complexity of statistical sum queries is known by the work of Kleinberg et al. (J. Comput. System Sci. 66 (2003) 244–253). We characterize all classes of statistical queries such that the auditing problem is polynomial-time solvable. We also prove that the problem is coNP-complete in all other cases under a plausible conjecture on the complexity of constraint satisfaction problems (CSP). The characterization is based on the complexity of certain CSP problems; the exact complexity for such problems is known in many cases. This result is obtained by exploiting connections between auditing and constraint satisfaction, and using certain algebraic techniques. We also study a generalization of the auditing problem where one asks if a set of statistical information imply that an attribute is restricted to K or less different values. We characterize all classes of polynomial-time solvable problems in this case, too.     

Spines of random constraint satisfaction problems: definition and connection with computational complexity

We study the connection between the order of phase transitions in combinatorial problems and the complexity of decision algorithms for such problems. We rigorously show that, for a class of random constraint satisfaction problems, a limited connection between the two phenomena indeed exists. Specifically, we extend the definition of the spine order parameter of Bollobás et al. [10] to random constraint satisfaction problems, rigorously showing that for such problems a discontinuity of the spine is associated with a 2^Ω(n) resolution complexity (and thus a 2^Ω(n) complexity of DPLL algorithms) on random instances. The two phenomena have a common underlying cause: the emergence of “large” (linear size) minimally unsatisfiable subformulas of a random formula at the satisfiability phase transition.We present several further results that add weight to the intuition that random constraint satisfaction problems with a sharp threshold and a continuous spine are “qualitatively similar to random 2-SAT”. Finally, we argue that it is the spine rather than the backbone parameter whose continuity has implications for the decision complexity of combinatorial problems, and we provide experimental evidence that the two parameters can behave in a different manner.AMS subject classification 68Q25, 82B27     

Error-Free and Best-Fit Extensions of Partially Defined Boolean Functions

In this paper, we address a fundamental problem related to the induction of Boolean logic: Given a set of data, represented as a set of binary “truen-vectors” (or “positive examples”) and a set of “falsen-vectors” (or “negative examples”), we establish a Boolean function (or an extension)f, so thatfis true (resp., false) in every given true (resp., false) vector. We shall further require that such an extension belongs to a certain specified class of functions, e.g., class of positive functions, class of Horn functions, and so on. The class of functions represents our a priori knowledge or hypothesis about the extensionf, which may be obtained from experience or from the analysis of mechanisms that may or may not cause the phenomena under consideration. The real-world data may contain errors, e.g., measurement and classification errors might come in when obtaining data, or there may be some other influential factors not represented as variables in the vectors. In such situations, we have to give up the goal of establishing an extension that is perfectly consistent with the given data, and we are satisfied with an extensionfhaving the minimum number of misclassifications. Both problems, i.e., the problem of finding an extension within a specified class of Boolean functions and the problem of finding a minimum error extension in that class, will be extensively studied in this paper. For certain classes we shall provide polynomial algorithms, and for other cases we prove their NP-hardness.     

Equivalence of models for polynomial learnability

In this paper we consider several variants of Valiant's learnability model that have appeared in the literature. We give conditions under which these models are equivalent in terms of the polynomially learnable concept classes they define. These equivalences allow comparisons of most of the existing theorems in Valiant-style learnability and show that several simplifying assumptions on polynomial learning algorithms can be made without loss of generality. We also give a useful reduction of learning problems to the problem of finding consistent hypotheses, and give comparisons and equivalences between Valiant's model and the prediction learning models of Haussler, Littlestone, and Warmuth (in “29th Annual IEEE Symposium on Foundations of Computer Science,” 1988).     

Complexity theoretic hardness results for query learning

We investigate the complexity of learning for the well-studied model in which the learning algorithm may ask membership and equivalence queries. While complexity theoretic techniques have previously been used to prove hardness results in various learning models, these techniques typically are not strong enough to use when a learning algorithm may make membership queries. We develop a general technique for proving hardness results for learning with membership and equivalence queries (and for more general query models). We apply the technique to show that, assuming , no polynomial-time membership and (proper) equivalence query algorithms exist for exactly learning read-thrice DNF formulas, unions of halfspaces over the Boolean domain, or some other related classes. Our hardness results are representation dependent, and do not preclude the existence of representation independent algorithms.?The general technique introduces the representation problem for a class F of representations (e.g., formulas), which is naturally associated with the learning problem for F. This problem is related to the structural question of how to characterize functions representable by formulas in F, and is a generalization of standard complexity problems such as Satisfiability. While in general the representation problem is in , we present a theorem demonstrating that for "reasonable" classes F, the existence of a polynomial-time membership and equivalence query algorithm for exactly learning F implies that the representation problem for F is in fact in co-NP. The theorem is applied to prove hardness results such as the ones mentioned above, by showing that the representation problem for specific classes of formulas is NP-hard. Received: December 6, 1994     

Trivial Reals

Solovay showed that there are noncomputable reals α such that H(α n) ≤ H(1ⁿ)+O(1), where H is prefix-free Kolmogorov complexity. Such H-trivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an H-trivial real. We also analyze various computability-theoretic properties of the H-trivial reals, showing for example that no H-trivial real can compute the halting problem (which means that our construction of an H-trivial computably enumerable set is a particularly easy, injury-free construction of an incomplete c.e. set). Finally, we relate the H-trivials to other classes of “highly nonrandom” reals that have been previously studied.     

Combined super-/substring and super-/subsequence problems

Super-/substring problems and super-/subsequence problems are well-known problems in stringology that have applications in a variety of areas, such as manufacturing systems design and molecular biology. Here we investigate the complexity of a new type of such problem that forms a combination of a super-/substring and a super-/subsequence problem. Moreover we introduce different types of minimal superstring and maximal substring problems. In particular, we consider the following problems: given a set L of strings and a string S, (i) find a minimal superstring (or maximal substring) of L that is also a supersequence (or a subsequence) of S, (ii) find a minimal supersequence (or maximal subsequence) of L that is also a superstring (or a substring) of S. In addition some non-super-/non-substring and non-super-/non-subsequence variants are studied. We obtain several NP-hardness or even MAX SNP-hardness results and also identify types of “weak minimal” superstrings and “weak maximal” substrings for which (i) is polynomial-time solvable.     

Algorithmic and Complexity Issues of Three Clustering Methods in Microarray Data Analysis

The complexity, approximation and algorithmic issues of several clustering problems are studied. These non-traditional clustering problems arise from recent studies in microarray data analysis. We prove the following results. (1) Two variants of the Order-Preserving Submatrix problem are NP-hard. There are polynomial-time algorithms for the Order-Preserving Submatrix problem when the condition or gene sets are fixed. (2) Three variants of the Smooth Clustering problem are NP-hard. The Smooth Subset problem is approximable with ratio 0.5, but it cannot be approximable with ratio 0.5 + δ for any δ > 0 unless NP = P. (3) The inferring plaid model problem is NP-hard.     

The air traffic flow control problem as an application of network theory

In this paper the air traffic flow control is approached as a constrained optimization problem on a multicommodity network. The proposed dynamic mathematical model, by means of a suitable time discretization, has been changed into a “static” one, in order to use static network flow algorithms while taking into account the “unsteady” nature of the air traffic congestion problems.The complexity of the model requires some preliminary effort, such as the identification of some characteristic parameters of the system.In this paper, the network theory is applied to evaluate the influence of the time discretization interval on the model significancy with respect to the actual traffic situation. In particular, a computational example concerning the Rome air traffic control region is presented and the relative results are discussed.     

Remarks on nonlinear adaptive observer design

A solution to the problem of state estimation for systems with unknown parameters is to use some on-line adaptation of the observer parameters. On the basis of various existing results for such adaptive observer designs, a unifying “adaptive observer form” is proposed in this paper. This form indeed emphasizes properties allowing some asymptotic state estimation in spite of unknown parameters, as well as additional properties which further allow parameter estimation. As an example, it is shown how an adaptive observer can be designed for a class of state affine systems.     

Structure and importance of logspace-MOD class

We refine the techniques of Beigelet al. [4] who investigated polynomial-time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes and demonstrate their significance by proving that all standard problems of linear algebra over the finite ringsZ/kZ are complete for these classes. We then define new complexity classes LogFew and LogFew and identify them as adequate logspace versions of Few and Few. We show that LogFew is contained in and that LogFew is contained in for allk. Also an upper bound for in terms of computation of integer determinants is given from which we conclude that all logspace counting classes are contained in.     

Parameterized complexity of candidate control in elections and related digraph problems

There are different ways for an external agent to influence the outcome of an election. We concentrate on “control” by adding or deleting candidates. Our main focus is to investigate the parameterized complexity of various control problems for different voting systems. To this end, we introduce natural digraph problems that may be of independent interest. They help in determining the parameterized complexity of control for different voting systems including Llull, Copeland, and plurality voting. Devising several parameterized reductions, we provide an overview of the parameterized complexity of the digraph and control problems with respect to natural parameters such as adding/deleting only a bounded number of candidates or having only few voters.     

Time-Space Tradeoffs for Undirected Graph Traversal by Graph Automata

We investigate time-space tradeoffs for traversing undirected graphs, using a variety of structured models that are all variants of Cook and Rackoff's “Jumping Automata for Graphs.” Our strongest tradeoff is a quadratic lower bound on the product of time and space for graph traversal. For example, achieving linear time requires linear space, implying that depth-first search is optimal. Since our bound in fact applies to nondeterministic algorithms fornonconnectivity, it also implies that closure under complementation of nondeterministic space-bounded complexity classes is achieved only at the expense of increased time. To demonstrate that these structured models are realistic, we also investigate their power. In addition to admitting well known algorithms such as depth-first search and random walk, we show that one simple variant of this model is nearly as powerful as a Turing machine. Specifically, for general undirected graph problems, it can simulate a Turing machine with only a constant factor increase in space and a polynomial factor increase in time.     

Schnorr Randomness

Schnorr randomness is a notion of algorithmic randomness for real numbers closely related to Martin-Löf randomness. Since its initial development in the 1970s, the notion has received considerably less attention than Martin-Löf randomness. In this article, we explore the properties of Schnorr random reals, and in particular the c.e. Schnorr random reals. We show that there are c.e. reals that are Schnorr random but not Martin-Löf random, and provide a new characterization of Schnorr random real numbers in terms of prefix-free machines. We prove that unlike Martin-Löf random c.e. reals, not all Schnorr random c.e. reals are Turing complete, though all are in high Turing degrees. We use the machine characterization to define a notion of “Schnorr reducibility” which allows us to calibrate the Schnorr complexity of reals. We define the class of “Schnorr trivial” reals, which are ones whose initial segment complexity is identical with the computable reals, and demonstrate that this class has non-computable members.     

Complexity theory of parallel time and hardware

The parallel resources time and hardware and the complexity classes defined by them are studied using the aggregate model. The equivalence of complexity classes defined by sequential space and uniform aggregate hardware is established. Aggregate time is related to (bounded fanin) circuit depth and, similarly, aggregate hardware is related to circuit width. Interelationships between aggregate time and hardware follow as corollaries. Aggregate time is related to the sequential resource reversal. Simultaneous relationships from aggregate hardware and time to sequential space and reversal are shown (and conversely), and these are used as evidence for an “extended parallel computation thesis.” These simultaneous relationships provide new characterizations for the simultaneous parallel complexity class NC and for the complementary class SC. The evaluation of monotone planar circuits is shown to be in NC, in fact in LOGCFL.     

Extracting a polyhedron from a single-view sketch: Topological construction of a wireframe sketch with minimal hidden elements

An essential prerequisite to construct a manifold trihedral polyhedron from a given natural (or partial-view) sketch is solution of the “wireframe sketch from a single natural sketch (WSS)” problem, which is the subject of this paper. Published solutions view WSS as an “image-processing”/“computer vision” problem where emphasis is placed on analyzing the given input (natural sketch) using various heuristics. This paper proposes a new WSS method based on robust tools from graph theory, solid modeling and Euclidean geometry. Focus is placed on producing a minimal wireframe sketch that corresponds to a topologically correct polyhedron.     

Dynamic Dictionary Matching in External Memory

In thedynamic dictionary matchingproblem, a dictionaryDcontains a set of patterns that can change over time under insertion and deletion of individual patterns. Given an arbitrary textT, we must efficiently list all the dictionary patterns that occur at each text position. We investigate the I/O complexity of this problem for a large dictionary that must be stored in external storage devices. By following a completely new approach, we devise an efficient solution which is based upon the SB-tree data structure (P. Ferragina and R. Grossi, 1995,in“Proc. ACM Symposium on Theory of Computing,” pp. 693–702), and a novel notion of certificate for the dictionary matching problem. Our data structure can be adapted to efficiently work in main memory and to solve other problems, thus providing a new insight into the nature of the dictionary matching problem.     

Validating for Liveness in Hidden Adversary Systems

Multi-stream interactive systems can be seen as “hidden adversary” systems (HAS), where the observable behaviour on any interaction channel is affected by interactions happening on other channels. One way of modelling HAS is in the form of a multi-process I/O automata, where each interacting process appears as a token in a shared state space. Constraints in the state space specify how the dynamics of one process affects other processes. We define the “liveness criterion” of each process as the end objective to be achieved by the process. The problem now for each process is to achieve this objective in the face of unforeseen interferences from other processes. In an earlier paper, it was proposed that this uncertainty can be mitigated by collaboration among the disparate processes. Two types of collaboration philosophies were also suggested: altruistic collaboration and pragmatic collaboration. This paper addresses the HAS validation problem where processes collaborate altruistically.