Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

Self-intersection local time for Gaussian ′( )-processes: Existence, path continuity and examples

In this paper we develop a criterion for existence or non-existence of self-intersection local time (SILT) for a wide class of Gaussian ′( ^d)-valued processes, we show that quite generally the SILT process has continuous paths, and we give several examples which illustrate existence of SILT for different ranges of dimensions (e.g., d ≤ 3, d ≤ 7 and 5 ≤ d ≤ 11 in the Brownian case). Some of the examples involve branching and exhibit “dimension gaps”. Our results generalize the work of Adler and coauthors, who studied the special case of “density processes” and proved that SILT paths are cadlag in the Brownian case making use of a “particle picture” approximation (this technique is not available for our general formulation).     

Induced disjoint paths problem in a planar digraph

As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}. The objective is to find k paths P₁,…,P_k such that P_i is a path from s_i to t_i and P_i and P_j have neither common vertices nor adjacent vertices for any distinct i,j.The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We investigate the computational complexity of several variants of the induced disjoint paths problem. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed (or undirected) planar graph, (ii) NP-hard when k=2 and G is an acyclic directed graph, (iii) NP-hard when k=2 and G is an undirected general graph.As an application of our first result, we show that we can find in polynomial time certain structures called a “hole” and a “theta” in a planar graph.     

Homotopic Fréchet distance between curves or, walking your dog in the woods in polynomial time

The Fréchet distance between two curves in the plane is the minimum length of a leash that allows a dog and its owner to walk along their respective curves, from one end to the other, without backtracking. We propose a natural extension of Fréchet distance to more general metric spaces, which requires the leash itself to move continuously over time. For example, for curves in the punctured plane, the leash cannot pass through or jump over the obstacles (“trees”). We describe a polynomial-time algorithm to compute the homotopic Fréchet distance between two given polygonal curves in the plane minus a given set of polygonal obstacles.     

The Enumeration of Permutations with a Prescribed Number of “Forbidden” Patterns

We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of “forbidden” patterns that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method completely in terms of the patterns “abc,” “cab,” and “abcd.”     

Optimal paths in bi-attribute networks with fractional cost functions

An important routing problem is to determine an optimal path through a multi-attribute network which minimizes a cost function of path attributes. In this paper, we study an optimal path problem in a bi-attribute network where the cost function for path evaluation is fractional. The problem can be equivalently formulated as the “bi-attribute rational path problem” which is known to be NP-complete. We develop an exact approach to find an optimal simple path through the network when arc attributes are non-negative. The approach uses some path preference structures and elimination techniques to discard, from further consideration, those (partial) paths that cannot be parts of an optimal path. Our extensive computational results demonstrate that the proposed method can find optimal paths for large networks in very attractive times.     

Solution of a problem of Ulam

In this paper we solve the following Ulam problem: “Give conditions in order for a linear mapping near an approximately linear mapping to exist” and establish results involving a product of powers of norms [[5.]; [5.]; [5.]]. There has been much activity on a similar “ -isometry” problem of Ulam [ [1.], 633–636; [2.], 263–277; [4.]]. This work represents an improvement and generalization of the work of [3.], 222–224].     

Orthogonal Rational Functions and Nested Disks

In Akhiezer's book [“The Classical Moment Problem and Some Related Questions in Analysis,” Oliver & Boyd, Edinburghasol;London, 1965] the uniqueness of the solution of the Hamburger moment problem, if a solution exists, is related to a theory of nested disks in the complex plane. The purpose of the present paper is to develop a similar nested disk theory for a moment problem that arises in the study of certain orthogonal rational functions. Let {α_n}^∞_n=0be a sequence in the open unit disk in the complex plane, let

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( /|α_k|=−1 whenα_k=0), and let

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We consider the following “moment” problem: Given a positive-definite Hermitian inner product ·, · on × , find a non-decreasing functionμon [−π, π] (or a positive Borel measureμon [−π,π)) such that

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In particular we give necessary and sufficient conditions for the uniqueness of the solution in the case that If this series diverges the solution is always unique.     

Enumeration of permutations containing a prescribed number of occurrences of a pattern of length three

We consider the problem of enumerating the permutations containing exactly k occurrences of a pattern of length 3. This enumeration has received a lot of interest recently, and there are a lot of known results. This paper presents an alternative approach to the problem, which yields a proof for a formula which so far only was conjectured (by Noonan and Zeilberger). This approach is based on bijections from permutations to certain lattice paths with “jumps,” which were first considered by Krattenthaler.     

A graph search algorithm for indoor pursuit/evasion

Using concepts from both robotics and graph theory, we formulate the problem of indoor pursuit/evasion in terms of searching the nodes of a graph for a mobile evader. We present the IGNS (Iterative Greedy Node Search) algorithm, which performs offline guaranteed search (i.e. no matter how the evader moves, it will eventually be captured). Furthermore, the algorithm produces an internal search (the searchers move only along the edges of the graph; “teleporting” is not used) and exploits non-monotonicity, extended visibility and finite evader speed to reduce the number of searchers required to clear an environment. We present search experiments for several indoor environments, in all of which the algorithm succeeds in clearing the graph (i.e. capturing the evader).     

k-Best Solutions under Distance Constraints in Valuated -Matroids

In discrete maximization problems one usually wants to find an optimal solution. However, in several topics like “alignments,” “automatic speech recognition,” and “computer chess” people are interested to find thekbest solutions for somek ≥ 2. We demand that theksolutions obey certain distance constraints to avoid that thekalternatives are too similar. Several results for valuated -matroids are presented, some of them concerning time complexity of algorithms.     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

Māl, enunciations, and the prehistory of Arabic algebra

Medieval Arabic algebra books intended for practical training generally have in common a first “book” which is divided into two sections: one on the methods of solving simplified equations and manipulating expressions, followed by one consisting of worked-out problems. By paying close attention to the wording of the problems in the books of al-Khwārizmī, Abū Kāmil, and Ibn Badr, we reveal the different ways the word māl was used. In the enunciation of a problem it is a common noun meaning “quantity,” while in the solution it is the proper noun naming the square of “thing” (shay '). We then look into the differences between the wording of enunciations and equations, which clarify certain problems solved without “thing,” and help explain the development of algebra before the time of al-Khwārizmī.     

Existence of nonstationary bubbles in higher dimensions

We are interested in the existence of travelling-waves for the nonlinear Schrödinger equation in R^N with “ψ³−ψ⁵”-type nonlinearity. First, we prove an abstract result in critical point theory (a local variant of the classical saddle-point theorem). Using this result, we get the existence of travelling-waves moving with sufficiently small velocity in space dimension N4.     

The b-transform

The b-transform is used to convert entire functions into “primary b-functions” by replacing the powers and factorials in the Taylor series of the entire function with corresponding “generalized powers” (which arise from a polynomial function with combinatorial applications) and “generalized factorials.” The b-transform of the exponential function turns out to be a generalization of the Euler partition generating function, and partition generating functions play a key role in obtaining results for the b-transforms of the elementary entire transcendental functions. A variety of normal-looking results arise, including generalizations of Euler's formula and De Moivre's theorem. Applications to discrete probability and applied mathematics (i.e., damped harmonic motion) are indicated. Also, generalized derivatives are obtained by extending the concept of a b-transform.     

AIC, Overfitting Principles, and the Boundedness of Moments of Inverse Matrices for Vector Autotregressions and Related Models

In his somewhat informal derivation, Akaike (in “Proceedings of the 2nd International Symposium Information Theory” (C. B. Petrov and F. Csaki, Eds.), pp. 610–624, Academici Kiado, Budapest, 1973) obtained AIC's parameter-count adjustment to the log-likelihood as a bias correction: it yields an asymptotically unbiased estimate of the quantity that measures the average fit of the estimated model to an independent replicate of the data used for estimation. We present the first mathematically complete derivation of an analogous property of AIC for comparing vector autoregressions fit to weakly stationary series. As a preparatory result, we derive a very general “overfitting principle,” first formulated in a more limited context in Findley (Ann. Inst. Statist. Math.43, 509–514, 1991), asserting that a natural measure of an estimated model's overfit due to parameter estimation is equal, asymptotically, to a measure of its accuracy loss with independent replicates. A formal principle of parsimony for fitted models is obtained from this, which for nested models, covers the situation in which all models considered are misspecified. To prove these results, we establish a set of general conditions under which, for each τ1, the absolute τth moments of the entries of the inverse matrices associated with least squares estimation are bounded for sufficiently large sample sizes.     

Computing homotopic shortest paths in the plane

We address the problem of computing homotopic shortest paths in the presence of obstacles in the plane. Problems on homotopy of paths received attention very recently [Cabello et al., in: Proc. 18th Annu. ACM Sympos. Comput. Geom., 2002, pp. 160–169; Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. We present two output-sensitive algorithms, for simple paths and non-simple paths. The algorithm for simple paths improves the previous algorithm [Efrat et al., in: Proc. 10th Annu. European Sympos. Algorithms, 2002, pp. 411–423]. The algorithm for non-simple paths achieves O(log²n) time per output vertex which is an improvement by a factor of O(n/log²n) of the previous algorithm [Hershberger, Snoeyink, Comput. Geom. Theory Appl. 4 (1994) 63–98], where n is the number of obstacles. The running time has an overhead O(n²⁺) for any positive constant . In the case k<n²⁺, where k is the total size of the input and output, we improve the running to O((n+k+(nk)^2/3)log^O(1)n).     

Instability of FIFO in session-oriented networks

We show that the First-In-First-Out (FIFO) scheduling discipline can be unstable in the (σ,ρ)-regulated session model for packet-switched networks. In this model packets are injected into the network in fixed sessions. The total size of the session-i packets injected during the time interval [x,y) is at most σ_i+ρ_i(y−x) for some burst parameter σ_i and rate ρ_i. The sum of the rates of sessions passing through a server is at most the server speed.Previous work on FIFO stability either allowed for dynamically changing session paths or else assumed that session-i packets are injected at a constant rate. Our result shows that FIFO can be unstable for static paths as long as the injections into a session can be temporarily suspended.     

Implication in fuzzy logic

Intermediate truth values and the order relation “as true as” are interpreted. The material implication A → B quantifies the degree by which “B is at least as true as A.” Axioms for the → operator lead to a representation of → by the pseudo-Lukasiewicz model. A canonical scale for the truth value of a fuzzy proposition is selected such that the → operator is the Lukasiewicz operator and the negation is the classical 1−. operator. The mathematical structure of some conjunction and disjunction operators related to → are derived.     

Maximizing Queueing Network Utility Subject to Stability: Greedy Primal-Dual Algorithm

We study a model of controlled queueing network, which operates and makes control decisions in discrete time. An underlying random network mode determines the set of available controls in each time slot. Each control decision “produces” a certain vector of “commodities”; it also has associated “traditional” queueing control effect, i.e., it determines traffic (customer) arrival rates, service rates at the nodes, and random routing of processed customers among the nodes. The problem is to find a dynamic control strategy which maximizes a concave utility function H(X), where X is the average value of commodity vector, subject to the constraint that network queues remain stable.We introduce a dynamic control algorithm, which we call Greedy Primal-Dual (GPD) algorithm, and prove its asymptotic optimality. We show that our network model and GPD algorithm accommodate a wide range of applications. As one example, we consider the problem of congestion control of networks where both traffic sources and network processing nodes may be randomly time-varying and interdependent. We also discuss a variety of resource allocation problems in wireless networks, which in particular involve average power consumption constraints and/or optimization, as well as traffic rate constraints.