We consider the Directed Steiner Network problem, also called the POINT-TO-POINT CONNECTION problem. Given a directed graph G and p pairs {(s(1), t(1)),..., (s(p), t(p))} of nodes in the graph, one has to find the sma...
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We consider the Directed Steiner Network problem, also called the POINT-TO-POINT CONNECTION problem. Given a directed graph G and p pairs {(s(1), t(1)),..., (s(p), t(p))} of nodes in the graph, one has to find the smallest subgraph H of G that contains paths from s(i) to t(i) for all i. The problem is NP-hard for general p, since the DIRECTED STEINER Tree problem is a special case. Until now, the complexity was unknown for constant p >= 3. We prove that the problem is polynomially solvable if p is any constant number, even if nodes and edges in G are weighted and the goal is to minimize the total weight of the subgraph H. In addition, we give an efficient algorithm for the STRONGLY CONNECTED STEINER SUBGRAPH problem for any constant p, where given a directed graph and p nodes in the graph, one has to compute the smallest strongly connected subgraph containing the p nodes.
This paper studies the convergence properties of a general class of decomposition algorithms for support vector machines (SVMs). We provide a model algorithm for decomposition, and prove necessary and sufficient condi...
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This paper studies the convergence properties of a general class of decomposition algorithms for support vector machines (SVMs). We provide a model algorithm for decomposition, and prove necessary and sufficient conditions for stepwise improvement of this algorithm. We introduce a simple "rate certifying" condition and prove a polynomial-time bound on the rate of convergence of the model algorithm when it satisfies this condition. Although it is not clear that existing SVM algorithms satisfy this condition, we provide a version of the model algorithm that does. For this algorithm we show that when the slack multiplier C satisfies root1/2 less than or equal to Cless than or equal to mL, where m is the number of samples and L is a matrix norm, then it takes no more than 4LC(2)m(4)/epsilon iterations to drive the criterion to within epsilon of its optimum.
We study a generalization of the constraint satisfaction problem (CSP), the periodic constraint satisfaction problem. An input instance of the periodic CSP is a finite set of "generating" constraints over a ...
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We study a generalization of the constraint satisfaction problem (CSP), the periodic constraint satisfaction problem. An input instance of the periodic CSP is a finite set of "generating" constraints over a structured variable set that implicitly specifies a larger, possibly infinite set of constraints;the problem is to decide whether or not the larger set of constraints has a satisfying assignment. This model is natural for studying constraint networks consisting of constraints obeying a high degree of regularity or symmetry. Our main contribution is the identification of two broad polynomial-time tractable subclasses of the periodic CSP.
We study a generalization of the constraint satisfaction problem (CSP), the periodic constraint satisfaction problem. An input instance of the periodic CSP is a finite set of "generating" constraints over a ...
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We study a generalization of the constraint satisfaction problem (CSP), the periodic constraint satisfaction problem. An input instance of the periodic CSP is a finite set of "generating" constraints over a structured variable set that implicitly specifies a larger, possibly infinite set of constraints;the problem is to decide whether or not the larger set of constraints has a satisfying assignment. This model is natural for studying constraint networks consisting of constraints obeying a high degree of regularity or symmetry. Our main contribution is the identification of two broad polynomial-time tractable subclasses of the periodic CSP.
Given graphs G, H, and lists L(v) subset of or equal to V(H), v is an element of V(G), a list homomorphism of G to H with respect to the lists L is a mapping f: V(G) --> V(H) such that uv is an element of E(G) impl...
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Given graphs G, H, and lists L(v) subset of or equal to V(H), v is an element of V(G), a list homomorphism of G to H with respect to the lists L is a mapping f: V(G) --> V(H) such that uv is an element of E(G) implies f(u)f(v) is an element of E(H), and f(v) is an element of (v) for all v is an element of V(G). The list homomorphism problem for a fixed graph H asks whether or not an input graph G, together with lists L(v) subset of or equal to V(H), v is an element of V(G), admits a list homomorphism with respect to L. In two earlier papers, we classified the complexity of the list homomorphism problem in two important special cases: When H is a reflexive graph (every vertex has a loop), the problem is polynomialtime solvable if H is an interval graph, and is NP-complete otherwise. When H is an irreflexive graph (no vertex has a loop), the problem is polynomialtime solvable if H is bipartite and (H) over bar is a circular arc graph, and is NP-complete otherwise. In this paper, we extend these classifications to arbitrary graphs H (each vertex may or may not have a loop). We introduce a new class of graphs, called bi-arc graphs, which contains both reflexive interval graphs (and no other reflexive graphs), and bipartite graphs with circular arc complements (and no other irreflexive graphs). We show that the problem is polynomialtime solvable when H is a bi-arc graph, and is NP-complete otherwise. In the case when H is a tree (with loops allowed), we give a simpler algorithm based on a structural characterization. (C) 2002 Wiley Periodicals, Inc.
Random Duplicated Assignment (RDA) is an approach in which video data is stored by assigning a number of copies of each data block to different, randomly chosen disks. It has been shown that this approach results in s...
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Random Duplicated Assignment (RDA) is an approach in which video data is stored by assigning a number of copies of each data block to different, randomly chosen disks. It has been shown that this approach results in smaller response times and lower disk and RAM costs compared to the well-known disk stripping techniques. Based on this storage approach, one has to determine, for each given batch of data blocks, from which disk each of the data blocks is to be retrieved. This is to be done in such a way that the maximum load of the disks is minimized. The problem is called the Retrieval Selection Problem (RSP). In this paper, we propose a new efficient algorithm for RSP. This algorithm is based on the breadth-first search approach and is able to guarantee optimal solutions for RSP in O(n(2) + mn), where m and n correspond to the number of data blocks and the number of disks, respectively. We will show that our proposed algorithm has a lower time complexity than an existing algorithm, called the MFS algorithm. (C) 2002 Elsevier Science B.V. All rights reserved.
A siphon-trap of a Petri net N is defined as a place set S with S-. = S-., where S-.={u\ N has an edge from u to a vertex of S} and S-. = {v\ N has an edge from a vertex of S to v}. A minimal siphon-trap is a siphon-t...
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A siphon-trap of a Petri net N is defined as a place set S with S-. = S-., where S-.={u\ N has an edge from u to a vertex of S} and S-. = {v\ N has an edge from a vertex of S to v}. A minimal siphon-trap is a siphon-trap such that any proper subset is not a siphon-trap. The following polynomial-time algorithms are proposed: 1. FDST for finding, if any, a minimal siphon-trap or even a maximal class of mutually disjoint minimal siphon-traps of a given Petri net;2. FDSTi that repeats FDST i times in order to extract more minimal siphon-traps than FDST. 3. STFM-T (STFM-T-i, respectively) which is a combination of the Fourier-Motzkin method and FDST (FDSTi) and which has high possibility of finding, if any, at least one minimal-support nonnegative integer invariant.
We study the computational complexity of linear programs with coefficients that are real algebraic numbers under a Turing machine model of computation. After reviewing a method for exact representation of algebraic nu...
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We study the computational complexity of linear programs with coefficients that are real algebraic numbers under a Turing machine model of computation. After reviewing a method for exact representation of algebraic numbers under the Turing model, we show that the fundamental tasks of comparison and arithmetic can be performed in polynomialtime. Our technique for establishing polynomial-time algorithms for comparison and arithmetic is distinct from the usual resultant-based approaches, and has the advantage that it provides a natural framework for analysis of the complexity of computational tasks, such as Gaussian elimination, that involve a sequence of arithmetic operations. Our main contribution is to show that a variant of the ellipsoid method can be used to solve linear programming in timepolynomial in the encoding size of the problem coefficients and the degree of any algebraic extension that contains those coefficients.
This paper presents a polynomial-time algorithm for equivalence under certain semiring congruences. These congruences arise when studying the isomorphism of state spaces for a class of hybrid systems. The area of hybr...
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This paper presents a polynomial-time algorithm for equivalence under certain semiring congruences. These congruences arise when studying the isomorphism of state spaces for a class of hybrid systems. The area of hybrid systems concerns issues of modeling, computation, and control for systems which combine discrete and continuous components. The subclass of piecewise linear (PL) systems provides one systematic approach to discrete-time hybrid systems, naturally blending switching mechanisms with classical linear components. PL systems model arbitrary interconnections of finite automata and linear systems. Tools from automata theory, logic, and related areas of computer science and finite mathematics are used in the study of PL systems, in conjunction with linear algebra techniques, all in the context of a "PL algebra" formalism. PL systems are of interest as controllers as well as identification models. Basic questions for any class of systems are those of equivalence, and, in particular, whether state spaces are equivalent under a change of variables. This paper studies this state-space equivalence problem for PL systems. The problem was known to be decidable, but its computational complexity was potentially exponential;here it is shown to be solvable in polynomialtime. (C) 2001 Elsevier Science B.V. All rights reserved.
Let G be a labeled directed graph with are labels drawn from alphabet Sigma, R be a regular expression over Sigma, and x and y be a pair of nodes from G. The regular simple path (RSP) problem is to determine whether t...
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Let G be a labeled directed graph with are labels drawn from alphabet Sigma, R be a regular expression over Sigma, and x and y be a pair of nodes from G. The regular simple path (RSP) problem is to determine whether there is a simple path p in G from x to y, such that the concatenation of are labels along p satisfies R. Although RSP is known to be NP-hard in general, we show that it is solvable in polynomialtime when G is outerplanar. The proof proceeds by presenting an algorithm which gives a polynomial-time reduction of RSP for outerplanar graphs to RSP for directed acyclic graphs, a problem which has been shown to be solvable in polynomialtime. (C) 2000 Academic Press.
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