Polynomial-time algorithm

An i-triangulated graph is a graph in which every odd cycle has two non-crossing chords; i-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite graph or a clique joined to an arbitrary bipartite graph. We exhibit a polynomial time algorithm for finding the maximum-weight induced k-partite subgraph of an i-triangulated graph, and show that the same problem is NP -complete for clique-separable graphs, even in the unweighted case when k = 2.

An i-triangulated graph is a graph in which every odd cycle has two non-crossing chords; i-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite graph or a clique joined to an arbitrary bipartite graph. We exhibit a polynomial time algorithm for finding the maximum-weight induced k-partite subgraph of an i-triangulated graph, and show that the same problem is NP -complete for clique-separable graphs, even in the unweighted case when k = 2.

Abstract. We consider the problem of assigning radii to a given set of points in the plane, such that the resulting set of circles is connected, and the sum of radii is minimized. We show that the problem is polynomially solvable if a connectivity tree is given. If the connectivity tree is unknown, the problem is NP-hard if there are upper bounds on the radii and open otherwise. We give approximation guarantees for a variety of polynomial-time algorithms, describe upper and lower bounds (which are matching in some of the cases), provide polynomial-time approximation schemes, and conclude with experimental results and open problems.

It is a simple fact that cubic Hamiltonian graphs have at least two Hamiltonian cycles. Finding such a cycle is N P -hard in general, and no polynomial time algorithm is known for the problem of finding a second Hamiltonian cycle when one such cycle is given as part of the input. We investigate the complexity of approximating this problem where by a feasible solution we mean a(nother) cycle in the graph. First we prove a negative result showing that the Longest Path problem is not constant approximable in cubic Hamiltonian graphs unless P = N P . No such negative result was previously known for this problem in Hamiltonian graphs. In strong opposition with this result we show that there is a polynomial time approximation scheme for finding another cycle in cubic Hamiltonian graphs if a Hamiltonian cycle is given in the input. ⋆

We consider m machines in parallel with each machine capable of producing one specific product type. There are n orders with each one requesting specific quantities of the various different product types. Order j has a release date r j and a due date d j . The different product types for order j can be produced at the same time. We consider various due date related objectives such as the minimization of the maximum lateness L max and the total number of late orders P U j . We present polynomial time algorithms for the easy cases and heuristics for NP-hard cases. For minimizing P U j , we also propose an exact algorithm based on Constraint Propagation and bounding strategy. The effectiveness of the algorithms is demonstrated through an empirical study.

proved that if every vertex v in a graph G has degree d(v) ≥ a(v)+ b(v) + 1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition ( and strengthened this result, proving that it suffices to assume d(v) ≥ a+b (a, b ≥ 1) or just d(v) ≥ a+b−1 (a, b ≥ 2) if G contains no cycles shorter than 4 or 5, respectively.

Many governmental agencies and businesses organizations use networked systems to provide a number of services. Such a service-oriented network can be implemented as an overlay on top of the physical network. It is well recognized that the performance of many of the networked computer systems is severely degraded under node and edge failures. The focus of our work is on the resilience of service-oriented networks. We develop a graph theoretic model for service-oriented networks. Using this model, we propose metrics that quantify the resilience of such networks under node and edge failures. These metrics are based on the topological structure of the network and the manner in which services are distributed over the network. Based on this framework, we address two types of problems. The first type involves the analysis of a given network to determine its resilience parameters. The second type involves the design of networks with a given degree of resilience. We present efficient algorithms for both types of problems. Our approach for solving analysis problems relies on known algorithms for computing minimum cuts in graphs. Our algorithms for the design problem are based on a careful analysis of the decomposition of the given graph into appropriate types of connected components.

Due to the increasing popularity of alternative-fuel (AF) vehicles in the last two decades, several models and solution techniques have been recently published in the literature to solve AF refueling station location problems. These problems can be classified depending on the set of candidate sites: when a (finite) set of candidate sites is predetermined, the problem is called discrete; when stations can be located anywhere along the network, the problem is called continuous. Most researchers have focused on the discrete version of the problem, but solutions to the discrete version are suboptimal to its continuous counterpart. This study addresses the continuous version of the problem for an AF refueling station on a tree-type transportation network when a portion of drivers are willing to deviate from their preplanned simple paths to receive refueling service. A polynomial time solution approach is proposed to solve the problem. We first present a new algorithm that identifies all possible deviation options for each travel path. Then, an efficient algorithm is used to determine the set of optimal locations for the refueling station that maximizes the total traffic flow covered. A numerical example is solved to illustrate the proposed solution approach.

For a class of permutations X the LXS problem is to identify in a given permutation σ of length n its longest subsequence that is isomorphic to a permutation of X. In general LXS is NP-hard. A general construction that produces polynomial time algorithms for many classes X is given. More efficient algorithms are given when X is defined by avoiding some set of permutations of length 3.

We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. [Phys. Rev. A 73, 012320 (2006)]. The conjecture states that a multipartite quantum state is separable if and only if the degree matrix of the graph associated with the state is equal to the degree matrix of the partial transpose of this graph. We call this statement to be the strong form of the conjecture. In its weak version, the conjecture requires only the necessity, that is, if the state is separable, the corresponding degree matrices match. We prove the strong form of the conjecture for pure multipartite quantum states, using the modified tensor product of graphs defined in [J. Phys. A: Math. Theor. 40, 10251 (2007)], as both necessary and sufficient condition for separability. Based on this proof, we give a polynomial-time algorithm for completely factorizing any pure multipartite quantum state. By polynomial-time algorithm we mean that the execution time of this algorithm increases as a polynomial in m, where m is the number of parts of the quantum system. We give a counterexample to show that the conjecture fails, in general, even in its weak form, for multipartite mixed states. Finally, we prove this conjecture, in its weak form, for a class of multipartite mixed states, giving only a necessary condition for separability.

We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. [Phys. Rev. A 73, 012320 (2006)]. The conjecture states that a multipartite quantum state is separable if and only if the degree matrix of the graph associated with the state is equal to the degree matrix of the partial transpose of this graph. We call this statement to be the strong form of the conjecture. In its weak version, the conjecture requires only the necessity, that is, if the state is separable, the corresponding degree matrices match. We prove the strong form of the conjecture for pure multipartite quantum states, using the modified tensor product of graphs defined in [J. Phys. A: Math. Theor. 40, 10251 (2007)], as both necessary and sufficient condition for separability. Based on this proof, we give a polynomial-time algorithm for completely factorizing any pure multipartite quantum state. By polynomial-time algorithm we mean that the execution time of this algorithm increases as a polynomial in m, where m is the number of parts of the quantum system. We give a counterexample to show that the conjecture fails, in general, even in its weak form, for multipartite mixed states. Finally, we prove this conjecture, in its weak form, for a class of multipartite mixed states, giving only a necessary condition for separability.

Scheduling Two Jobs with Fixed and Nonflxed Routes. The shop-scheduling problem with two jobs and m machines is considered under the condition that the machine order is fixed in advance for the first job and nonfixed for the second job. The problems ofmakespan and mean flow time minimization are proved to be NP-hard if operation preemption is forbidden. In the case of preemption allowance for any given regular criterion the O(n,) algorithm is proposed. Here, n, is the maximum number of operations per job.

The entropy of a set of data is a measure of the amount of information contained in it. Entropy calculations for fully specified data have been used to get a theoretical bound on how much that data can be compressed. This paper extends the concept of entropy for incompletely specified test data (i.e., that has unspecified or don't care bits) and explores the use of entropy to show how bounds on the maximum amount of compression for a particular symbol partitioning can be calculated. The impact of different ways of partitioning the test data into symbols on entropy is studied. For a class of partitions that use fixed-length symbols, a greedy algorithm for specifying the don't cares to reduce entropy is described. It is shown to be equivalent to the minimum entropy set cover problem and thus is within an additive constant error with respect to the minimum entropy possible among all ways of specifying the don't cares. A polynomial time algorithm that can be used to approximate the calculation of entropy is described. Different test data compression techniques proposed in the literature are analyzed with respect to the entropy bounds. The limitations and advantages of certain types of test data encoding strategies are studied using entropy theory.

GRASP (Greedy Randomized Adaptive Search Procedures) is a multistart metaheuristic for producing good-quality solutions of combinatorial optimization problems. Each GRASP iteration is usually made up of a construction phase, where a feasible solution is constructed, and a local search phase which starts at the constructed solution and applies iterative improvement until a locally optimal solution is found. While, in general, the construction phase of GRASP is a randomized greedy algorithm, other types of construction procedures have been proposed. Repeated applications of a construction procedure yields diverse starting solutions for the local search. This chapter gives an overview of GRASP describing its basic components and enhancements to the basic procedure, including reactive GRASP and intensification strategies.

We settle the so-called degree conjecture for the separability of multipartite quantum states, which are normalized graph Laplacians, first given by Braunstein et al. [Phys. Rev. A 73, 012320 (2006)]. The conjecture states that a multipartite quantum state is separable if and only if the degree matrix of the graph associated with the state is equal to the degree matrix of the partial transpose of this graph. We call this statement to be the strong form of the conjecture. In its weak version, the conjecture requires only the necessity, that is, if the state is separable, the corresponding degree matrices match. We prove the strong form of the conjecture for pure multipartite quantum states, using the modified tensor product of graphs defined in [J. Phys. A: Math. Theor. 40, 10251 (2007)], as both necessary and sufficient condition for separability. Based on this proof, we give a polynomial-time algorithm for completely factorizing any pure multipartite quantum state. By polynomial-time algorithm we mean that the execution time of this algorithm increases as a polynomial in m, where m is the number of parts of the quantum system. We give a counterexample to show that the conjecture fails, in general, even in its weak form, for multipartite mixed states. Finally, we prove this conjecture, in its weak form, for a class of multipartite mixed states, giving only a necessary condition for separability.

Motivated by a problem commonly found in electronic assembly lines, this paper deals with the problem of scheduling jobs and a rate-modifying activity on a single machine. A rate-modifying activity is an activity that changes the production rate of the equipment under consideration. Hence the processing times of jobs vary depending on whether the job is scheduled before or after the rate-modifying activity. The decisions under consideration are when to schedule the rate-modifying activity and the sequence of jobs to optimize some performance measure. In this paper, we develop polynomial algorithms for solving problems of minimizing makespan, and total completion time respectively. We also develop pseudo-polynomial algorithms for solving problems of total weighted completion time under the agreeable ratio assumption. We prove that the problem of minimizing maximum lateness is NP-hard and also provide a pseudo-polynomial time algorithm to solve it optimally.

At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor.

In this paper we study the problem of motion planning in the presence of time dependent, i.e. moving, obstacles. More specifically, we will consider the problem: given a body B and a collection of moving obstacles in D-dimensional space decide whether there is a continuous, collision-free movement of B from a given initial position to a target position subject to the condition that B cannot move any faster than some fixed top-speed c. As a discrete, combinatorial model for the continuous, geometric motion planning problem we introduce time-dependent graphs. It is shown that a path existence problem in time-dependent graphs is PSPACE-complete. Using this result we will demonstrate that a version of the motion planning problem (where the obstacles are allowed to move periodically) is PSPACEhard, even if D = 2, B is a square and the obstacles have only translational movement. For D = 1 it is shown that motion planning is NP-hard. Furthermore we introduce the notion of the c-hull of an obstacle: the c-hull is the collection of all positions in space-time at which a future collision with an obstacle cannot be avoided. In the low-dimensional situation D = 1 and D =2 we develop polynomial-time algorithms for the computation of the c-hull as well as for the motion planning problem in the special case where the obstacles are polyhedral. In particular for D = 1 there is a O(n lg n) time algorithm for the motion planning problem where n is the number of edges of the obstacle.

Coloring a k-colorable graph using k colors (k ≥ 3) is a notoriously hard problem. Considering average case analysis allows for better results. In this work we consider the uniform distribution over k-colorable graphs with n vertices and exactly cn edges, c greater than some sufficiently large constant. We rigorously show that all proper k-colorings of most such graphs lie in a single "cluster", and agree on all but a small, though constant, portion of the vertices. We also describe a polynomial time algorithm that whp finds a proper k-coloring of such a random k-colorable graph, thus asserting that most such graphs are easy to color. This should be contrasted with the setting of very sparse random graphs (which are k-colorable whp), where experimental results show some regime of edge density to be difficult for many coloring heuristics.

In this paper, we consider the recognition problem on the HHDS-free graphs, a class of homogeneously orderable graphs, and we show that it has polynomial time complexity. In particular, we describe a simple O(n 2 m)-time algorithm which determines whether a graph G on n vertices and m edges is HHDS-free. To the best of our knowledge, this is the first polynomial-time algorithm for recognizing this class of graphs.

Abstract: We consider ‘source location problems ’ in undirected graphs motivated by localization problems in sensor networks. In such a network the fundamental problem is to determine the locations of the sensors in the plane from a subset of pairwise distances. To achieve unique localizability it is necessary to designate a set of sensors, called anchors, for which the exact location is known. We consider the problem of finding a smallest set of anchors which make the network uniquely localizable, provided that the coordinates are ‘generic’. We give polynomial time algorithms for two relaxations of the problem. By combining these algorithms we obtain a 2-approximation algorithm for the anchor minimization problem.

For a multi-tiered logistics network containing the manufacturers, distributors and consumers, in this paper we present a fast polynomial time algorithm to find product flow with the minimum congestion by shipping products from manufacturers to distributors and from distributors to consumers. Global convergence is delivered for the proposed algorithm. The values of congestion and the potential function can be successfully decreased by conducting the proposed algorithm in logistics networks. Numerical computations and comparisons with well-known linear programming package CPLEX on large-scale logistics networks are performed where significant savings in computations are gained from the proposed algorithm.

For a multi-tiered logistics network containing the manufacturers, distributors and consumers, in this paper we present a fast polynomial time algorithm to find product flow with the minimum congestion by shipping products from manufacturers to distributors and from ...

Given a finite set V , and a hypergraph H ⊆ 2 V , the hypergraph transversal problem calls for enumerating all minimal hitting sets (transversals) for H. This problem plays an important role in practical applications as many other problems were shown to be polynomially equivalent to it. Fredman and Khachiyan (1996) gave an incremental quasi-polynomial time algorithm for solving the hypergraph transversal problem [9]. In this paper, we present an efficient implementation of this algorithm. While we show that our implementation achieves the same bound on the running time as in [9], practical experience with this implementation shows that it can be substantially faster. We also show that a slight modification of the algorithm in [9] can be used to give a stronger bound on the running time.

We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers n, d, k where k ≤ d, and a set P of n points in Rd. The goal is to compute the outer k-radius of P, denoted by Rk(P), which is the minimum, over all (d−k)-dimensional flats F, of maxp∈P d(p, F), where d(p, F) is the Euclidean distance between the point p and flat F. Computing the radii of point sets is a fundamental problem in computational convexity with significantly many applications. The problem admits a polynomial time algorithm when the dimension d is constant [9]. Here we are interested in the general case when the dimension d is not fixed and can be as large as n, where the problem becomes NP-hard even for k = 1. It has been known that Rk(P) can be approximated in polynomial time by a factor of (1 + ε), for any ε> 0, when d − k is a fixed constant [15, 2]. A factor of O(√logn) approx-imation for R1(P), the width of the point set P, is implied from the results of Nemi...

In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: Given a set of t topics and a collection of n users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is N P-hard and approximable within O(log t) in polynomial time. In this paper, we further investigate the problem and some of its special instances. We give various hardness results for instances where the number of topics in which an user is interested in is bounded by a constant, and also for the instances where the number of users interested in a common topic is constant. For the latter case, we present a first constant approximation algorithm. We also present some polynomial-time algorithms for very restricted instances of Min-TCO.

The authors describe the use of bounded model checking (BMC) for verifying Web application code. Vulnerable sections of code are patched automatically with runtime guards, allowing both verification and assurance to occur without user intervention. Model checking techniques have relatively complexity compared to the typestate-based polynomial-time algorithm (TS) we adopted in an earlier paper, but they offer three benefits-they provide counterexamples, more precise models, and sound and complete verification. Compared to conventional model checking techniques, BMC offers a more practical approach to verifying programs containing large numbers of variables, but requires fixed program diameters to be complete. Formalizing Web application vulnerabilities as a secure information flow problem with fixed diameter allows for BMC application without drawback. Using BMC-produced counterexamples, errors that result from propagations of the same initial error can be reported as a single group rather than individually. This offers two distinct benefits. First, together with the counterexamples themselves, they allow for more descriptive and precise error reports. Second, it allows for automated patching at locations where errors are initially introduced rather than at locations where the propagated errors cause problems. Results from a TS-BMC comparison test using 230 opensource Web applications showed a 41.0% decrease in runtime instrumentations when BMC was used. In the 38 vulnerable projects identified by TS, BMC classified the TS-reported 980 individual errors into 578 groups, with each group requiring a minimal set of patches for repair.

Failure diagnosis is an important task in large complex systems and as such this problem has received in the last years considerable attention in the literature. The rst step to diagnose failure occurrences in discrete event systems is the veri cation of the system diagnosability. Several works in the literature have addresses this problem using either diagnosers or veri ers for the centralized and decentralized architectures. In this paper a new polynomial time algorithm to verify the decentralized diagnosability property of a discrete event system is proposed. The algorithm has lower computational complexity than other methods proposed in the literature and can also be applied to the centralized case.

A key problem in networks that support advance reservations is the routing and time scheduling of connections with flexible starting time. In this paper we present a multicost routing and scheduling algorithm for selecting the path to be followed by such a connection and the time the data should start so as to minimize the reception time at the destination, or some other QoS requirement. The utilization profiles of the network links, the link propagation delays, and the parameters of the connection to be scheduled form the inputs to the algorithm. We initially present a scheme of non-polynomial complexity to compute a set of so-called non-dominated candidate paths, from which the optimal path can be found. By appropriately pruning the set of candidate paths using path pseudo-domination relationships, we also find multicost routing and scheduling algorithms of polynomial complexity. We examine the performance of the algorithms in the special case of an Optical Burst Switched network. Our results indicate that the proposed polynomial time algorithms have performance that it is very close to that of the optimal algorithm.

Sorting a permutation by block moves is a task that every bridge player has to solve every time she picks up a new hand of cards. It is also a problem for the computational biologist, for block moves are a fundamental type of mutation that can explain why genes common to two species do not occur in the same order in the chromosome. It is not known whether there exists an optimal sorting procedure running in polynomial time. Bafna and Pevzner gave a polynomial time algorithm that sorts any permutation of length n in at most 3n=4 moves. Our new algorithm improves this to (2n − 2)=3 for n ¿ 9. For the reverse permutation, we give an exact expression for the number of moves needed, namely (n + 1)=2. Computations of Bafna and Pevzner up to n = 10 seemed to suggest that this is the worst case; but as it turns out, a ÿrst counterexample occurs for n = 13, i.e. the bridge player's case. Professional card players never sort by rank, only by suit. For this case, we give a complete answer to the optimal sorting problem.

We present a new method of identifying a class of asymmetric matrices for which an optimal traveling salesman tour exists that is pyramidal. The new class generalizes two previously known classes of matrices and includes some new matrices as well.

Tracking of movements such as that of people, animals, vehicles, or of phenomena such as fire, can be achieved by deploying a wireless sensor network. So far only prototype systems have been deployed and hence the issue of scale has not become critical. Real-life deployments, however, will be at large scale and achieving this scale will become prohibitively expensive if we require every point in the region to be covered (i.e., full coverage), as has been the case in prototype deployments. In this paper we therefore propose a new model of coverage, called Trap Coverage, that scales well with large deployment regions. A sensor network providing Trap Coverage guarantees that any moving object or phenomena can move at most a (known) displacement before it is guaranteed to be detected by the network, for any trajectory and speed. Applications aside, trap coverage generalizes the de-facto model of full coverage by allowing holes of a given maximum diameter. From a probabilistic analysis perspective, the trap coverage model explains the continuum between percolation (when coverage holes become finite) and full coverage (when coverage holes cease to exist). We take first steps toward establishing a strong foundation for this new model of coverage. We derive reliable, explicit estimates for the density needed to achieve trap coverage with a given diameter when sensors are deployed randomly. Our density estimates are more accurate than those obtained using asymptotic critical conditions. We show by simulation that our analytical predictions of density are quite accurate even for small networks. We then propose polynomial-time algorithms to determine the level of trap coverage achieved once sensors are deployed on the ground. Finally, we point out several new research problems that arise by the introduction of the trap coverage model.

We give a polynomial time algorithm to find the population variance of tour costs over the solution space of the symmetric Traveling Salesman Problem (TSP). In practical terms the algorithm provides a linear time method, on the number of edges of the problem, for determining the standard deviation of these costs. Application of the algorithm has produced empirical evidence that there is a clear relationship between the optimal tour cost and the standard deviation. This suggests that there may be a polynomial time algorithm to estimate the likely optimal tour cost of a TSP. The method for finding the variance also shows promise of being generalizable to higher statistical moments.

This paper deals with the packing of a grid by horizontal bars while respecting given orthogonal projections and several constraints of distance between the consecutive bars. We show that packing under a maximal or uniform distance is an NP-complete problem. We also give a polynomial time algorithm to solve the packing problem under a minimal distance.

The Grundy number of a graph G is the largest number of colors used by any execution of the greedy algorithm to color G. The problem of determining the Grundy number of G is polynomial if G is a P 4-free graph and N P-hard if G is a P 5-free graph. In this article, we define a new class of graphs, the fat-extended P 4-laden graphs, and we show a polynomial time algorithm to determine the Grundy number of any graph in this class. Our class intersects the class of P 5-free graphs and strictly contains the class of P 4-free graphs. More precisely, our result implies that the Grundy number can be computed in polynomial time for any graph of the following classes: P 4-reducible, extended P 4-reducible, P 4-sparse, extended P 4-sparse, P 4-extendible, P 4-lite, P 4-tidy, P 4laden and extended P 4-laden, which are all strictly contained in the fat-extended P 4-laden class.

In this paper we consider the problem of no-wait cyclic scheduling of identical parts in an m-machine production line in which a robot is responsible for moving each part from a machine to another. The aim is to find the minimum cycle time for the so-called 2-cyclic schedules, in which exactly two parts enter and two parts leave the production line during each cycle. The earlier known polynomial-time algorithms for this problem are applicable only under the additional assumption that the robot travel times satisfy the triangle inequalities. We lift this assumption on robot travel times and present a polynomial-time algorithm with the same time complexity as in the metric case, O(m 5 log m).

The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values is #P-complete and computing Shapley-Shubik indices or values is NP-hard for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. This answers (positively) an open question of whether computing Shapley-Shubik indices for a simple game represented by the set of minimal winning coalitions is NP-hard.

Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics and threshold phenomena. Recent work on heuristics, and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and stconnectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side-which includes but is not limited to all problems with polynomial time algorithms for satisfiability-is in P for the st-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, small diameter and tractability of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary.

A centromere is a special region in the chromosome that plays a vital role during cell division. Every new chromosome created by a genome rearrangement event must have a centromere in order to survive. This constraint has been ignored in the computational modeling and analysis of genome rearrangements to date. Unlike genes, the different centromeres are indistinguishable, they have no

Given a permutation π, the application of prefix reversal f (i) to π reverses the order of the first i elements of π. The problem of Sorting By Prefix Reversals (also known as pancake flipping), made famous by Gates and Papadimitriou (Bounds for sorting by prefix reversal, Discrete Mathematics 27, pp. 47-57), asks for the minimum number of prefix reversals required to sort the elements of a given permutation. In this paper we study a variant of this problem where the prefix reversals act not on permutations but on strings over a fixed size alphabet. We determine the minimum number of prefix reversals required to sort binary and ternary strings, with polynomial-time algorithms for these sorting problems as a result; demonstrate that computing the minimum prefix reversal distance between two binary strings is NP-hard; give an exact expression for the prefix reversal diameter of binary strings, and give bounds on the prefix reversal diameter of ternary strings. We also consider a weaker form of sorting called grouping (of identical symbols) and give polynomial-time algorithms for optimally grouping binary and ternary strings. A number of intriguing open problems are also discussed.

The distortion of a message due to channel noise can be alleviated significantly without redundant error control bits by judicious assignment of binary indices to message symbols. The nonredundant coding gain relies only on a notion of distance between symbols. In this paper, we consider the index assignment problem and adopt a minimax design criterion instead of the usual mean squared error (MSE) measure. The problem is found to be NPhard, and an effective, heuristic, polynomial-time algorithm is presented for computing approximate solutions. The minimax criterion yields greatly improved worst case performance while maintaining good average performance. In addition, the familiar MSE criterion is shown likewise to yield an NP-hard index assignment task. The MSE problem is a special case of the classical Quadratic Assignment Problem, for which computationally and theoretically useful results are available from the discrete mathematics literature.

Due to the increasing popularity of alternative-fuel (AF) vehicles in the last two decades, several models and solution techniques have been recently published in the literature to solve AF refueling station location problems. These problems can be classified depending on the set of candidate sites: when a (finite) set of candidate sites is predetermined, the problem is called discrete; when stations can be located anywhere along the network, the problem is called continuous. Most researchers have focused on the discrete version of the problem, but solutions to the discrete version are suboptimal to its continuous counterpart. This study addresses the continuous version of the problem for an AF refueling station on a tree-type transportation network when a portion of drivers are willing to deviate from their preplanned simple paths to receive refueling service. A polynomial time solution approach is proposed to solve the problem. We first present a new algorithm that identifies all ...

We consider the generalized version of the classical Minimum Spanning Tree problem where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. We present a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. Two of them work in complementary ways in order to maximize search effectivity. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply Mixed Integer Programming to optimize local parts within candidate solution trees. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches.

We s h o w : f o r mJe~j constant m i t can be decided i n p o l y n o m i a l time w h e t h e r or not two m-ambiguous finite tree automata are equivalent. In general, inequivalence for finite tree automata is DEXPTIfdE-complete v).r.t, logspace reductions, and PSPACE-complete w.r.t, logspace reductions, i f the automata in question are supposed to accept only finite languages. For finite tree automata u i t h coefficients in a field R we give a polynomial time algorithm for deciding ambiguity-equivalence provided R-operations and R-tests for 0 can be perfor-,ned i n constant time. We apply this algorithm for deciding ambiguityinequivalence of finite tree automata in randomized polynomial time. FUrthermore, for every constant m we show that it can be decided in polynomial time whether or not a given finite tree automaton is m-ambiguous.

A bound quiver is a digraph together with a collection of specified directed walks. Given an undirected graph G, and a collection of walks I , the bound quiver recognition problem asks: Is there an orientation of G such that each walk in I is directed? We present a polynomial time algorithm for this problem, and a structural characterization of which trees can be oriented as bound quivers. These results are developed using homomorphisms of edge-coloured graphs. Our work includes a classification of the computational complexity of edge-coloured homomorphism problems where the target is of order at most three.