Polynomial-time algorithm

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.

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...

We address a variant of the unbounded knapsack problem (UKP) into which the processing time of each item is also put and considered, referred as MMPTUKP. The MMPTUKP is a decision problem of allocating amount of n items, such that the maximum processing time of the selected items is minimized and the total profit is gained as at least as determined without exceeding capacity of knapsack. In this study, we proposed a new exact algorithm for this problem, called MMPTUKP algorithm. This pseudo polynomial time algorithm solves the bounded knapsack problem (BKP) sequentially with the updated bounds until reaching an optimal solution. We present computational experience with various data instances randomly generated to validate our ideas and demonstrate the efficiency of the proposed algorithm.

A Two Stage Interval Time Minimizing Transportation Problem, where total availability of a homogeneous product at various sources is known to lie in a specified interval, is studied in the present paper. In the first stage, the sources ship all of their on-hand material to the demand points, while a second-stage delivery covers the demand that is not fulfilled in the first shipment. In each stage, the objective is to minimize the shipment time, and the overall goal is to find a solution that minimizes the sum of the firstand secondstage shipment times. A polynomial time algorithm is proposed to solve the problem to optimality, where at various steps of the algorithm lexicographic optimal solutions of restricted versions of a related standard time minimizing transportation problem are examined and finally the global optimal solution is determined.

In this paper, we consider a large class of vertex partitioning problems and apply to them the theory of algorithm design for problems restricted to partial k-trees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications. We give a precise characterization of vertex partitioning problems,

We analyze how an observer synchronizes to the internal state of a finite-state information source, using the-machine causal representation. Here, we treat the case of exact synchronization, when it is possible for the observer to synchronize completely after a finite number of observations. The more difficult case of strictly asymptotic synchronization is treated in a sequel. In both cases, we find that an observer, on average, will synchronize to the source state exponentially fast and that, as a result, the average accuracy in an observer's predictions of the source output approaches its optimal level exponentially fast as well. Additionally, we show here how to analytically calculate the synchronization rate for exact-machines and provide an efficient polynomial-time algorithm to test-machines for exactness.

The theory of multidimensional persistence captures the topology of a multifiltration-a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence.

We consider the classical problem of scheduling a set of independent jobs on a set of unrelated machines with costs. We are given a set of n monoprocessor jobs and m machines where each job is to be processed without preemptions. Executing job j on machine i requires time p ij≥ 0 and incurs cost c ij. Our objective is to find a schedule obtaining a tradeoff between the makespan and the total cost.

In this paper, we examine algorithmic issues related to the delayed multicast technique for video-ondemand delivery. We introduce the minimum total memory (MTM), minimum total traffic (MTT) and the minimum maximum memory per node (MMMN) delayed-multicast allocation problems. We examine these problems on two networks of practical interest, namely, the chandelier and the broom networks. We provide polynomial time algorithms for solving the MTM and the MTT problems on the chandelier network and the MTM problem on the broom network. We also show that a version of the decision-MMMN problem on a general graph is NP-complete. Finally, we present a heuristic method for obtaining a solution for the MTM problem on tree networks.

We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various elds such as machine learning, operations research and pattern recognition. In the rst class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to nd a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, , > and 6 = are considered. While Min RVLS with equations was known to be NP-hard in 27], we established in 2, 5] that Min ULR with equalities and inequalities are NP-hard even when restricted to homogeneous systems with bipolar coe cients. The latter problems have been shown hard to approximate in 7]. In this paper we determine strong bounds on the approximability of various variants of Min RVLS and Min ULR, including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional. The various NP-hard versions turn out to have di erent approximability properties depending on the type of relations and the additional constraints, but none of them can be approximated within any constant factor, unless P=NP. Particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning. In particular, we disprove a conjecture in 64] regarding the existence of a polynomial time algorithm to design linear classi ers (or perceptrons) that involve a close-to-minimum number of features.

The subgraph isomorphism problem, that of finding a copy of one graph in another, has proved to be intractable except when certain restrictions are placed on the inputs. In this paper, we introduce a new property for graphs along with an associated graph class (a generalization on bounded degree graphs) and extend the known classes of inputs for which polynomial-time subgraph isomorphism algorithms are attainable. In particular, if the removal of any set of at most k vertices from an n-vertex graph results in O(k log n) connected components, we say that the graph is a log-bounded fragmentation graph. We present a polynomial-time algorithm for finding a subgraph of H isomorphic to a graph G when G is a log-bounded fragmentation graph and H has bounded treewidth; these results are extended to handle graphs of locally bounded treewidth (a generalization of treewidth) when G is a log-bounded fragmentation graph and has constant diameter.

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.

This paper considers the problem of radiation therapy treatment planning for cancer patients. During radiation therapy, beams of radiation pass through a patient. This radiation kills both cancerous and normal cells, so the radiation therapy must be carefully planned to deliver a clinically prescribed dose to certain targets while sparing nearby organs and tissues. Currently, a technique called intensity modulated radiation therapy (IMRT) is considered to be the most effective radiation therapy for many forms of cancer. In IMRT, the patient is irradiated from several different directions. From each direction, one or more irregularly shaped radiation beams of uniform intensity are used to deliver the treatment. This paper deals with the problem of designing a treatment plan for IMRT that determines an optimal set of such shapes (called apertures) and their corresponding intensities. This is in contrast with established two-stage approaches where, in the first phase, each radiation beam is viewed as consisting of a set of individual beamlets, each with its own intensity. A second phase is then needed to approximate and decompose the optimal intensity profile into a set of apertures with corresponding intensities. The problem is formulated as a large-scale convex programming problem, and a column generation approach to deal with its dimensionality is developed. The associated pricing problem determines, in each iteration, one or more apertures to be added to our problem. Several variants of this pricing problem are discussed, each corresponding to a particular set of constraints that the apertures must satisfy in one or more of the currently available types of commercial IMRT equipment. Polynomial-time algorithms are presented for solving each of these variants of the pricing problem to optimality. Finally, the effectiveness of our approach is demonstrated on clinical data.

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.

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.

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.

Casting is a manufacturing process in which liquid is poured into a cast (mould) that has a cavity with the shape of the object to be manufactured. The liquid then hardens, after which the cast is removed. We address geometric problems concerning the removal of the cast. A cast consists of two parts, one of which retracts in a given direction carrying the object with it. Afterwards, the object will be ejected from the retracted cast part. In this paper, we give necessary and sufficient conditions to test the feasibility of the cast part retraction and object ejection, where retraction and ejection directions need not be the same. For polyhedral objects, we show that the test can be performed in O(n 2 log 2 n) time and the cast parts can be constructed within the same time bound. The complexity of the cast parts constructed is worst-case optimal. We also give a polynomial time algorithm for finding a feasible pair of retraction and ejection directions for a given polyhedral object.

We consider the problem of identifying an unknown Boolean function f by asking an oracle the functional values f (a) for a selected set of test vectors a ∈ {0, 1} n. Furthermore, we assume that f is a positive (or monotone) function of n variables. It is not known yet whether the whole task of generating test vectors and checking if the identification is completed can be carried out in polynomial time in n and m or not, where m = | min T (f)| + | max F (f)| and min T (f) (respectively, max F (f)) denotes the set of minimal true (respectively, maximal false) vectors of f. To partially answer this question, we propose here two polynomial time algorithms that, given an unknown positive function f of n variables, decide whether f is 2-monotonic or not, and if f is 2-monotonic, output both sets min T (f) and max F (f). The first algorithm uses O(nm 2 + n 2 m) time and O(nm) queries, while the second one uses O(n 3 m) time and O(n 3 m) queries.

A minimum degree spanning tree of a graph G is a spanning tree of G whose maximum degree is minimum among all spanning trees of G. The minimum degree spanning tree problem (MDST) is to construct such a spanning tree of a graph. In this paper, we propose a polynomial-time algorithm for solving the MDST problem on series-parallel graphs. Our algorithm runs in linear time for series-parallel graphs with small degrees. By applying this algorithm, we also give an approximation algorithm for solving the minimum edge-ranking spanning tree problem on series-parallel graphs.

We consider an extension of the popular matching problem in this paper. The input to the popular matching problem is a bipartite graph G = (A ∪ B, E), where A is a set of people, B is a set of items, and each person a ∈ A ranks a subset of items in an order of preference, with ties allowed. The popular matching problem seeks to compute a matching M * between people and items such that there is no matching M where more people are happier with M than with M *. Such a matching M * is called a popular matching. However, there are simple instances where no popular matching exists. Here we consider the following natural extension to the above problem: associated with each item b ∈ B is a non-negative price cost(b), that is, for any item b, new copies of b can be added to the input graph by paying an amount of cost(b) per copy. When G does not admit a popular matching, the problem is to "augment" G at minimum cost such that the new graph admits a popular matching. We show that this problem is NP-hard; in fact, it is NP-hard to approximate it within a factor of √ n1/2, where n1 is the number of people. This problem has a simple polynomial time algorithm when each person has a preference list of length at most 2. However, if we consider the problem of constructing a graph at minimum cost that admits a popular matching that matches all people, then even with preference lists of length 2, the problem becomes NP-hard. On the other hand, when the number of copies of each item is fixed, we show that the problem of computing a minimum cost popular matching or deciding that no popular matching exists can be solved in O(mn1) time, where m is the number of edges.

We propose a novel framework for location detection with sensor networks, based on the theory of identifying codes. The key idea of this approach is to allow sensor coverage areas to overlap so that each resolvable position is covered by a unique set of sensors. In this setting, determining a sensor-placement with a minimum number of sensors is equivalent to constructing an optimal identifying code, an NP-complete problem in general. We thus propose and analyze new polynomial-time algorithms for generating irreducible (but not necessarily optimal) codes for arbitrary topologies. Our algorithms incorporate robustness properties that are critically needed in harsh environments. We further introduce distributed versions of these algorithms, allowing sensors to selforganize and determine a (robust) identifying code without any central coordination. Through analysis and simulation, we show that our algorithms produce nearly optimal solutions for a wide range of parameters. In addition, we demonstrate a tradeoff between system robustness and the number of active sensors (which is related to the expected lifetime of the system). Finally, we present experimental results, obtained on a small testbed, that demonstrate the feasibility of our approach.

The theory of multidimensional persistence captures the topology of a multifiltration-a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence.

Tasks' scheduling has always been a central problem in the embedded real-time systems community. As in general the scheduling problem is N P-hard, researchers have been looking for efficient heuristics to solve the scheduling problem in polynomial time. One of the most important scheduling strategies is the Earliest Deadline First (EDF). It is known that EDF is optimal for uniprocessor platforms for many cases, such as: non-preemptive synchronous tasks (i.e., all tasks have the same starting time and cannot be interrupted), and preemptive asynchronous tasks (i.e., the tasks may be interrupted and may have arbitrary starting time). However, Mok showed that EDF is not optimal in multiprocessor platforms. In fact, for the multiprocessor platforms, the scheduling problem is N P-complete in most of the cases where the corresponding scheduling problem can be solved by a polynomial-time algorithm for uniprocessor platforms. Coffman and Graham identified a class of tasks for which the scheduling problem can be solved by a polynomialtime algorithm, that is, two-processor platform, no resources, arbitrary partial order relations, and every task is nonpreemptive and has a unit computation time. Our paper introduces a new non-trivial and practical subclass of tasks, called urgent tasks. Briefly, a task is urgent if it is executed right after it is ready or it can only wait one unit time after it is ready. Practical examples of embedded realtime systems dealing with urgent tasks are all modern building alarm systems, as these include urgent tasks such as 'checking for intruders', 'sending a warning signal to the security office', 'informing the building's owner about a potential intrusion', and so on. By using propositional logic, we prove a new result in schedulability theory, namely that the scheduling problem for asynchronous and preemptive urgent tasks can be solved in polynomial time.

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.

Fast and frugal heuristics are well studied models of bounded rationality. Psychological research has proposed the take-the-best heuristic as a successful strategy in decision making with limited resources. Takethe-best searches for a sufficiently good ordering of cues (or features) in a task where objects are to be compared lexicographically. We investigate the computational complexity of finding optimal cue permutations for lexicographic strategies and prove that the problem is NP-complete. It follows that no efficient (that is, polynomial-time) algorithm computes optimal solutions, unless P = NP. We further analyze the complexity of approximating optimal cue permutations for lexicographic strategies. We show that there is no efficient algorithm that approximates the optimum to within any constant factor, unless P = NP. The results have implications for the complexity of learning lexicographic strategies from examples. They show that learning them in polynomial time within the model of agnostic probably approximately correct (PAC) learning is impossible, unless RP = NP. We further consider greedy approaches for building lexicographic strategies and determine upper and lower bounds for the performance ratio of simple algorithms. Moreover, we present a greedy algorithm that performs provably better than take-thebest. Tight bounds on the sample complexity for learning lexicographic strategies are also given in this article.

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.

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 ...

This paper starts by settling the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We explore our algorithms with some experiments. We conclude the article with extensions to other polytopes and discussion of other available methods.

In this paper we study the computational complexity of the capacitated lot size problem with a particular cost structure that is likely to be used in practical settings. For the single item case new properties are introduced, classes of problems solvable by polynomial time algorithms are i.dentified, and efficient solution procedures are given. We show that special classes are N-hard, and that the problem with two items and independent setups is NP-hard under conditions similar to those where the single item problem is easy. Topics for further research are discussed in the last section, On leave from INPE, Brazil.

In this paper we present a class of polynomial primal-dual interior-point algorithms for linear optimization based on a new class of kernel functions. This class is fairly general and includes the class of finite kernel functions by Y.Q. Bai, M.El Ghami and C. Roos [Y.Q. Bai, M. El Ghami, and C. Roos. A new efficient large-update primal-dual interior-point method based on a finite barrier, SIAM Journal on Optimization, 13 (3) (2003) 766-782]. The proposed functions have a finite value at the boundary of the feasible region. They are not exponentially convex and also not strongly convex like the usual barrier functions. The goal of this paper is to investigate such a class of kernel functions and to show that the interior-point methods based on these functions have favorable complexity results. In order to achieve these complexity results, several new arguments had to be used for the analysis. The iteration bound of large-update interior-point methods based on these functions and analyzed in this paper, is shown to be O(√ n log n log n). For small-update interior-point methods the iteration bound is O √ n log n , which is currently the bestknown bound for primal-dual IPMs. We also present some numerical results which show that by using a new kernel function, the best iteration numbers were achieved in most of the test problems.

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.

The mazimum agreement subtree approach is one method of reconciling different evolutionary trees for the same set of species. An agreement subtree enables choosing a subset of the species for whom the restricted subtree is equivalent (under a suitable definition) in all given evolutionary trees. Recently, dynamic programming ideas were used to provide polynomial time algorithms for finding a maximum homeomorphic agreement subtree of two trees. Generalizing these methods to sets of more than two trees yields algorithms that are exponential in the number of trees. Unfortunately, it turns out that in reality one is usually presented with more than two trees, sometimes as many as thousands of trees. In this paper we prove that the maximum homeomorphic agreement subtree problem is NP-complete for three trees with unbounded degrees. We then show an approximation algorithm of time O(kn5) for choosing the species that are not in a maximum agreement subtree of a set of k trees. Our approximation is guaranteed to provide a set that is no more than 4 times the optimum solution. While the set of evolutionary trees may be large in practice, the trees usually have very small degrees, typically no larger than three. We develop a new method for finding a maximum agreement subtree of k trees, of which one has degree bounded by d. This new method enables us to find a maximum agreement subtree in time O(knd+').

We introduce an algorithm for cooperative planning in multi-agent systems. The algorithm enables the agents to combine (fuse) their plans in order to increase their joint profits. A computational resources and skills framework is developed for representing the planned activities of an agent under time constraints. Using this resource-skill framework, we present an efficient (polynomial time) algorithm that fuses the plans of a group of agents in such a way that their joint profits improve. The framework and the algorithm are illustrated ...

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.

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.

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.

In this paper we present a collection of results pertaining to haplotyping. The first set of results concerns the combinatorial problem of reconstructing haplotypes from incomplete and/or imperfectly sequenced haplotype data. More specifically, we show that an interesting, restricted case of Minimum Error Correction (MEC) is NP-hard, point out problems in earlier claims about a related problem, and present a polynomial-time algorithm for the ungapped case of Longest Haplotype Reconstruction (LHR). Secondly, we present a polynomial time algorithm for the problem of resolving genotype data using as few haplotypes as possible (the Pure Parsimony Haplotyping Problem, PPH) where each genotype has at most two ambiguous positions, thus solving an open problem posed by Lancia et al in "Haplotyping Populations by Pure Parsimony: Complexity of Exact and Approximation Algorithms."

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.

The complexity of a computational problem is the order of computational resources which are necessary and sufficient to solve the problem. The algorithm complexity is the cost of a particular algorithm. We say that a problem has polynomial complexity if its computational complexity is a polynomial in the measure of input size. We introduce polynomial time algorithms based in generating functions for computing the Myerson value in weighted voting games restricted by a tree. Moreover, we apply the new generating algorithm for computing the Myerson value in the Council of Ministers of the European Union restricted by a communication structure.

The paper concerns the application of a non-classical performance measure, a late work criterion (Y, Y w), to scheduling problems. It estimates the quality of the obtained solution with regard to the duration of the late parts of tasks not taking into account the quantity of this delay. The paper provides the formal definition of the late work parameter, especially in the shop environment, together with its practical motivation. It contains general complexity studies and the results of investigating open-shop scheduling cases, i.e. two polynomial time algorithms for problems O | pmtn, r i | Y w and O2 | d i = d | Y, as well as the binary NP-hardness proof for O2 | d i = d | Y w .

This contribution is deliberately application-oriented. Conversely, the considered application introduces new circumstances and helpful designs for the graph techniques and interpreta- tions. This study seeks to improve the knowledge of the the cyclical structure of macro-econometric models. The retained application is a recent aca- demic model from Fair '94 for the United States. This empirical economic model of 130 equations is highly interdependent with exactly 7930 elementary circuits.

This paper examines the complexity of global verification for MAX-SAT, MAX-k-SAT (for k 3), Vertex Cover, and Traveling Salesman Problem. These results are obtained by adaptations of the transformations that prove such problems to be NP-complete. The class of problems PGS is defined to be those discrete optimization problems for which there exists a polynomial time algorithm such that given any solution ω, either a solution can be found with a better objective function value or it can be concluded that no such solution exists and ω is a global optimum. This paper demonstrates that if any one of MAX-SAT, MAX-k-SAT (for k 3), Vertex Cover, or Traveling Salesman Problem are in PGS, then P=NP.

A general technique is described for solving certain NP-hard graph problems in time that is exponential in a parameter k defined as the maximum, over all nonseparable components C of the graph, of the number of edges that must be added to a tree to produce C; for a connected graph, k is no more than the number of edges of the graph minus the number of vertices plus one. The technique is illustrated in detail for the following facility location problem: Given a connected graph G(V, E) such that each edge has an associated positive integer length and given a positive integer r, place the minimum number of centers on points of the graph such that every point of the graph is within distance r from some center (a "point" is either a vertex or a point on some edge). An algorithm of time complexity O(I El. (6r) rkm) is given. A parallel implementation of the algorithm, with optimal speedup over the sequential version for a fairly wide range for the number of processors, is presented.

Constraint satisfaction problems (CSPs) play a central role in the real world and in computer science. CSPs are in general NP-hard and a general deterministic polynomial time algorithm is not known. CSPs with finite domains for the variables (finite constraint satisfaction problems) are considered. They (and all NP-complete problems) can be reduced in polynomial time to the satisfaction of a

We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various elds such as machine learning, operations research and pattern recognition. In the rst class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to nd a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, , > and 6 = are considered. While Min RVLS with equations was known to be NP-hard in 27], we established in 2, 5] that Min ULR with equalities and inequalities are NP-hard even when restricted to homogeneous systems with bipolar coe cients. The latter problems have been shown hard to approximate in 7]. In this paper we determine strong bounds on the approximability of various variants of Min RVLS and Min ULR, including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional. The various NP-hard versions turn out to have di erent approximability properties depending on the type of relations and the additional constraints, but none of them can be approximated within any constant factor, unless P=NP. Particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning. In particular, we disprove a conjecture in 64] regarding the existence of a polynomial time algorithm to design linear classi ers (or perceptrons) that involve a close-to-minimum number of features.

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.

This paper describes a pseudo-polynomial time algorithm for timing analysis of a class of choice-free asynchronous systems, called tightly coupled systems, with both min-and max-type timing constraints and bounded component delays. The algorithm consists of two phases:(1) long-term behavior analysis, that computes bounds on the time separation of events after the system has run for a sufficiently long period of time, and (2) startup behavior analysis, that computes time separations between events during an initial startup period ...