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We consider the homogenized linear feasibility problem, to find an $x$ on the unit sphere, satisfying $n$ line ar inequalities $a_i^Tx\ge 0$. To solve this problem we consider the centers of the insphere of spherical simpl ices, whose facets are determined by a subset of the constraints. As a result we find a new combinatorial algor ithm for the linear feasibility problem. If we allow rescaling this algorithm becomes polynomial. We point out that the algorithm solves as well the more general convex feasibility problem. Moreover numerical experiments s how that the algorithm could be of practical interest.

The subject of this report is an analysis of the expected, or average case performance of local improvement algorithms. The first chapter presents the basic model, defines the combinatorial structures which are the basis for the analysis, and describes the randomness assumptions upon which the expectation are based. The second chapter examines these structures in more detail, including an analysis of both best and worst case performance. The third chapter discusses simulation results which predict an approximately linear average case performance, and proves an O(n2 log n) upper bound for two of the random distributions assumed. Chapter Four proves some extensions and sharper versions of this upper bound. The fifth chapter applies the model to principal pivoting algorithms for the linear complementarity problem, and to the simplex method. Although local improvement is not guaranteed to find a global optimum for all problems, most notably those that are NP-complete, it is nonetheless often used in these cases. Chapter Six discusses these appllications.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.

Combinatorial games lead to several interesting, clean problems in algorithms and complexity theory, many of which remain open. The purpose of this paper is to provide an overview of the area to encourage further research. In particular, we begin with general background in Combinatorial Game Theory, which analyzes ideal play in perfect-information games, and Constraint Logic, which provides a framework for showing hardness. Then we survey results about the complexity of determining ideal play in these games, and the related problems of solving puzzles, in terms of both polynomial-time algorithms and computational intractability results. Our review of background and survey of algorithmic results are by no means complete, but should serve as a useful primer.

We study the {\em $k$-route} generalizations of various cut problems, the most general of which is \emph{$k$-route multicut} ($k$-MC) problem, wherein we have $r$ source-sink pairs and the goal is to delete a minimum-cost set of edges to reduce the edge-connectivity of every source-sink pair to below $k$. The $k$-route extensions of multiway cut ($k$-MWC), and the minimum $s$-$t$ cut problem ($k$-$(s,t)$-cut), are similarly defined. We present various approximation and hardness results for these $k$-route cut problems that improve the state-of-the-art for these problems in several cases. (i) For {\em $k$-route multiway cut}, we devise simple, but surprisingly effective, combinatorial algorithms that yield bicriteria approximation guarantees that markedly improve upon the previous-best guarantees. (ii) For {\em $k$-route multicut}, we design algorithms that improve upon the previous-best approximation factors by roughly an $O(\sqrt{\log r})$-factor, when $k=2$, and for general $k$ and unit costs and any fixed violation of the connectivity threshold $k$. The main technical innovation is the definition of a new, powerful \emph{region growing} lemma that allows us to perform region-growing in a recursive fashion even though the LP solution yields a {\em different metric} for each source-sink pair. (iii) We complement these results by showing that the {\em $k$-route $s$-$t$ cut} problem is at least as hard to approximate as the {\em densest-$k$-subgraph} (DkS) problem on uniform hypergraphs.

We study the {\em $k$-route} generalizations of various cut problems, the most general of which is \emph{$k$-route multicut} ($k$-MC) problem, wherein we have $r$ source-sink pairs and the goal is to delete a minimum-cost set of edges to reduce the edge-connectivity of every source-sink pair to below $k$. The $k$-route extensions of multiway cut ($k$-MWC), and the minimum $s$-$t$ cut problem ($k$-$(s,t)$-cut), are similarly defined. We present various approximation and hardness results for these $k$-route cut problems that improve the state-of-the-art for these problems in several cases. (i) For {\em $k$-route multiway cut}, we devise simple, but surprisingly effective, combinatorial algorithms that yield bicriteria approximation guarantees that markedly improve upon the previous-best guarantees. (ii) For {\em $k$-route multicut}, we design algorithms that improve upon the previous-best approximation factors by roughly an $O(\sqrt{\log r})$-factor, when $k=2$, and for general $k$ and unit costs and any fixed violation of the connectivity threshold $k$. The main technical innovation is the definition of a new, powerful \emph{region growing} lemma that allows us to perform region-growing in a recursive fashion even though the LP solution yields a {\em different metric} for each source-sink pair. (iii) We complement these results by showing that the {\em $k$-route $s$-$t$ cut} problem is at least as hard to approximate as the {\em densest-$k$-subgraph} (DkS) problem on uniform hypergraphs.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

The primary activities supported by the grant during the reporting period include a new result showing the hardness of the single-source robust network design and an invitation to include this in the special issue devoted to selected papers in FOCS 2005. A summer intern was hosted, Andrew McGregor from UPenn, who worked with Shepherd on recognizing Hilbert Bases and other theoretical topics in Math Programming. A visit was also supported for Gianpaolo Oriolo (Rome), which resulted in some new joint work on robust network design. In addition, there was a week visit from Seffi Naor (Technicion). Travel supported during this period includes trips by Shepherd to UPenn to work with Sanjeev Khanna and C. Chekuri on the mutlicommodity flow problem. This work has resulted in the FOCS 2005 paper, which in addition was invited into a special issue of selected papers. Conferences attended were the 2004 APPROX/RANDOM (Chekuri) and CORC 4th Optimization Day (Shepherd).

The project dealt with two classes of core decision-making algorithms related to operator-UV collaboration; the first class involves the routing of UVs through the set of targets nominated by the operator and the second class of problems involves decision-making algorithms for UVs to accommodate uncertainty. We have developed approximation, lower bounding and exact algorithms to address the two classes of problems. We have also implemented these algorithms in simulations to corroborate the performance of these algorithms. In the ensuing discussion, we will summarize our work for the project, and our main results.

Biological processes, including cell differentiation, organism development, and disease progression, can be interpreted as attractors (fixed points or limit cycles) of an underlying networked dynamical system. In this paper, we study the problem of computing a minimum-size subset of control nodes that can be used to steer a given biological network towards a desired attractor, when the networked system has Boolean dynamics. We first prove that this problem cannot be approximated to any nontrivial factor unless P=NP. We then formulate a sufficient condition and prove that the sufficient condition is equivalent to a target set selection problem, which can be solved using integer linear programming. Furthermore, we show that structural properties of biological networks can be exploited to reduce the computational complexity. We prove that when the network nodes have threshold dynamics and certain topological structures, such as block cactus topology and hierarchical organization, the input selection problem can be solved or approximated in polynomial time. For networks with nested canalyzing dynamics, we propose polynomial-time algorithms that are within a polylogarithmic bound of the global optimum. We validate our approach through numerical study on real-world gene regulatory networks.

Biological processes, including cell differentiation, organism development, and disease progression, can be interpreted as attractors (fixed points or limit cycles) of an underlying networked dynamical system. In this paper, we study the problem of computing a minimum-size subset of control nodes that can be used to steer a given biological network towards a desired attractor, when the networked system has Boolean dynamics. We first prove that this problem cannot be approximated to any nontrivial factor unless P=NP. We then formulate a sufficient condition and prove that the sufficient condition is equivalent to a target set selection problem, which can be solved using integer linear programming. Furthermore, we show that structural properties of biological networks can be exploited to reduce the computational complexity. We prove that when the network nodes have threshold dynamics and certain topological structures, such as block cactus topology and hierarchical organization, the input selection problem can be solved or approximated in polynomial time. For networks with nested canalyzing dynamics, we propose polynomial-time algorithms that are within a polylogarithmic bound of the global optimum. We validate our approach through numerical study on real-world gene regulatory networks.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Combinatorial optimization problems are problems that have a large number of discrete solutions and a cost function for evaluating those solutions in comparison to one another. With the vital need of solving the combinatorial problem, several research efforts have been concentrated on the biological entities behaviors to utilize such behaviors in population-based metaheuristic. This paper presents bee colony algorithms which is one of the sophisticated biological nature life. A brief detail of the nature of bee life has been presented with further classification of its behaviors. Furthermore, an illustration of the algorithms that have been derived from bee colony which are bee colony optimization, and artificial bee colony. Finally, a comparative analysis has been conducted between these algorithms according to the results of the traveling salesman problem solution. Where the bee colony optimization (BCO) rendered the best performance in terms of computing time and results.

Combinatorial optimization problems are problems that have a large number of discrete solutions and a cost function for evaluating those solutions in comparison to one another. With the vital need of solving the combinatorial problem, several research efforts have been concentrated on the biological entities behaviors to utilize such behaviors in population-based metaheuristic. This paper presents bee colony algorithms which is one of the sophisticated biological nature life. A brief detail of the nature of bee life has been presented with further classification of its behaviors. Furthermore, an illustration of the algorithms that have been derived from bee colony which are bee colony optimization, and artificial bee colony. Finally, a comparative analysis has been conducted between these algorithms according to the results of the traveling salesman problem solution. Where the bee colony optimization (BCO) rendered the best performance in terms of computing time and results.

Combinatorial optimization problems are problems that have a large number of discrete solutions and a cost function for evaluating those solutions in comparison to one another. With the vital need of solving the combinatorial problem, several research efforts have been concentrated on the biological entities behaviors to utilize such behaviors in population-based metaheuristic. This paper presents bee colony algorithms which is one of the sophisticated biological nature life. A brief detail of the nature of bee life has been presented with further classification of its behaviors. Furthermore, an illustration of the algorithms that have been derived from bee colony which are bee colony optimization, and artificial bee colony. Finally, a comparative analysis has been conducted between these algorithms according to the results of the traveling salesman problem solution. Where the bee colony optimization (BCO) rendered the best performance in terms of computing time and results.

Combinatorial optimization problems are problems that have a large number of discrete solutions and a cost function for evaluating those solutions in comparison to one another. With the vital need of solving the combinatorial problem, several research efforts have been concentrated on the biological entities behaviors to utilize such behaviors in population-based metaheuristic. This paper presents bee colony algorithms which is one of the sophisticated biological nature life. A brief detail of the nature of bee life has been presented with further classification of its behaviors. Furthermore, an illustration of the algorithms that have been derived from bee colony which are bee colony optimization, and artificial bee colony. Finally, a comparative analysis has been conducted between these algorithms according to the results of the traveling salesman problem solution. Where the bee colony optimization (BCO) rendered the best performance in terms of computing time and results.

[CITATION] Solving combinatorial optimization problems by simulated annealing, genetic algorithms, and neural networks Y Lu - 1991 - University of Minnesota Cited by 4

[CITATION] Solving combinatorial optimization problems by simulated annealing, genetic algorithms, and neural networks Y Lu - 1991 - University of Minnesota Cited by 4

We re-examine a practical aspect of combinatorial fuzzy problems of various types, including search, counting, optimization, and decision problems. We are focused only on those fuzzy problems that take series of fuzzy input objects and produce fuzzy values. To solve such problems efficiently, we design fast fuzzy algorithms, which are modeled by polynomial-time deterministic fuzzy Turing machines equipped with read-only auxiliary tapes and write-only output tapes and also modeled by polynomial-size fuzzy circuits composed of fuzzy gates. We also introduce fuzzy proof verification systems to model the fuzzification of nondeterminism. Those models help us identify four complexity classes: Fuzzy-FPA of fuzzy functions, Fuzzy-PA and Fuzzy-NPA of fuzzy decision problems, and Fuzzy-NPAO of fuzzy optimization problems. Based on a relative approximation scheme targeting fuzzy membership degree, we formulate two notions of "reducibility" in order to compare the computational complexity of two fuzzy problems. These reducibility notions make it possible to locate the most difficult fuzzy problems in Fuzzy-NPA and in Fuzzy-NPAO.

We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the fixed-charge counterpart of the problem. For many practical concave cost problems, the fixed-charge counterpart is a well-studied combinatorial optimization problem. Our technique preserves constant factor approximation ratios, as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. Using our technique, we obtain a new 1.61-approximation algorithm for the concave cost facility location problem. For inventory problems, we obtain a new exact algorithm for the economic lot-sizing problem with general concave ordering costs, and a 4-approximation algorithm for the joint replenishment problem with general concave individual ordering costs.

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This paper examines recent work on the complexity of combinatorial algorithms, highlighting the aims of the work, the mathematical tools used, and the important results. Included are sections discussing ways to measure the complexity of an algorithm, methods for proving that certain problems are very hard to solve, tools useful in the design of good algorithms, and recent improvements in algorithms for solving ten representative problems. The final section suggests some directions for future research. (Author)

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

We study practically efficient methods for performing combinatorial group testing. We present efficient non-adaptive and two-stage combinatorial group testing algorithms, which identify the at most d items out of a given set of n items that are defective, using fewer tests for all practical set sizes. For example, our two-stage algorithm matches the information theoretic lower bound for the number of tests in a combinatorial group testing regimen.

Estimation of Distribution Algorithms (EDAs) require flexible probability models that can be efficiently learned and sampled. Deep Boltzmann Machines (DBMs) are generative neural networks with these desired properties. We integrate a DBM into an EDA and evaluate the performance of this system in solving combinatorial optimization problems with a single objective. We compare the results to the Bayesian Optimization Algorithm. The performance of DBM-EDA was superior to BOA for difficult additively decomposable functions, i.e., concatenated deceptive traps of higher order. For most other benchmark problems, DBM-EDA cannot clearly outperform BOA, or other neural network-based EDAs. In particular, it often yields optimal solutions for a subset of the runs (with fewer evaluations than BOA), but is unable to provide reliable convergence to the global optimum competitively. At the same time, the model building process is computationally more expensive than that of other EDAs using probabilistic models from the neural network family, such as DAE-EDA.

We extend an hypergraph representation, introduced by Finkelstein and Roytberg, to unify dynamic programming algorithms in the context of RNA folding with pseudoknots. Classic applications of RNA dynamic programming energy minimization, partition function, base-pair probabilities...) are reformulated within this framework, giving rise to very simple algorithms. This reformulation allows one to conceptually detach the conformation space/energy model -- captured by the hypergraph model -- from the specific application, assuming unambiguity of the decomposition. To ensure the latter property, we propose a new combinatorial methodology based on generating functions. We extend the set of generic applications by proposing an exact algorithm for extracting generalized moments in weighted distribution, generalizing a prior contribution by Miklos and al. Finally, we illustrate our full-fledged programme on three exemplary conformation spaces (secondary structures, Akutsu's simple type pseudoknots and kissing hairpins). This readily gives sets of algorithms that are either novel or have complexity comparable to classic implementations for minimization and Boltzmann ensemble applications of dynamic programming.

We extend an hypergraph representation, introduced by Finkelstein and Roytberg, to unify dynamic programming algorithms in the context of RNA folding with pseudoknots. Classic applications of RNA dynamic programming energy minimization, partition function, base-pair probabilities...) are reformulated within this framework, giving rise to very simple algorithms. This reformulation allows one to conceptually detach the conformation space/energy model -- captured by the hypergraph model -- from the specific application, assuming unambiguity of the decomposition. To ensure the latter property, we propose a new combinatorial methodology based on generating functions. We extend the set of generic applications by proposing an exact algorithm for extracting generalized moments in weighted distribution, generalizing a prior contribution by Miklos and al. Finally, we illustrate our full-fledged programme on three exemplary conformation spaces (secondary structures, Akutsu's simple type pseudoknots and kissing hairpins). This readily gives sets of algorithms that are either novel or have complexity comparable to classic implementations for minimization and Boltzmann ensemble applications of dynamic programming.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

Since October 1, 1979, research in Complexity Theory and Combinatorial Algorithms at the Department of Computer Science at the University of Illinois was supported by the Office of Naval Research. During this period of time, research work was carried out in the areas of Computational Complexity Theory, Scheduling Algorithms, Graph Algorithms, Dynamic Programming, and Fault- Tolerance Computing. We summarize here our accomplishments and our future plans, and we wish to request continued support for the period of October 1, 1980 - September 30, 1982 from ONR for research in these areas. Scheduling to meet deadlines -- The problem of scheduling jobs to meet their deadlines was studied. Given a set of jobs each of which is specified by three parameters, ready time, deadline, and computation time, we want to schedule them on a computer system so that, if possible, all deadlines will be met. Furthermore, if indeed all deadlines can be met, we want to know the possibility of completing the executing of each job so that there will be a 'slack time' between the time of completion and the deadline. In particular, the following model is used: There is a single processor in the computing system. Each job consists of an infinite stream of periodic and identical requests. A request is ready when it arrives and should be completed prior to the arrival of the next request of the same job. The execution of a job can be interrupted and be resumed later on.

This project focused on designing fast algorithms for basic combinational optimization problems, including maximum flow, matching, multicommodity flow, and generalized flow. Many important applications are naturally stated as variants of these problems, and hence improved algorithms for these problems immediately lead to improved algorithms for a wide variety of applications. Our goal was to improve both sequential and parallel complexity. In many applications, solving a multicommodity or a generalized flow problem is only a first step in approximately solving an NP-complete problem; in the majority of such cases there is no need to have an exact solution of the problem. One of the focuses of the project was design of efficient approximation algorithms.

This project focused on designing fast algorithms for basic combinational optimization problems, including maximum flow, matching, multicommodity flow, and generalized flow. Many important applications are naturally stated as variants of these problems, and hence improved algorithms for these problems immediately lead to improved algorithms for a wide variety of applications. Our goal was to improve both sequential and parallel complexity. In many applications, solving a multicommodity or a generalized flow problem is only a first step in approximately solving an NP-complete problem; in the majority of such cases there is no need to have an exact solution of the problem. One of the focuses of the project was design of efficient approximation algorithms.

This thesis investigates parallel algorithms for a small, but representative, subclass of graph and matrix problems. In some cases, we develop new algorithms which we analyze for theoretical efficiency. In other cases, we modify and implement existing algorithms which we analyze for practical efficiency. We show how n-node, e-edge graphs can be contracted in a manner similar to the parallel tree contraction algorithm due to Miller and Reif. We give an O((n+e)/lgn)-processor deterministic algorithm that contracts a graph in O(lg squared n) time in the EREW PRAM model.

The first methods for tolerating more than a small number of worst- case faults in commonly-Lised networks such as the butterfly, the mesh of trees, and other hypercubic networks. Previously, work on the fault-tolerance properties of these networks was limited to showing how to overcome a single fault by adding an extra stage of hardware to the networks.

This project has completed. The PI considered optimization problems with uncertainty in the data. Specifically, algorithms and analysis methods were developed for several optimization scenarios where the random parameters were coming from some unknown probability distribution. One notable result efficiently calculates the average optimal solution for a combinatorial regret problem and provides new bounds for how for this may be from the worst case. For details, see the final report. Further, the project produced 8 published papers and 4 more under review or preparation. The specifics of these references are included in uploaded final report. This work has help to advance the state-of-the-art in robust optimization.

This project has completed. The PI considered optimization problems with uncertainty in the data. Specifically, algorithms and analysis methods were developed for several optimization scenarios where the random parameters were coming from some unknown probability distribution. One notable result efficiently calculates the average optimal solution for a combinatorial regret problem and provides new bounds for how for this may be from the worst case. For details, see the final report. Further, the project produced 8 published papers and 4 more under review or preparation. The specifics of these references are included in uploaded final report. This work has help to advance the state-of-the-art in robust optimization.

We introduce a new class of combinatorial markets in which agents have covering constraints over resources required and are interested in delay minimization. Our market model is applicable to several settings including scheduling, cloud computing, and communicating over a network. This model is quite different from the traditional models, to the extent that neither do the classical equilibrium existence results seem to apply to it nor do any of the efficient algorithmic techniques developed to compute equilibria seem to apply directly. We give a proof of existence of equilibrium and a polynomial time algorithm for finding one, drawing heavily on techniques from LP duality and submodular minimization. We observe that in our market model, the set of equilibrium prices could be a connected, non-convex set. To the best of our knowledge, this is the first natural example of the phenomenon where the set of solutions could have such complicated structure, yet there is a combinatorial polynomial time algorithm to find one. Finally, we show that our model inherits many of the fairness properties of traditional equilibrium models.

This paper considers an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0-1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function. The performance and utility of the proposed algorithms are demonstrated with the ON/OFF control problems of interdependent refrigeration systems.

Combinatorial optimization is widely applied in a number of areas nowadays. Unfortunately, many combinatorial optimization problems are NP-hard which usually means that they are unsolvable in practice. However, it is often unnecessary to have an exact solution. In this case one may use heuristic approach to obtain a near-optimal solution in some reasonable time. We focus on two combinatorial optimization problems, namely the Generalized Traveling Salesman Problem and the Multidimensional Assignment Problem. The first problem is an important generalization of the Traveling Salesman Problem; the second one is a generalization of the Assignment Problem for an arbitrary number of dimensions. Both problems are NP-hard and have hosts of applications. In this work, we discuss different aspects of heuristics design and evaluation. A broad spectrum of related subjects, covered in this research, includes test bed generation and analysis, implementation and performance issues, local search neighborhoods and efficient exploration algorithms, metaheuristics design and population sizing in memetic algorithm. The most important results are obtained in the areas of local search and memetic algorithms for the considered problems. In both cases we have significantly advanced the existing knowledge on the local search neighborhoods and algorithms by systematizing and improving the previous results. We have proposed a number of efficient heuristics which dominate the existing algorithms in a wide range of time/quality requirements. Several new approaches, introduced in our memetic algorithms, make them the state-of-the-art metaheuristics for the corresponding problems. Population sizing is one of the most promising among these approaches; it is expected to be applicable to virtually any memetic algorithm.

We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the constructible classes of Flajolet and Sedgewick with Joyal's species theory. We extend the implicit species theorem to structures of size zero. A quadratic iterative Newton method is shown to solve well-founded systems combinatorially. From there, truncations of the corresponding generating series are obtained in quasi-optimal complexity. This iteration transfers to a numerical scheme that converges unconditionally to the values of the generating series inside their disk of convergence. These results provide important subroutines in random generation. Finally, the approach is extended to combinatorial differential systems.

We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the constructible classes of Flajolet and Sedgewick with Joyal's species theory. We extend the implicit species theorem to structures of size zero. A quadratic iterative Newton method is shown to solve well-founded systems combinatorially. From there, truncations of the corresponding generating series are obtained in quasi-optimal complexity. This iteration transfers to a numerical scheme that converges unconditionally to the values of the generating series inside their disk of convergence. These results provide important subroutines in random generation. Finally, the approach is extended to combinatorial differential systems.

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be finite if alpha lies below an algorithm--dependent threshold alpha\_A, and exponentially small in N above. The critical behaviour is universal for all algorithms based on the widely-used unitary propagation rule: P[ (1 + epsilon) alpha\_A, N] ~ exp[-N^(1/6) Phi(epsilon N^(1/3)) ]. Exponents are related to the critical behaviour of random graphs, and the scaling function Phi is exactly calculated through a mapping onto a diffusion-and-death problem.

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be finite if alpha lies below an algorithm--dependent threshold alpha\_A, and exponentially small in N above. The critical behaviour is universal for all algorithms based on the widely-used unitary propagation rule: P[ (1 + epsilon) alpha\_A, N] ~ exp[-N^(1/6) Phi(epsilon N^(1/3)) ]. Exponents are related to the critical behaviour of random graphs, and the scaling function Phi is exactly calculated through a mapping onto a diffusion-and-death problem.

We focus on designing combinatorial algorithms for the Capacitated Network Design problem (Cap-SNDP). The Cap-SNDP is the problem of satisfying connectivity requirements when edges have costs and hard capacities. We begin by showing that the Group Steiner tree problem (GST) is a special case of Cap-SNDP even when there is connectivity requirement between only one source-sink pair. This implies the first poly-logarithmic lower bound for the Cap-SNDP. We next provide combinatorial algorithms for several special cases of this problem. The Cap-SNDP is equivalent to its special case when every edge has either zero cost or infinite capacity. We consider a special case, called Connected Cap-SNDP, where all infinite-capacity edges in the solution are required to form a connected component containing the sinks. This problem is motivated by its similarity to the Connected Facility Location problem [G+01,SW04]. We solve this problem by reducing it to Submodular tree cover problem, which is a common generalization of Connected Cap-SNDP and Group Steiner tree problem. We generalize the recursive greedy algorithm [CEK] achieving a poly-logarithmic approximation algorithm for Submodular tree cover problem. This result is interesting in its own right and gives the first poly-logarithmic approximation algorithms for Connected hard capacities set multi-cover and Connected source location. We then study another special case of Cap-SNDP called Unbalanced point-to-point connection problem. Besides its practical applications to shift design problems [EKS], it generalizes many problems such as k-MST, Steiner Forest and Point-to-Point Connection. We give a combinatorial logarithmic approximation algorithm for this problem by reducing it to degree-bounded SNDP.

Two general algorithms based on opportunity costs are given for approximating a revenue-maximizing set of bids an auctioneer should accept, in a combinatorial auction in which each bidder offers a price for some subset of the available goods and the auctioneer can only accept non-intersecting bids. Since this problem is difficult even to approximate in general, the algorithms are most useful when the bids are restricted to be connected node subsets of an underlying object graph that represents which objects are relevant to each other. The approximation ratios of the algorithms depend on structural properties of this graph and are small constants for many interesting families of object graphs. The running times of the algorithms are linear in the size of the bid graph, which describes the conflicts between bids. Extensions of the algorithms allow for efficient processing of additional constraints, such as budget constraints that associate bids with particular bidders and limit how many bids from a particular bidder can be accepted.