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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.

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)

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.

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.

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.

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.

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.

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.

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.

In theoretical computer science, combinatorial optimization problems are about finding an optimal item from a finite set of objects. Combinatorial optimization is the process of searching for maxima or minima of an unbiased function whose domain is a discrete and large configuration space. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems. Applications for combinatorial optimization include determining the optimal way to deliver packages in logistics applications, determining taxis best route to reach a destination address, and determining the best allocation of jobs to people. Some common problems involving combinatorial optimizations are the Knapsack problem, the Job Assignment problem, and the Travelling Salesman problem. This paper proposes three new optimized algorithms for solving three combinatorial optimization problems namely the Knapsack problem, the Job Assignment problem, and the Traveling Salesman respectively. The Knapsack problem is about finding the most valuable subset of items that fit into the knapsack. The Job Assignment problem is about assigning a person to a job with the lowest total cost possible. The Traveling Salesman problem is about finding the shortest tour to a destination city through travelling a given set of cities. Each problem is to be tackled separately. First, the design is proposed, then the pseudo code is created along with analyzing its time complexity. Finally, the algorithm is implemented using a high level programming language. As future work, the proposed algorithms are to be parallelized so that they can execute on multiprocessing environments making their execution time faster and more scalable.  by Youssef Bassil "Implementation of Combinatorial Algorithms using Optimization Techniques"  Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019,  URL: https://www.ijtsrd.com/papers/ijtsrd22925.pdf Paper URL: https://www.ijtsrd.com/computer-science/data-processing/22925/implementation-of-combinatorial-algorithms-using-optimization-techniques/youssef-bassil

In theoretical computer science, combinatorial optimization problems are about finding an optimal item from a finite set of objects. Combinatorial optimization is the process of searching for maxima or minima of an unbiased function whose domain is a discrete and large configuration space. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems. Applications for combinatorial optimization include determining the optimal way to deliver packages in logistics applications, determining taxis best route to reach a destination address, and determining the best allocation of jobs to people. Some common problems involving combinatorial optimizations are the Knapsack problem, the Job Assignment problem, and the Travelling Salesman problem. This paper proposes three new optimized algorithms for solving three combinatorial optimization problems namely the Knapsack problem, the Job Assignment problem, and the Traveling Salesman respectively. The Knapsack problem is about finding the most valuable subset of items that fit into the knapsack. The Job Assignment problem is about assigning a person to a job with the lowest total cost possible. The Traveling Salesman problem is about finding the shortest tour to a destination city through travelling a given set of cities. Each problem is to be tackled separately. First, the design is proposed, then the pseudo code is created along with analyzing its time complexity. Finally, the algorithm is implemented using a high level programming language. As future work, the proposed algorithms are to be parallelized so that they can execute on multiprocessing environments making their execution time faster and more scalable.  by Youssef Bassil "Implementation of Combinatorial Algorithms using Optimization Techniques"  Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019,  URL: https://www.ijtsrd.com/papers/ijtsrd22925.pdf Paper URL: https://www.ijtsrd.com/computer-science/data-processing/22925/implementation-of-combinatorial-algorithms-using-optimization-techniques/youssef-bassil

In theoretical computer science, combinatorial optimization problems are about finding an optimal item from a finite set of objects. Combinatorial optimization is the process of searching for maxima or minima of an unbiased function whose domain is a discrete and large configuration space. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems. Applications for combinatorial optimization include determining the optimal way to deliver packages in logistics applications, determining taxis best route to reach a destination address, and determining the best allocation of jobs to people. Some common problems involving combinatorial optimizations are the Knapsack problem, the Job Assignment problem, and the Travelling Salesman problem. This paper proposes three new optimized algorithms for solving three combinatorial optimization problems namely the Knapsack problem, the Job Assignment problem, and the Traveling Salesman respectively. The Knapsack problem is about finding the most valuable subset of items that fit into the knapsack. The Job Assignment problem is about assigning a person to a job with the lowest total cost possible. The Traveling Salesman problem is about finding the shortest tour to a destination city through travelling a given set of cities. Each problem is to be tackled separately. First, the design is proposed, then the pseudo code is created along with analyzing its time complexity. Finally, the algorithm is implemented using a high level programming language. As future work, the proposed algorithms are to be parallelized so that they can execute on multiprocessing environments making their execution time faster and more scalable.  by Youssef Bassil "Implementation of Combinatorial Algorithms using Optimization Techniques"  Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019,  URL: https://www.ijtsrd.com/papers/ijtsrd22925.pdf Paper URL: https://www.ijtsrd.com/computer-science/data-processing/22925/implementation-of-combinatorial-algorithms-using-optimization-techniques/youssef-bassil

In theoretical computer science, combinatorial optimization problems are about finding an optimal item from a finite set of objects. Combinatorial optimization is the process of searching for maxima or minima of an unbiased function whose domain is a discrete and large configuration space. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems. Applications for combinatorial optimization include determining the optimal way to deliver packages in logistics applications, determining taxis best route to reach a destination address, and determining the best allocation of jobs to people. Some common problems involving combinatorial optimizations are the Knapsack problem, the Job Assignment problem, and the Travelling Salesman problem. This paper proposes three new optimized algorithms for solving three combinatorial optimization problems namely the Knapsack problem, the Job Assignment problem, and the Traveling Salesman respectively. The Knapsack problem is about finding the most valuable subset of items that fit into the knapsack. The Job Assignment problem is about assigning a person to a job with the lowest total cost possible. The Traveling Salesman problem is about finding the shortest tour to a destination city through travelling a given set of cities. Each problem is to be tackled separately. First, the design is proposed, then the pseudo code is created along with analyzing its time complexity. Finally, the algorithm is implemented using a high level programming language. As future work, the proposed algorithms are to be parallelized so that they can execute on multiprocessing environments making their execution time faster and more scalable.  by Youssef Bassil "Implementation of Combinatorial Algorithms using Optimization Techniques"  Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019,  URL: https://www.ijtsrd.com/papers/ijtsrd22925.pdf Paper URL: https://www.ijtsrd.com/computer-science/data-processing/22925/implementation-of-combinatorial-algorithms-using-optimization-techniques/youssef-bassil

In theoretical computer science, combinatorial optimization problems are about finding an optimal item from a finite set of objects. Combinatorial optimization is the process of searching for maxima or minima of an unbiased function whose domain is a discrete and large configuration space. It often involves determining the way to efficiently allocate resources used to find solutions to mathematical problems. Applications for combinatorial optimization include determining the optimal way to deliver packages in logistics applications, determining taxis best route to reach a destination address, and determining the best allocation of jobs to people. Some common problems involving combinatorial optimizations are the Knapsack problem, the Job Assignment problem, and the Travelling Salesman problem. This paper proposes three new optimized algorithms for solving three combinatorial optimization problems namely the Knapsack problem, the Job Assignment problem, and the Traveling Salesman respectively. The Knapsack problem is about finding the most valuable subset of items that fit into the knapsack. The Job Assignment problem is about assigning a person to a job with the lowest total cost possible. The Traveling Salesman problem is about finding the shortest tour to a destination city through travelling a given set of cities. Each problem is to be tackled separately. First, the design is proposed, then the pseudo code is created along with analyzing its time complexity. Finally, the algorithm is implemented using a high level programming language. As future work, the proposed algorithms are to be parallelized so that they can execute on multiprocessing environments making their execution time faster and more scalable.  by Youssef Bassil "Implementation of Combinatorial Algorithms using Optimization Techniques"  Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-3 , April 2019,  URL: https://www.ijtsrd.com/papers/ijtsrd22925.pdf Paper URL: https://www.ijtsrd.com/computer-science/data-processing/22925/implementation-of-combinatorial-algorithms-using-optimization-techniques/youssef-bassil

This paper is devoted to the solving of graph partition task. We consider five algorithms: iterative, evolutionary, genetic, ant colony and bee colony. A computer program was created during this work. This program realizes the described model of biologically inspired algorithms. Experimental researches have proved efficiency of the bee algorithm in comparison with other algorithms. 

This paper is devoted to the solving of graph partition task. We consider five algorithms: iterative, evolutionary, genetic, ant colony and bee colony. A computer program was created during this work. This program realizes the described model of biologically inspired algorithms. Experimental researches have proved efficiency of the bee algorithm in comparison with other algorithms. 

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.

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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.

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.

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.

Evolutionary algorithms are popular heuristics for solving various combinatorial problems as they are easy to apply and often produce good results. Island models parallelize evolution by using different populations, called islands, which are connected by a graph structure as communication topology. Each island periodically communicates copies of good solutions to neighboring islands in a process called migration. We consider the speedup gained by island models in terms of the parallel running time for problems from combinatorial optimization: sorting (as maximization of sortedness), shortest paths, and Eulerian cycles. Different search operators are considered. The results show in which settings and up to what degree evolutionary algorithms can be parallelized efficiently. Along the way, we also investigate how island models deal with plateaus. In particular, we show that natural settings lead to exponential vs. logarithmic speedups, depending on the frequency of migration.

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 work of Avestimehr et al. '07 has recently proposed a deterministic model for wireless networks and characterized the unicast capacity C of such networks as the minimum rank of the adjacency matrices describing all possible source-destination cuts. Amaudruz & Fragouli first proposed a polynomial-time algorithm for finding the unicast capacity of a linear deterministic wireless network in their 2009 paper. In this work, we improve upon Amaudruz & Fragouli's work and further reduce the computational complexity of the algorithm by fully exploring the useful combinatorial features intrinsic in the problem. Our improvement applies generally with any size of finite fields associated with the channel model. Comparing with other algorithms on solving the same problem, our improved algorithm is very competitive in terms of complexity.

The work of Avestimehr et al. '07 has recently proposed a deterministic model for wireless networks and characterized the unicast capacity C of such networks as the minimum rank of the adjacency matrices describing all possible source-destination cuts. Amaudruz & Fragouli first proposed a polynomial-time algorithm for finding the unicast capacity of a linear deterministic wireless network in their 2009 paper. In this work, we improve upon Amaudruz & Fragouli's work and further reduce the computational complexity of the algorithm by fully exploring the useful combinatorial features intrinsic in the problem. Our improvement applies generally with any size of finite fields associated with the channel model. Comparing with other algorithms on solving the same problem, our improved algorithm is very competitive in terms of complexity.

[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

Several major advances in the area of combinatorial algorithms include improved algorithms for matrix multiplication and maximum network flow, a polynomial-time algorithm for linear programming, and steps toward a polynomial-time algorithm for graph isomorphism. This paper surveys these results and suggests directions for future research. Included is a discussion of recent work by the author and his students on dynamic dictionaries, network flow problems, and related questions.

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.

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 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 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.

We discuss the theme on translating different descriptions of computative algorithms into high-level programming languages, enumerate some advantages of analytical descriptions and demonstrate that logical functions may be used effectively to create analytical formulae available for describing a set of combinatorial and number-theoretic computative algorithms.

We discuss the theme on translating different descriptions of computative algorithms into high-level programming languages, enumerate some advantages of analytical descriptions and demonstrate that logical functions may be used effectively to create analytical formulae available for describing a set of combinatorial and number-theoretic computative algorithms.

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.

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.

We consider two combinatorial problems. The first we call "search with wildcards": given an unknown n-bit string x, and the ability to check whether any subset of the bits of x is equal to a provided query string, the goal is to output x. We give a nearly optimal O(sqrt(n) log n) quantum query algorithm for search with wildcards, beating the classical lower bound of Omega(n) queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most k special items out of a set of n items, given the ability to make queries of the form "does the set S contain any special items?" for any subset S of the n items. We give a simple quantum algorithm which uses O(k log k) queries to solve this problem, as compared with the classical lower bound of Omega(k log(n/k)) queries.

We consider two combinatorial problems. The first we call "search with wildcards": given an unknown n-bit string x, and the ability to check whether any subset of the bits of x is equal to a provided query string, the goal is to output x. We give a nearly optimal O(sqrt(n) log n) quantum query algorithm for search with wildcards, beating the classical lower bound of Omega(n) queries. Rather than using amplitude amplification or a quantum walk, our algorithm is ultimately based on the solution to a state discrimination problem. The second problem we consider is combinatorial group testing, which is the task of identifying a subset of at most k special items out of a set of n items, given the ability to make queries of the form "does the set S contain any special items?" for any subset S of the n items. We give a simple quantum algorithm which uses O(k log k) queries to solve this problem, as compared with the classical lower bound of Omega(k log(n/k)) queries.

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.

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.

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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.

We consider a generalization of the maximum network flow problem in which the amounts of flow entering and leaving an arc are linearly related. More precisely, if x(e) units of flow enter an arc e, x(e) gamma (e) units arrive at the other end. For instance, nodes of the graph can correspond to different currencies, with the multipliers being the exchange rates. We require conservation of flow at every node except a given source node. The goal is to maximize the amount of flow excess at the source. This problem is a special case of linear programming, and therefore can be solved in polynomial time. In this paper we present the first polynomial time combinatorial optimization algorithms for this problem. The algorithms are simple and intuitive.