Downloads & Free Reading Options - Results

A new method for detecting negative cycles in a graph is proposed. This method is based upon the primal-dual relationships of a linear program formulated from an assignment problem type network. A computer program is developed for this new method to include the complete solution of the assignment problem. Results are given on program efficiency.

Method and Validation for Optimal Lineup Creation for Daily Fantasy Football Using Machine Learning and Linear Programming

Method and Validation for Optimal Lineup Creation for Daily Fantasy Football Using Machine Learning and Linear Programming

A complete computation procedure is given for a special class of two- stage linear programming models in which allocations in the first stage are made to meet an uncertain but known distribution of demands occurring in the second stage. This case, applicable to many practical problems constitutes the principal part of the paper. Next, a class of models is considered where the activities are divided into two or more stages. The quantities of activities in the first stage are the only ones that can be determined in advance because those in the second and latter stages depend on the outcome of random events. Theorems on convexity of the objective (cost) functions are established for the general m-stage case.

A complete computation procedure is given for a special class of two- stage linear programming models in which allocations in the first stage are made to meet an uncertain but known distribution of demands occurring in the second stage. This case, applicable to many practical problems constitutes the principal part of the paper. Next, a class of models is considered where the activities are divided into two or more stages. The quantities of activities in the first stage are the only ones that can be determined in advance because those in the second and latter stages depend on the outcome of random events. Theorems on convexity of the objective (cost) functions are established for the general m-stage case.

In this paper we describe SYNERGY, which is a highly parallelizable, linear planning system that is based on the genetic programming paradigm. Rather than reasoning about the world it is planning for, SYNERGY uses artificial selection, recombination and fitness measure to generate linear plans that solve conjunctive goals. We ran SYNERGY on several domains (e.g., the briefcase problem and a few variants of the robot navigation problem), and the experimental results show that our planner is capable of handling problem instances that are one to two orders of magnitude larger than the ones solved by UCPOP. In order to facilitate the search reduction and to enhance the expressive power of SYNERGY, we also propose two major extensions to our planning system: a formalism for using hierarchical planning operators, and a framework for planning in dynamic environments.

In this paper we describe SYNERGY, which is a highly parallelizable, linear planning system that is based on the genetic programming paradigm. Rather than reasoning about the world it is planning for, SYNERGY uses artificial selection, recombination and fitness measure to generate linear plans that solve conjunctive goals. We ran SYNERGY on several domains (e.g., the briefcase problem and a few variants of the robot navigation problem), and the experimental results show that our planner is capable of handling problem instances that are one to two orders of magnitude larger than the ones solved by UCPOP. In order to facilitate the search reduction and to enhance the expressive power of SYNERGY, we also propose two major extensions to our planning system: a formalism for using hierarchical planning operators, and a framework for planning in dynamic environments.

In this paper we describe SYNERGY, which is a highly parallelizable, linear planning system that is based on the genetic programming paradigm. Rather than reasoning about the world it is planning for, SYNERGY uses artificial selection, recombination and fitness measure to generate linear plans that solve conjunctive goals. We ran SYNERGY on several domains (e.g., the briefcase problem and a few variants of the robot navigation problem), and the experimental results show that our planner is capable of handling problem instances that are one to two orders of magnitude larger than the ones solved by UCPOP. In order to facilitate the search reduction and to enhance the expressive power of SYNERGY, we also propose two major extensions to our planning system: a formalism for using hierarchical planning operators, and a framework for planning in dynamic environments.

We consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form. Each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing. Our main contribution is the synthesis of a distributed dynamics and a set of state-based rules, termed triggers, that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program. Our technical approach to the algorithm design and analysis overcomes a number of challenges, including establishing convergence in the absence of a common smooth Lyapunov function, ensuring that the triggers are detectable by agents using only local information, accounting for asynchronism in the state broadcasts, and ruling out various causes of arbitrarily fast state broadcasting. Various simulations illustrate our results.

We consider a network of agents whose objective is for the aggregate of their states to converge to a solution of a linear program in standard form. Each agent has limited information about the problem data and can communicate with other agents at discrete time instants of their choosing. Our main contribution is the synthesis of a distributed dynamics and a set of state-based rules, termed triggers, that individual agents use to determine when to opportunistically broadcast their state to neighboring agents to ensure asymptotic convergence to a solution of the linear program. Our technical approach to the algorithm design and analysis overcomes a number of challenges, including establishing convergence in the absence of a common smooth Lyapunov function, ensuring that the triggers are detectable by agents using only local information, accounting for asynchronism in the state broadcasts, and ruling out various causes of arbitrarily fast state broadcasting. Various simulations illustrate our results.

Given a collection of discrete random variables representing outcomes of learned local predictors in natural language. e.g.. named entities and relations. we seek an optimal global assignment to the variables in the presence of general (non-sequential) constraints. Examples of these constraints include the type of arguments a relation can take, and the mutual activity of different relations. etc. We develop a linear programing formulation for this problem and evaluate it in the context of simultaneously learning named entities and relations. Our approach allows us to efficiently incorporate domain and task specific constraints at decision time, resulting in significant improvements in the accuracy and the human-like quality of the inferences.

Given a collection of discrete random variables representing outcomes of learned local predictors in natural language. e.g.. named entities and relations. we seek an optimal global assignment to the variables in the presence of general (non-sequential) constraints. Examples of these constraints include the type of arguments a relation can take, and the mutual activity of different relations. etc. We develop a linear programing formulation for this problem and evaluate it in the context of simultaneously learning named entities and relations. Our approach allows us to efficiently incorporate domain and task specific constraints at decision time, resulting in significant improvements in the accuracy and the human-like quality of the inferences.

In future conflicts, the projected expenditure rates of artillery ammunition greatly exceed the ability of the Marine direct support artillery battery's ammunition transportation assets. It is therefore vital that the artillery battery commander be able to select the most effective mix of ammunition to carry on his organic transportation in a given tactical situation. Linear programing is a tool which the batten' commander can use to help solve this important problem. This thesis provides a linear program to assist him in this solution. In addition, with slight modification, this linear program can be of use to commanders at all levels of the Marine artillerv organization.

The motivation for this work has been the need for a practical procedure to solve the maximum-weight cut problem (MCP) in undirected graphs. Our primary focus has been on problems arising from considerations in statistical mechanics. These problems are typically posed on grid graphs and some natural variants. There has been significant progress in two areas: solution of the maximum-weight cut problem and development of simplex-based tools for integer programming. Codes developed have been widely used to improve solution time.

The motivation for this work has been the need for a practical procedure to solve the maximum-weight cut problem (MCP) in undirected graphs. Our primary focus has been on problems arising from considerations in statistical mechanics. These problems are typically posed on grid graphs and some natural variants. There has been significant progress in two areas: solution of the maximum-weight cut problem and development of simplex-based tools for integer programming. Codes developed have been widely used to improve solution time.

A nonlinear programming technique was developed for the automated design and optimization of autopilots for large flexible launch vehicles. This technique, which resulted in the COEBRA program, uses the iterative application of linear programming. The method deals directly with the three main requirements of booster autopilot design: to provide (1) good response to guidance commands; (2) response to external disturbances (e.g. wind) to minimize structural bending moment loads and trajectory dispersions; and (3) stability with specified tolerances on the vehicle and flight control system parameters. The method is applicable to very high order systems (30th and greater per flight condition). Examples are provided that demonstrate the successful application of the employed algorithm to the design of autopilots for both single and multiple flight conditions.

An efficient method for computing solutions to the Optimal Transportation (OT) problem with a wide class of cost functions is presented. The standard linear programming (LP) discretization of the continuous problem becomes intractible for moderate grid sizes. A grid refinement method results in a linear cost algorithm. Weak convergence of solutions is stablished. Barycentric projection of transference plans is used to improve the accuracy of solutions. The method is applied to more general problems, including partial optimal transportation, and barycenter problems. Computational examples validate the accuracy and efficiency of the method. Optimal maps between nonconvex domains, partial OT free boundaries, and high accuracy barycenters are presented.

An isomorphism between two graphs is a bijection between their vertices that preserves the edges. We consider the problem of determining whether two finite undirected weighted graphs are isomorphic, and finding an isomorphism relating them if the answer is positive. In this paper we introduce effective probabilistic linear programming (LP) heuristics to solve the graph isomorphism problem. We motivate our heuristics by showing guarantees under some conditions, and present numerical experiments that show effectiveness of these heuristics in the general case.

In this paper, the linear programming (LP) decoder for binary linear codes, introduced by Feldman, et al. is extended to joint-decoding of binary-input finite-state channels. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the LP joint decoder also has the maximum-likelihood (ML) certificate property and all integer valued solutions are codewords. In this case, the performance loss relative to ML decoding can be explained completely by fractional valued JD-PCWs.

In this paper, the linear programming (LP) decoder for binary linear codes, introduced by Feldman, et al. is extended to joint-decoding of binary-input finite-state channels. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the LP joint decoder also has the maximum-likelihood (ML) certificate property and all integer valued solutions are codewords. In this case, the performance loss relative to ML decoding can be explained completely by fractional valued JD-PCWs.

ADA001247

We show how categorial deduction can be implemented in higher-order (linear) logic programming, thereby realising parsing as deduction for the associative and non-associative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions.

An introduction to Dantzig's simplex method of finding optimal solutions.

We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. For example, we prove that if $H_1$ is a finitely generated factor-free noncyclic subgroup of the free product $G_1 * G_2$ of two finite groups $G_1$, $G_2$, then the Walter Neumann coefficient $\sigma(H_1)$ of $H_1$ is rational and can be computed. This coefficient $\sigma(H_1)$ is the minimal positive real number such that, for every finitely generated factor-free subgroup $H_2$ of $G_1 * G_2$, it is true that $\bar {\rm r}(H_1, H_2) \le \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar {\rm r} (H) = \max ( {\rm r} (H)-1,0)$ is reduced rank of $H$, ${\rm r}(H)$ is rank of $H$, and $\bar {\rm r}(H_1, H_2)$ is reduced rank of a generalized intersection of $H_1, H_2$.

We study the intersection of finitely generated factor-free subgroups of free products of groups by utilizing the method of linear programming. For example, we prove that if $H_1$ is a finitely generated factor-free noncyclic subgroup of the free product $G_1 * G_2$ of two finite groups $G_1$, $G_2$, then the Walter Neumann coefficient $\sigma(H_1)$ of $H_1$ is rational and can be computed. This coefficient $\sigma(H_1)$ is the minimal positive real number such that, for every finitely generated factor-free subgroup $H_2$ of $G_1 * G_2$, it is true that $\bar {\rm r}(H_1, H_2) \le \sigma(H_1) \bar {\rm r}(H_1) \bar {\rm r}(H_2)$, where $\bar {\rm r} (H) = \max ( {\rm r} (H)-1,0)$ is reduced rank of $H$, ${\rm r}(H)$ is rank of $H$, and $\bar {\rm r}(H_1, H_2)$ is reduced rank of a generalized intersection of $H_1, H_2$.

The constraint on the $R$-parity violating supersymmetric interactions is discussed in the light of current experimental data of the electric dipole moment of neutron, $^{129}$Xe , $^{205}$Tl, and $^{199}$Hg atoms, and YbF and ThO molecules. To investigate the constraints without relying upon the assumption of the dominance of a particular combination of couplings over all the rest, an extensive use is made of the linear programming method in the scan of the parameter space. We give maximally possible values for the EDMs of the proton, deuteron, $^3$He nucleus, $^{211}$Rn, $^{225}$Ra, $^{210}$Fr, and the $R$-correlation of the neutron beta decay within the constraints from the current experimental data of the EDMs of neutron, $^{129}$Xe, $^{205}$Tl, and $^{199}$Hg atoms, and YbF and ThO molecules using the linear programming method. It is found that the $R$-correlation of the neutron beta decay and hadronic EDMs are very useful observables to constrain definite regions of the parameter space of the $R$-parity violating supersymmetry.

An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author)

An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author)

An algorithm is described for determining the optimal solution of parametric linear and quadratic programming problems as an explicit piecewise linear function of the parameter. Each linear function is uniquely determined by an appropriate subset of active constraints. For every critical value of the parameter a new subset has to be determined. A simple rule is given for adding and deleting constraints from this subset. (Author)

The main purpose of this work is to give explicit sparsity-preserving SOR(Successive Overrelaxation) algorithms for the solution of separable quadratic and linear programming problems. The principal and computationally distinguishing feature of the present SOR algorithms is that they preserve the sparsity structure of the problem and do not require the computation of the product of the constraint matrix by its transpose as is the case in earlier SOR algorithms for linear and quadratic programming. (Author)

The performance of two different approaches to solving the non-linear control allocation problem is presented. The non-linear control allocation problem is formulated using piecewise linear functions to approximate the control moments produced by a set of control effectors. when the control allocation problem is formulated as a piecewise linear program, an additional set of constraints enter into the problem formation. One approach is to introduce a set of binary variables to enforce these constraints. The result is a mixed- integer linear programming problem that can be solved using any branch-and-bound software. A second approach is to solve the piecewise linear programming problem using a modified simplex method. The simplex algorithm is modified to enforce a subset of the decision variables to enter into the basis only if certain conditions are met. We will show that solving the optimization problem using the simplex based approach is significantly faster than solving the same problem using a mixed-integer formulation. We will then compare the closed-loop performance of a re-entry vehicle using both approaches.

An algorithm to solve linear programming problems is presented in this thesis which is based on Karmarkar's projective method. The algorithm includes a practical method to project a general linear programming problem onto a unit simplex and eliminates the a priori need to know the optimal value of the objective function. The implementation conserves sparsity. The key part of the implementation is the solution of a linear least-squares problem to find an improving direction: a direct and an iterative method are implemented to solve this problem. The direct method employs the minimum-degree heuristic to reorder the system of normal equations, and thus conserve sparsity during the following Cholesky factor of the normal equation matrix as a preconditioner for conjugate gradient iterations which are performed implicity on the preconditioned matrix. The study concludes with implementation remarks, and computational results.

Since 1988, the United States Army has closed 112 and has completed or will soon complete realignment of another 27 of its domestic installations. The Army estimates the total cost (between 1988 and 2001) of these closures and realignments to be $5.3 billion, of which about $2.3 billion (43%) is associated with environmental cleanup. Beyond 2001, the Army expects to spend an additional $ 1.09 billion to complete cleanup and continue restoration. The Army Base Realignment and Closure Office (BRACO) is currently funding environmental cleanup at 649 sites on 39 current and former Army installations. BRACO' 5 environmental restoration budget from 2001 to 2007 to support cleanup at these installations (totaling over $620 million) is not sufficient to support each installation's requirement for those years. Considering environmental policies and yearly funding requests from 2001 to 2015 for each site, this thesis develops optimization models and a spreadsheet interface to help BRACO allocate its budget. Model results prescribe either funding each site as requested or delaying cleanup by one to five years. Extensive model use helped BRACO analyze alternate yearly budgets, suggest alternate site funding and determine site funding for 2001 to 2007.

This report examines the role played by several kinds of ceiling points in solving the pure, general integer linear programming problem (ILP). While no assumptions are made concerning the structure or signs of the data of the problem, it is assumed that the feasible region for (ILP) is non-empty and bounded. A ceiling point with respect to a single constraint maybe thought of as an integer solution on or close to the boundary of the feasible region defined by the constraint. The definition of a ceiling point with respect to a single constraint is extended to take multiple constraints into consideration simultaneously, defining what is called a feasible ceiling point. It is shown that the set all feasible ceiling points contains at least one optimal solution for (ILP). A related class of solutions called feasible 1-ceiling points is also characterized and shown to contain all optimal solutions for (ILP). Moreover, 1- ceiling points are computationally easier to identify than ordinary ceiling points and may be sought with respect to one constant at a time. It is also demonstrated that solving (ILP) requires only enumerating feasible 1-ceiling points with respect to a subset of all functional constraints. Keywords: Integer variables; Enumeration algorithms.

Considering the intermittency of renewable energy systems, a sizing and scheduling model is proposed for a finite number of static electric loads. The model objective is to maximize solar energy utilization with and without storage. For the application of optimal load size selection, the energy production of a solar photovoltaic is assumed to be consumed by a finite number of discrete loads in an off-grid system using mixed-integer linear programming. Additional constraints are battery charge and discharge limitations and minimum uptime and downtime for each unit. For a certain solar power profile the model outputs optimal unit size as well as the optimal scheduling for both units and battery charge and discharge (if applicable). The impact of different solar power profiles and minimum up and down time constraints on the optimal unit and battery sizes are studied. The battery size required to achieve full solar energy utilization decreases with the number of units and with increased flexibility of the units (shorter on and off-time). A novel formulation is introduced to model quasi-dynamic units that gradually start and stop and the quasi-dynamic units increase solar energy utilization. The model can also be applied to search for the optimal number of units for a given cost function.

We consider the problem of the computation of $\inf_p \theta p$ over the set of exponent pairs $P \ni p$ under linear constraints for a certain class of objective functions $\theta$. An effective algorithm is presented. The output of the algorithm leads to the improvement and establishing new estimates in the various divisor problems in the analytic number theory.

In the paper we consider the linear underdetermined system of a special type. Systems of this type appear in non-homogeneous network flow programming problems in the form of systems of constraints and can be characterized as systems with a large sparse submatrix representing the embedded network structure. We develop a direct method for finding solutions of the system. The algorithm is based on the theoretic-graph specificities for the structure of the support and properties of the basis of a solution space of a homogeneous system. One of the key steps is decomposition of the system. A simple example is regarded at the end of the paper.

We introduce a linear programming method to obtain bounds on the cardinality of codes in Grassmannian spaces for the chordal distance. We obtain explicit bounds, and an asymptotic bound that improves on the Hamming bound. Our approach generalizes the approach originally developed by P. Delsarte and Kabatianski-Levenshtein for compact two-point homogeneous spaces.

We consider the problem of tracking a target whose dynamics is modeled by a continuous It\=o semi-martingale. The aim is to minimize both deviation from the target and tracking efforts. We establish the existence of asymptotic lower bounds for this problem, depending on the cost structure. These lower bounds can be related to the time-average control of Brownian motion, which is characterized as a deterministic linear programming problem. A comprehensive list of examples with explicit expressions for the lower bounds is provided.

We introduce a new approach for the numerical pricing of American options. The main idea is to choose a finite number of suitable excessive functions (randomly) and to find the smallest majorant of the gain function in the span of these functions. The resulting problem is a linear semi-infinite programming problem, that can be solved using standard algorithms. This leads to good upper bounds for the original problem. For our algorithms no discretization of space and time and no simulation is necessary. Furthermore it is applicable even for high-dimensional problems. The algorithm provides an approximation of the value not only for one starting point, but for the complete value function on the continuation set, so that the optimal exercise region and e.g. the Greeks can be calculated. We apply the algorithm to (one- and) multidimensional diffusions and to L\'evy processes, and show it to be fast and accurate.

We examine inhomogeneous primordial nucleosynthesis for {\it arbitrary} distributions $f$ of the baryon-to-photon ratio $\eta$, in the limit where neither particle diffusion nor gravitational collapse is important. By discretizing $f(\eta)$ and using linear programming, we show that for a set of $m$ constraints on the primordial element abundances, the maximum and minimum possible values of $\bar \eta$ (the final mean value of $\eta$) are given when $f(\eta)$ is a sum of at most $m+1$ delta functions. Our linear programming results indicate that when $f$ is taken to be an arbitrary function, there is no lower bound on $\bar \eta$, while the upper bound is essentially the homogeneous upper bound.

This work deals with a class of problems under interval data uncertainty, namely interval robust-hard problems, composed of interval data min-max regret generalizations of classical NP-hard combinatorial problems modeled as 0-1 integer linear programming problems. These problems are more challenging than other interval data min-max regret problems, as solely computing the cost of any feasible solution requires solving an instance of an NP-hard problem. The state-of-the-art exact algorithms in the literature are based on the generation of a possibly exponential number of cuts. As each cut separation involves the resolution of an NP-hard classical optimization problem, the size of the instances that can be solved efficiently is relatively small. To smooth this issue, we present a modeling technique for interval robust-hard problems in the context of a heuristic framework. The heuristic obtains feasible solutions by exploring dual information of a linearly relaxed model associated with the classical optimization problem counterpart. Computational experiments for interval data min-max regret versions of the restricted shortest path problem and the set covering problem show that our heuristic is able to find optimal or near-optimal solutions and also improves the primal bounds obtained by a state-of-the-art exact algorithm and a 2-approximation procedure for interval data min-max regret problems.

The operation of an elaborate set of computer codes raises problems of its own which can only be appreciated from experience. Nevertheless, certain general principles for designing such a system can be set forth. The activities which are engaged in during the evolution of such a system are not simple sequence of events but there is feedback from later steps to earlier ones. How easily the resulting changes can be handled is dependent on the organization of the codes and on the assembly program used. An appendix discusses some shortcomings of the latter together with suggested improvements.

The operation of an elaborate set of computer codes raises problems of its own which can only be appreciated from experience. Nevertheless, certain general principles for designing such a system can be set forth. The activities which are engaged in during the evolution of such a system are not simple sequence of events but there is feedback from later steps to earlier ones. How easily the resulting changes can be handled is dependent on the organization of the codes and on the assembly program used. An appendix discusses some shortcomings of the latter together with suggested improvements.

In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef et al., FOCS, 2006). We show a polynomial time algorithm that, given an n vertex graph G with minrank k, finds a linear index code for G of length $\widetilde{O}(n^{f(k)})$, where f(k) depends only on k. For example, for k=3 we obtain f(3) ~ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovasz theta-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef et al., FOCS, 2006). We show a polynomial time algorithm that, given an n vertex graph G with minrank k, finds a linear index code for G of length $\widetilde{O}(n^{f(k)})$, where f(k) depends only on k. For example, for k=3 we obtain f(3) ~ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovasz theta-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

In the index coding problem, introduced by Birk and Kol (INFOCOM, 1998), the goal is to broadcast an n bit word to n receivers (one bit per receiver), where the receivers have side information represented by a graph G. The objective is to minimize the length of a codeword sent to all receivers which allows each receiver to learn its bit. For linear index coding, the minimum possible length is known to be equal to a graph parameter called minrank (Bar-Yossef et al., FOCS, 2006). We show a polynomial time algorithm that, given an n vertex graph G with minrank k, finds a linear index code for G of length $\widetilde{O}(n^{f(k)})$, where f(k) depends only on k. For example, for k=3 we obtain f(3) ~ 0.2574. Our algorithm employs a semidefinite program (SDP) introduced by Karger, Motwani and Sudan (J. ACM, 1998) for graph coloring and its refined analysis due to Arora, Chlamtac and Charikar (STOC, 2006). Since the SDP we use is not a relaxation of the minimization problem we consider, a crucial component of our analysis is an upper bound on the objective value of the SDP in terms of the minrank. At the heart of our analysis lies a combinatorial result which may be of independent interest. Namely, we show an exact expression for the maximum possible value of the Lovasz theta-function of a graph with minrank k. This yields a tight gap between two classical upper bounds on the Shannon capacity of a graph.

Scan of book Linear programming and applications, a course text by Will McLewin [1980] {519.72--oclc} Content material:- Conversion to specified form: Basic, Feasible and Optimum solutions The Simplex Method Duality Parametric linear programming and sensitivity analysis The Shor-Khachian ellipsoid method Transportation and similar problems Network flows The marriage problem Games theory: two person matrix games Quadratic programming Functional aproximation, Matrix eigenvalue perturbation analysis

It is shown that any communication system which admits a sum-product (SP) receiver also admits a corresponding linear-programming (LP) receiver. The two receivers have a relationship defined by the local structure of the underlying graphical model, and are inhibited by the same phenomenon, which we call 'pseudoconfigurations'. This concept is a generalization of the concept of 'pseudocodewords' for linear codes. It is proved that the LP receiver has the 'optimum certificate' property, and that the receiver output is the lowest cost pseudoconfiguration. Equivalence of graph-cover pseudoconfigurations and linear-programming pseudoconfigurations is also proved. While the LP receiver is generally more complex than the corresponding SP receiver, the LP receiver and its associated pseudoconfiguration structure provide an analytic tool for the analysis of SP receivers. As an example application, we show how the LP design technique may be applied to the problem of joint equalization and decoding.

We consider the stability analysis of a large class of linear 1-D PDEs with polynomial data. This class of PDEs contains, as examples, parabolic and hyperbolic PDEs, PDEs with boundary feedback and systems of in-domain/boundary coupled PDEs. Our approach is Lyapunov based which allows us to reduce the stability problem to the verification of integral inequalities on the subspaces of Hilbert spaces. Then, using fundamental theorem of calculus and Green's theorem, we construct a polynomial problem to verify the integral inequalities. Constraining the solution of the polynomial problem to belong to the set of sum-of-squares polynomials subject to affine constraints allows us to use semi-definite programming to algorithmically construct Lyapunov certificates of stability for the systems under consideration. We also provide numerical results of the application of the proposed method on different types of PDEs.