Certificate Of Optimality Research Articles

Weighted Sum-Rate Maximization for a Set of Interfering Links via Branch and Bound

We consider the problem of weighted sum-rate maximization (WSRMax) for a set of interfering links. It plays a central role in resource allocation, link scheduling or in finding achievable rate regions for both wireline and wireless networks. This problem is known to be NP-hard. We propose a solution method, based on the branch and bound technique, which solves globally the nonconvex WSRMax problem with an optimality certificate. Efficient analytic bounding techniques are introduced and their impact on the convergence is numerically evaluated. The considered link-interference model is general enough to model a wide range of network topologies with various node capabilities, e.g., single- or multipacket transmission (or reception), simultaneous transmission and reception. Several applications, including cross-layer network utility maximization and maximum weighted link scheduling for multihop wireless networks as well as finding achievable rate regions for singlecast/multicast wireless networks, are presented. The proposed algorithm can be further used to provide other performance benchmarks by back-substituting it into any network design method which relies on WSRMax. It is also very useful for evaluating the performance loss encountered by any heuristic algorithm.

A Branch-and-Bound Algorithm for Solving the Multiprocessor Scheduling Problem with Improved Lower Bounding Techniques

In branch-and-bound (B&B) schemes for solving a minimization problem, a better lower bound could prune many meaningless branches which do not lead to an optimum solution. In this paper, we propose several techniques to refine the lower bound on the makespan in the multiprocessor scheduling problem (MSP). The key idea of our proposed method is to combine an efficient quadratic-time algorithm for calculating the Fernandez's bound, which is known as the best lower bounding technique proposed in the literature with two improvements based on the notions of binary search and recursion. The proposed method was implemented as a part of a B&B algorithm for solving MSP, and was evaluated experimentally. The result of experiments indicates that the proposed method certainly improves the performance of the underlying B&B scheme. In particular, we found that it improves solutions generated by conventional heuristic schemes for more than 20 percent of randomly generated instances, and for more than 80 percent of instances, it could provide a certification of optimality of the resulting solutions, even when the execution time of the B&B scheme is limited by one minute.

An integer programming approach to DNA sequence assembly

De novo sequence assembly is a ubiquitous combinatorial problem in all DNA sequencing technologies. In the presence of errors in the experimental data, the assembly problem is computationally challenging, and its solution may not lead to a unique reconstruct. The enumeration of all alternative solutions is important in drawing a reliable conclusion on the target sequence, and is often overlooked in the heuristic approaches that are currently available. In this paper, we develop an integer programming formulation and global optimization solution strategy to solve the sequence assembly problem with errors in the data. We also propose an efficient technique to identify all alternative reconstructs. When applied to examples of sequencing-by-hybridization, our approach dramatically increases the length of DNA sequences that can be handled with global optimality certificate to over 10,000, which is more than 10 times longer than previously reported. For some problem instances, alternative solutions exhibited a wide range of different ability in reproducing the target DNA sequence. Therefore, it is important to utilize the methodology proposed in this paper in order to obtain all alternative solutions to reliably infer the true reconstruct. These alternative solutions can be used to refine the obtained results and guide the design of further experiments to correctly reconstruct the target DNA sequence.

A universally fastest algorithm for Max 2-Sat, Max 2-CSP, and everything in between

In this paper we introduce “hybrid” Max 2-CSP formulas consisting of “simple clauses”, namely conjunctions and disjunctions of pairs of variables, and general 2-variable clauses, which can be any integer-valued functions of pairs of boolean variables. This allows an algorithm to use both efficient reductions specific to AND and OR clauses, and other powerful reductions that require the general CSP setting. We use new reductions introduced here, and recent reductions such as “clause-learning” and “2-reductions” generalized to our settingʼs mixture of simple and general clauses. We parametrize a hybrid instance by the fraction p of non-simple clauses. We give an exact, exponential-time but polynomial-space algorithm that is the fastest known for p = 0 , which includes the well-studied Max 2-Sat problem but also instances with arbitrary mixtures of AND and OR clauses; for an m-clause instance it runs in time O ⋆ ( 2 m / 6.321 ) . The same algorithm is tied for fastest for general Max 2-CSP ( p = 1 ), with running time O ⋆ ( 2 m / 5.263 ) . The algorithm is the only one to treat mixtures of AND, OR, and general integer-valued clauses more efficiently than the general case, with intermediate running time bounds depending on the value of p. Since even a pure Max 2-Sat input instance may be transformed to a hybrid instance in the course of solving it, the algorithmʼs efficiency and generality go hand in hand. Our algorithm analysis and optimization use the familiar measure-and-conquer approach, but in a variation resulting in mathematical programs that are convex rather than quasi-convex, and can be solved efficiently and with a certificate of optimality. We produce a family of running-time upper-bound formulas, each optimized for instances with a particular value of p but valid for all instances.

Certificates of optimality: the third way to biproportional apportionment

The biproportional apportionment problem (BAP) must be faced in many proportional electoral systems where seats must be allocated to parties within regions. BAP is a non-trivial optimization problem, and only sophisticated algorithms are currently available for solving it. The issue is: are they “writable” as an actual law? Citizens rightly demand simple, easy to understand, voting systems. The alternative, though, seems to have simple, but unsound electoral laws. We propose the following way out of this dilemma: leave to a mathematically sophisticated algorithm the task of producing an optimal apportionment, but attach to it a “certificate of optimality”, that is, describe a simple procedure whereby anybody can check, through some elementary operations, that the seat allocation output by the algorithm is indeed an optimal apportionment. We discuss one such certificate, based on the Max flow- min cut Theorem, relative to a parametric max flow method of ours for BAP.

Certificate revocation release policies

Public key infrastructure provides a promising foundation for verifying the authenticity of communicating parties and transferring trust over the Internet. The key issue in public key infrastructure is how to process certificate revocations. Previous research in this area has concentrated on the tr adeoffs that can be made among different revocation options. No rigorous efforts have been made to understand the probability distribution of certificate revocation requests based on real empirical data. In this study, we first collect real data from VeriSign and suggest a functional form for the probability density function of certificate revocation requests. Exponential distribution function is chosen as it adequately approximates the real data. We then provide an economic model based on which a certificate authority can choose the optimal Certificate Revocation List (CRL) release interval considering the intrinsic properties among different types of certificate services. To conclude we draw some insights by comparing the performance of four different CRL strategies.

The [formula omitted]-Cardinality Tree Problem: Reformulations and Lagrangian Relaxation

Given an undirected defining graph for the k -Cardinality Tree Problem (KCTP), an associated directed graph involving two additional vertices is introduced in this paper and gives rise to two compact reformulations of the problem. For the first one, connectivity of feasible solutions is enforced through multicommodity flows while, for the other, lifted Miller–Tucker–Zemlin constraints are used. Comparing the two reformulations, much stronger Linear Programming relaxation bounds are obtained from the first one, albeit at much higher CPU times. However, a Branch-and-Bound algorithm based on the second reformulation proved much more effective and managed to obtain, for the first time, optimality certificates for a large number of KCTP instances from the literature. Additionally, for some instances where optimality could not be proven within the given pre-specified CPU time limit, new best upper bounds were generated. Finally, a Lagrangian heuristic based on the first reformulation was also implemented and proved capable of generating feasible KCTP solutions comparable in quality with the best overall results obtained by metaheuristic based heuristics found in the literature. For our test cases, Lagrangian upper bounds are no more than 3.8% away from the best upper bounds known. Additionally, several new best upper bounds and optimality certificates were obtained by the heuristic. Corresponding Lagrangian heuristic CPU times, however, are typically higher than those associated with their competitors.

On the feedback vertex set polytope of a series–parallel graph

The minimum weight feedback vertex set problem (FVS) on series–parallel graphs can be solved in O ( n ) time by dynamic programming. This solution, however, does not provide a “nice” certificate of optimality. We prove a min–max relation for FVS on series–parallel graphs with no induced subdivision of K 2 , 3 (a class of graphs containing the outerplanar graphs), thereby establishing the existence of nice certificates for these graphs. Our proof relies on the description of a complete set of inequalities defining the feedback vertex set polytope of a series–parallel graph with no induced subdivision of K 2 , 3 . We also prove that many of the inequalities described are facets of this polytope.

A heuristic approach based on dynamic programming and and/or-graph search for the constrained two-dimensional guillotine cutting problem

In this paper we present a heuristic method to generate constrained two-dimensional guillotine cutting patterns. This problem appears in different industrial processes of cutting rectangular plates to produce ordered items, such as in the glass, furniture and circuit board business. The method uses a state space relaxation of a dynamic programming formulation of the problem and a state space ascent procedure of subgradient optimization type. We propose the combination of this existing approach with an and/or-graph search and an inner heuristic that turns infeasible solutions provided in each step of the ascent procedure into feasible solutions. Results for benchmark and randomly generated instances indicate that the method’s performance is competitive compared to other methods proposed in the literature. One of its advantages is that it often produces a relatively tight upper bound to the optimal value. Moreover, in most cases for which an optimal solution is obtained, it also provides a certificate of optimality.

A spectral heuristic for bisecting random graphs

AbstractThe minimum bisection problem is to partition the vertices of a graph into two classes of equal size so as to minimize the number of crossing edges. Computing a minimum bisection is NP‐hard in the worst case. In this paper we study a spectral heuristic for bisecting random graphs Gn(p,p′) with a planted bisection obtained as follows: partition n vertices into two classes of equal size randomly, and then insert edges inside the two classes with probability p′ and edges crossing the partition with probability p independently. If $n(p'-p)\geq c_0\sqrt{np'\ln(np')}$, where c0 is a suitable constant, then with probability 1 − o(1) the heuristic finds a minimum bisection of Gn(p,p′) along with a certificate of optimality. Furthermore, we show that the structure of the set of all minimum bisections of Gn(p,p′) undergoes a phase transition as $n(p'-p)=\Theta(\sqrt{np'\ln n})$. The spectral heuristic solves instances in the subcritical, the critical, and the supercritical phases of the phase transition optimally with probability 1 − o(1). These results extend previous work of Boppana Proc. 28th FOCS (1987) 280–285. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006

An Optimal Certificate Dispersal Algorithm for Mobile Ad Hoc Networks

In this paper, we focus on the problem that in an ad hoc network, how to send a message securely between two users using the certificate dispersal system. In this system, special data called certificate is issued between two users and these issued certificates are stored among the network. Our final purpose on this certificate dispersal problem is to construct certificate graphs with lower dispersability cost which indicates the average number of certificates stored in each node in an ad hoc network. As our first step, when a certificate graph is given, we construct two efficient certificate dispersal algorithms for strongly connected graphs and directed graphs in this paper. We can show that for a strongly connected graph G = (V, E) and a directed graph H = (V' E'), new upper bounds on dispersability cost on the average number of certificates stored in one node are O(DG + |E'|/|V|) and O(pGdmax + |E'|/|V'|) respectively, where DG is the diameter of G, dmax is the maximum diameter of strongly connected components of H and pG is the number of strongly connected components of H. Furthermore, we give some new lower bounds for the problem and we also show that our algorithms are optimal for several graph classes.

GLOBAL OPTIMIZATION OF LINEAR HYBRID SYSTEMS WITH VARYING TIME EVENTS

Dynamic optimization problems with linear hybrid (discrete/continuous) systems embedded whose transition times vary are inherently nonconvex. For a wide variety of applications, a certificate of global optimality is essential, but this cannot be obtained using conventional numerical methods. We present a deterministic framework for the solution of such problems in the continuous time domain. First, the control parametrization enhancing transform is used to transform the embedded dynamic system from a linear hybrid system with scaled discontinuities and varying transition times into a nonlinear hybrid system with stationary discontinuities and fixed transition times. Next, a recently developed convexity theory is applied to construct a convex relaxation of the original nonconvex problem. This allows the problem to be solved in a branch-and-bound framework that can guarantee the global solution within epsilon optimality in a finite number of iterations.

On Primal and Dual Infeasibility Certificates in a Homogeneous Model for Convex Optimization

Andersen and Ye [ Math. Programming, 84 (1999), pp. 375--399] suggested a homogeneous formulation and an interior-point algorithm for solution of the monotone complementarity problem (MCP). The advantage of the homogeneous formulation is that it always has a solution. Moreover, in the case in which the MCP is solvable or is (strongly) infeasible, the solution provides a certificate of optimality or infeasibility. In this note we demonstrate that if the suggested formulation is applied to the Karush--Kuhn--Tucker optimality conditions corresponding to a convex optimization problem, then an infeasibility certificate provides information about whether the primal or dual problem is infeasible given certain assumptions.

Minimizing and Stationary Sequences of Constrained Optimization Problems

We study some fundamental asymptotic properties associated with the well-posedness of constrained optimization problems. Emphasis is placed on the relation between minimizing and stationary sequences and their characterizations in terms of a set of asymptotic Karush--Kuhn--Tucker (KKT) optimality conditions. Unlike the subdifferential approach used by Auslender, Cominetti, and Crouzeix, we use a residual function approach that is closely tied to the theory of error bounds; this approach handles constraints explicitly and allows the effective treatment of infeasible, stationary sequences. The asymptotic KKT conditions provide an asymptotic optimality certificate for inequality constrained programs. Specializations of the results to convex quadratically constrained convex quadratic spline minimization problems and convex programs with Hölderian minima are discussed. An application of the results to the convergence of a family of Newton-type iterative descent methods involving singular quasi-Newton matrices for solving a constrained minimization problem is also presented.

OR software - ORSEP operations research software exchange program Edited by Professor H.W. Hamacher K_TREE/K_SUBGRAPH: A program package for minimal weighted K-cardinality trees and subgraphs

Given an undirected defining graph for the k-Cardinality Tree Problem (KCTP), an associated directed graph involving two additional vertices is introduced in this paper and gives rise to two compact reformulations of the problem. For the first one, connectivity of feasible solutions is enforced through multicommodity flows while, for the other, lifted Miller–Tucker–Zemlin constraints are used. Comparing the two reformulations, much stronger Linear Programming relaxation bounds are obtained from the first one, albeit at much higher CPU times. However, a Branch-and-Bound algorithm based on the second reformulation proved much more effective and managed to obtain, for the first time, optimality certificates for a large number of KCTP instances from the literature. Additionally, for some instances where optimality could not be proven within the given pre-specified CPU time limit, new best upper bounds were generated. Finally, a Lagrangian heuristic based on the first reformulation was also implemented and proved capable of generating feasible KCTP solutions comparable in quality with the best overall results obtained by metaheuristic based heuristics found in the literature. For our test cases, Lagrangian upper bounds are no more than 3.8% away from the best upper bounds known. Additionally, several new best upper bounds and optimality certificates were obtained by the heuristic. Corresponding Lagrangian heuristic CPU times, however, are typically higher than those associated with their competitors.