Robust network optimization under polyhedral demand uncertainty is NP-hard

AbstractMinimum cost network design/dimensioning problems where feasibility has to be ensured w.r.t. a given (possibly infinite) set of scenarios of requirements form an important subclass of robust LP problems with right-hand side uncertainty. Such problems arise in many practical contexts such as Telecommunications, logistic networks, power distribution networks, etc. Though some evidence of the computational difficulty of such problems can be found in the literature, no formal NP-hardness proof was available up to now. In the present paper, this pending complexity issue is settled for all robust network optimization problems featuring polyhedral demand uncertainty, both for the single-commodity and multicommodity case, even if the corresponding deterministic versions are polynomially solvable as regular (continuous) linear programs. A new family of polynomially solvable instances is also discussed

An Approach to Computational Complexity in Membrane Computing

In this paper we present a theory of computational complexity in the framework of membrane computing. Polynomial complexity classes in recognizer membrane systems and capturing the classical deterministic and non-deterministic modes of computation, are introduced. In this context, a characterization of the relation P = NP is described.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

Optimal Recombination in Genetic Algorithms

This paper surveys results on complexity of the optimal recombination problem (ORP), which consists in finding the best possible offspring as a result of a recombination operator in a genetic algorithm, given two parent solutions. We consider efficient reductions of the ORPs, allowing to establish polynomial solvability or NP-hardness of the ORPs, as well as direct proofs of hardness results

New complexity results and algorithms for min-max-min robust combinatorial optimization

In this work we investigate the min-max-min robust optimization problem applied to combinatorial problems with uncertain cost-vectors which are contained in a convex uncertainty set. The idea of the approach is to calculate a set of k feasible solutions which are worst-case optimal if in each possible scenario the best of the k solutions would be implemented. It is known that the min-max-min robust problem can be solved efficiently if k is at least the dimension of the problem, while it is theoretically and computationally hard if k is small. While both cases are well studied in the literature nothing is known about the intermediate case, namely if k is smaller than but close to the dimension of the problem. We approach this open question and show that for a selection of combinatorial problems the min-max-min problem can be solved exactly and approximately in polynomial time if some problem specific values are fixed. Furthermore we approach a second open question and present the first implementable algorithm with oracle-pseudopolynomial runtime for the case that k is at least the dimension of the problem. The algorithm is based on a projected subgradient method where the projection problem is solved by the classical Frank-Wolfe algorithm. Additionally we derive a branch & bound method to solve the min-max-min problem for arbitrary values of k and perform tests on knapsack and shortest path instances. The experiments show that despite its theoretical impact the projected subgradient method cannot compete with an already existing method. On the other hand the performance of the branch & bound method scales very well with the number of solutions. Thus we are able to solve instances where k is above some small threshold very efficiently

A Linear-Time Solution to the Knapsack Problem Using P Systems with Active Membranes

Up to now, P systems dealing with numerical problems have been rarely considered in the literature. In this paper we present an effective solution to the Knapsack problem using a family of deterministic P systems with active membranes using 2-division. We show that the number of steps of any computation is of linear order, but polynomial time is required for pre-computing resources.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

Regret Models and Preprocessing Techniques for Combinatorial Optimization under Uncertainty

Ph.DDOCTOR OF PHILOSOPH

An Efficient Cellular Solution for the Partition Problem

Numerical problems are not very frequently addressed in the P sys- tems literature. In this paper we present an e®ective solution to the Partition problem via a family of deterministic P systems with active membranes using 2-division. The design of this solution is a sequel of several previous works on other problems, mainly the Subset-Sum and the Knapsack problems but also the VALIDITY and SAT. Several improvements are introduced and explained.Ministerio de Ciencia y Tecnología TIC2002-04220-C03-0

Computational efficiency of dissolution rules in membrane systems

Trading (in polynomial time) space for time in the framework of membrane systems is not sufficient to efficiently solve computationally hard problems. On the one hand, an exponential number of objects generated in polynomial time is not sufficient to solve NP-complete problems in polynomial time. On the other hand, when an exponential number of membranes is created and used as workspace, the situation is very different. Two operations in P systems (membrane division and membrane creation) capable of constructing an exponential number of membranes in linear time are studied in this paper. NP-complete problems can be solved in polynomial time using P systems with active membranes and with polarizations, but when electrical charges are not used, then dissolution rules turn out to be very important. We show that in the framework of P systems with active membranes but without polarizations and in the framework of P systems with membrane creation, dissolution rules play a crucial role from the computational efficiency point of view.Ministerio de Educación y Ciencia TIN2005-09345-C04-0