An Algorithmic Theory of Integer Programming

We study the general integer programming problem where the number of variables $n$ is a variable part of the input. We consider two natural parameters of the constraint matrix $A$: its numeric measure $a$ and its sparsity measure $d$. We show that integer programming can be solved in time $g(a,d)\textrm{poly}(n,L)$, where $g$ is some computable function of the parameters $a$ and $d$, and $L$ is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by $a$ and $d$, and is solvable in polynomial time for every fixed $a$ and $d$. Our results also extend to nonlinear separable convex objective functions. Moreover, for linear objectives, we derive a strongly-polynomial algorithm, that is, with running time $g(a,d)\textrm{poly}(n)$, independent of the rest of the input data. We obtain these results by developing an algorithmic framework based on the idea of iterative augmentation: starting from an initial feasible solution, we show how to quickly find augmenting steps which rapidly converge to an optimum. A central notion in this framework is the Graver basis of the matrix $A$, which constitutes a set of fundamental augmenting steps. The iterative augmentation idea is then enhanced via the use of other techniques such as new and improved bounds on the Graver basis, rapid solution of integer programs with bounded variables, proximity theorems and a new proximity-scaling algorithm, the notion of a reduced objective function, and others. As a consequence of our work, we advance the state of the art of solving block-structured integer programs. In particular, we develop near-linear time algorithms for $n$-fold, tree-fold, and $2$-stage stochastic integer programs. We also discuss some of the many applications of these classes.Comment: Revision 2: - strengthened dual treedepth lower bound - simplified proximity-scaling algorith

Routing in multi-class queueing networks

PhD ThesisWe consider the problem of routing (incorporating local scheduling) in a distributed network. Dedicated jobs arrive directly at their specified station for processing. The choice of station for generic jobs is open. Each job class has an associated holding cost rate. We aim to develop routing policies to minimise the long-run average holding cost rate. We first consider the class of static policies. Dacre, Glazebrook and Nifio-Mora (1999) developed an approach to the formulation of static routing policies, in which the work at each station is scheduled optimally, using the achievable region approach. The achievable region approach attempts to solve stochastic optimisation problems by characterising the space of all possible performances and optimising the performance objective over this space. Optimal local scheduling takes the form of a priority policy. Such static routing policies distribute the generic traffic to the stations via a simple Bernoulli routing mechanism. We provide an overview of the achievements made in following this approach to static routing. In the course of this discussion we expand upon the study of Becker et al. (2000) in which they considered routing to a collection of stations specialised in processing certain job classes and we consider how the composition of the available stations affects the system performance for this particular problem. We conclude our examination of static routing policies with an investigation into a network design problem in which the number of stations available for processing remains to be determined. The second class of policies of interest is the class of dynamic policies. General DP theory asserts the existence of a deterministic, stationary and Markov optimal dynamic policy. However, a full DP solution may be unobtainable and theoretical difficulties posed by simple routing problems suggest that a closed form optimal policy may not be available. This motivates a requirement for good heuristic policies. We consider two approaches to the development of dynamic routing heuristics. We develop an idea proposed, in the context of simple single class systems, by Krishnan (1987) by applying a single policy improvement step to some given static policy. The resulting dynamic policy is shown to be of simple structure and easily computable. We include an investigation into the comparative performance of the dynamic policy with a number of competitor policies and of the performance of the heuristic as the number of stations in the network changes. In our second approach the generic traffic may only access processing when the station has been cleared of all (higher priority) jobs and can be considered as background work. We deploy a prescription of Whittle (1988) developed for RBPs to develop a suitable approach to station indexation. Taking an approximative approach to Whittle's proposal results in a very simple form of index policy for routing the generic traffic. We investigate the closeness to optimality of the index policy and compare the performance of both of the dynamic routing policies developed here

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

A NOTE ON HARDNESS OF MULTIPROCESSOR SCHEDULING WITH SCHEDULING SOLUTION SPACE TREE

We study the computational complexity of the non-preemptive scheduling problem of a listof independent jobs on a set of identical parallel processors with a makespan minimizationobjective. We make a maiden attempt to explore the combinatorial structure showing theexhaustive solution space of the problem by defining the Scheduling Solution Space Tree(SSST) data structure. The properties of the SSST are formally defined and characterizedthrough our analytical results. We develop a unique technique to show the problemNP using the SSST and the Weighted Scheduling Solution Space Tree (WSSST) datastructures. We design the first non-deterministic polynomial-time algorithm named MagicScheduling (MS) for the problem based on the reduction framework. We also define anew variant of multiprocessor scheduling by including the user as an additional inputparameter. We formally establish the complexity class of the variant by the reductionprinciple. Finally, we conclude the article by exploring several interesting open problemsfor future research investigation