Broadcast and network optimization : essays on approximation algorithms and experiments on problems in latency cover, network fragmentation and routing on the internet

Ordering search results, network stability and routing on the Internet are applications of which the performance almost entirely depends of automated solution methods. This dissertation studies fundamental aspects of solution methods for these problems. The simplest solution methods that will always find the best solution is trying each possible solution one by one and selecting the best one. Even on the fastest computer this method takes too much time: years instead of days! Finding a solution method that is both fast and finds the best answer, appears unfortunately not always possible. This dissertation describes solution methods that are very fast, but find a very good answer instead of the best answer. Moreover, for various problems better and simpler solution methods are described

A branch and price procedure for the container premarshalling problem

During the loading phase of a vessel, only the containers that are on top of their stack are directly accessible. If the container that needs to be loaded next is not the top container, extra moves have to be performed, resulting in an increased loading time. One way to resolve this issue is via a procedure called premarshalling. The goal of premarshalling is to reshuffle the containers into a desired lay-out prior to the arrival of the vessel, in the minimum number of moves possible. This paper presents an exact algorithm based on branch and bound, that is evaluated on a large set of instances. The complexity of the premarshalling problem is also considered, and this paper shows that the problem at hand is NP-hard, even in the natural case of stacks with fixed height

Internet routing between autonomous systems: Fast algorithms for path trading

Routing traffic on the internet efficiently has become an important research topic over the past decade. In this article we consider a generalization of the shortest path problem, the path-trading problem, which has applications in inter-domain traffic routing. When traffic is forwarded between autonomous systems (ASes), such as competing internet providers, each AS selfishly routes the traffic inside its own network. Efficient solutions to the path trading problem can lead to higher global performance in such systems, while maintaining the objectives and costs of the individual ASes. First, we extend a previous hardness result for the path trading problem. Moreover, we provide an algorithm that finds all Pareto-optimal path trades for a pair of two ASes. While in principal the number of Pareto-optimal path trades can be exponential, in our experiments this number was typically small. We use the framework of smoothed analysis to give a theoretical explanation for that fact. The computational results show that our algorithm yields far superior running times and can solve considerably larger instances than a previously known algorithm

Path trading : fast algorithms, smoothed analysis, and hardness results

The Border Gateway Protocol (BGP) serves as the main routing protocol of the Internet and ensures network reachability among autonomous systems (ASes). When traffic is forwarded between the many ASes on the Internet according to that protocol, each AS selfishly routes the traffic inside its own network according to some internal protocol that supports the local objectives of the AS. We consider possibilities of achieving higher global performance in such systems while maintaining the objectives and costs of the individual ASes. In particular, we consider how path trading, i.e. deviations from routing the traffic using individually optimal protocols, can lead to a better global performance. Shavitt and Singer ("Limitations and Possibilities of Path Trading between Autonomous Systems", INFOCOM 2010) were the first to consider the computational complexity of finding such path trading solutions. They show that the problem is weakly NP-hard and provide a dynamic program to find path trades between pairs of ASes. In this paper we improve upon their results, both theoretically and practically. First, we show that finding path trades between sets of ASes is also strongly NP-hard. Moreover, we provide an algorithm that finds all Pareto-optimal path trades for a pair of two ASes. While in principal the number of Pareto-optimal path trades can be exponential, in our experiments this number was typically small. We use the framework of smoothed analysis to give theoretical evidence that this is a general phenomenon, and not only limited to the instances on which we performed experiments. The computational results show that our algorithm yields far superior running times and can solve considerably larger instances than the previous dynamic program

How to cut a graph into many pieces

In this paper we consider the problem of finding a graph separator of a given size that decomposes the graph into the maximum number of connected components. We present the picture of the computational complexity and the approximability of this problem for several natural classes of graphs. We first provide an overview of the hardness of approximation of this problem, which stems mainly from its close relation to the Independent Set and to the Maximum Clique problem. Next, we show that the problem is solvable in polynomial time for interval graphs and graphs of bounded treewidth. We also show that MaxiNum Components is fixed-parameter tractable on planar graphs with the size of the separator as the parameter. Our main contribution is the derivation of an efficient polynomial-time approximation scheme for the problem on planar graphs

Approximating vector scheduling: almost matching upper and lower bounds

We consider the vector scheduling problem, a natural generalization of the classical makespan minimization problem to multiple resources. Here, we are given $n$ jobs represented as $d$-dimensional vectors in $[0,1]^d$ and $m$ identical machines, and the goal is to assign the jobs to machines such that the maximum {\em load} of each machine over all the coordinates is at most $1$. For fixed $d$, the problem admits an approximation scheme, and the best known running time is $n^{f(\epsilon,d)}$ where $f(\epsilon,d) = (1/\epsilon)^{\tilde{O}(d)}$ ($\tilde{O}$ supresses polylogarithmic terms in $d$). In particular, the dependence on $d$ is doubly exponential. In this paper we show that a double exponential dependence on $d$ is necessary, and give an improved algorithm with essentially optimum running time. Specifically, we show that: \begin{itemize} \item For any $\epsilon0$. \item We complement these lower bounds with a $(1+\epsilon)$-approximation that runs in time $\exp((1/\epsilon)^{O(d \log \log d)}) + nd$. This gives the first efficient approximation scheme (EPTAS) for the problem. \end{itemize

Approximating vector scheduling: almost matching upper and lower bounds

We consider the vector scheduling problem, a natural generalization of the classical makespan minimization problem to multiple resources. Here, we are given $n$ jobs represented as $d$-dimensional vectors in $[0,1]^d$ and $m$ identical machines, and the goal is to assign the jobs to machines such that the maximum {\em load} of each machine over all the coordinates is at most $1$. For fixed $d$, the problem admits an approximation scheme, and the best known running time is $n^{f(\epsilon,d)}$ where $f(\epsilon,d) = (1/\epsilon)^{\tilde{O}(d)}$ ($\tilde{O}$ supresses polylogarithmic terms in $d$). In particular, the dependence on $d$ is doubly exponential. In this paper we show that a double exponential dependence on $d$ is necessary, and give an improved algorithm with essentially optimum running time. Specifically, we show that: \begin{itemize} \item For any $\epsilon0$. \item We complement these lower bounds with a $(1+\epsilon)$-approximation that runs in time $\exp((1/\epsilon)^{O(d \log \log d)}) + nd$. This gives the first efficient approximation scheme (EPTAS) for the problem. \end{itemize

Scheduling unit-length jobs with precedence constraints of small height

We consider the problem of scheduling unit-length jobs on identical machines subject to precedence constraints. We show that natural scheduling rules fail when the precedence constraints form a collection of stars or a collection of complete bipartite graphs. We prove that the problem is in fact NP-hard on collections of stars when the input is given in a compact encoding, whereas it can be solved in polynomial time with standard adjacency list encoding. On a subclass of collections of stars and on collections of complete bipartite graphs we show that the problem can be solved in polynomial time even when the input is given in compact encoding, in both cases via non-trivial algorithms. Keywords: Scheduling; Precedence constraints; Computational complexity; Polynomial-time algorithm