9 research outputs found
Buyback Problem - Approximate matroid intersection with cancellation costs
In the buyback problem, an algorithm observes a sequence of bids and must
decide whether to accept each bid at the moment it arrives, subject to some
constraints on the set of accepted bids. Decisions to reject bids are
irrevocable, whereas decisions to accept bids may be canceled at a cost that is
a fixed fraction of the bid value. Previous to our work, deterministic and
randomized algorithms were known when the constraint is a matroid constraint.
We extend this and give a deterministic algorithm for the case when the
constraint is an intersection of matroid constraints. We further prove a
matching lower bound on the competitive ratio for this problem and extend our
results to arbitrary downward closed set systems. This problem has applications
to banner advertisement, semi-streaming, routing, load balancing and other
problems where preemption or cancellation of previous allocations is allowed
Rigorous Runtime Analysis of MOEA/D for Solving Multi-Objective Minimum Weight Base Problems
We study the multi-objective minimum weight base problem, an abstraction of
classical NP-hard combinatorial problems such as the multi-objective minimum
spanning tree problem. We prove some important properties of the convex hull of
the non-dominated front, such as its approximation quality and an upper bound
on the number of extreme points. Using these properties, we give the first
run-time analysis of the MOEA/D algorithm for this problem, an evolutionary
algorithm that effectively optimizes by decomposing the objectives into
single-objective components. We show that the MOEA/D, given an appropriate
decomposition setting, finds all extreme points within expected fixed-parameter
polynomial time in the oracle model, the parameter being the number of
objectives. Experiments are conducted on random bi-objective minimum spanning
tree instances, and the results agree with our theoretical findings.
Furthermore, compared with a previously studied evolutionary algorithm for the
problem GSEMO, MOEA/D finds all extreme points much faster across all
instances.Comment: 12 page
Towards a Runtime Comparison of Natural and Artificial Evolution
Evolutionary algorithms (EAs) form a popular optimisation paradigm inspired by natural evolution. In recent years the field of evolutionary computation has developed a rigorous analytical theory to analyse the runtimes of EAs on many illustrative problems. Here we apply this theory to a simple model of natural evolution. In the Strong Selection Weak Mutation (SSWM) evolutionary regime the time between occurrences of new mutations is much longer than the time it takes for a mutated genotype to take over the population. In this situation, the population only contains copies of one genotype and evolution can be modelled as a stochastic process evolving one genotype by means of mutation and selection between the resident and the mutated genotype. The probability of accepting the mutated genotype then depends on the change in fitness. We study this process, SSWM, from an algorithmic perspective, quantifying its expected optimisation time for various parameters and investigating differences to a similar evolutionary algorithm, the well-known (1+1) EA. We show that SSWM can have a moderate advantage over the (1+1) EA at crossing fitness valleys and study an example where SSWM outperforms the (1+1) EA by taking advantage of information on the fitness gradient
Parameterized Complexity Analysis of Randomized Search Heuristics
This chapter compiles a number of results that apply the theory of
parameterized algorithmics to the running-time analysis of randomized search
heuristics such as evolutionary algorithms. The parameterized approach
articulates the running time of algorithms solving combinatorial problems in
finer detail than traditional approaches from classical complexity theory. We
outline the main results and proof techniques for a collection of randomized
search heuristics tasked to solve NP-hard combinatorial optimization problems
such as finding a minimum vertex cover in a graph, finding a maximum leaf
spanning tree in a graph, and the traveling salesperson problem.Comment: This is a preliminary version of a chapter in the book "Theory of
Evolutionary Computation: Recent Developments in Discrete Optimization",
edited by Benjamin Doerr and Frank Neumann, published by Springe
Random combinatorial structures and randomized search heuristics
This thesis is concerned with the probabilistic analysis of random combinatorial structures and the runtime analysis of randomized search heuristics.
On the subject of random structures, we investigate two classes of combinatorial objects. The first is the class of planar maps and the second is the class of generalized parking functions. We identify typical properties of these structures and show strong concentration results on the probabilities that these properties hold. To this end, we develop and apply techniques based on exact enumeration by generating functions. For several types of random planar maps, this culminates in concentration results for the degree sequence. For parking functions, we determine the distribution of the defect, the most characteristic parameter. On the subject of randomized search heuristics, we present, improve, and unify different probabilistic methods and their applications. In this, special focus is given to potential functions and the analysis of the drift of stochastic processes. We apply these techniques to investigate the runtimes of evolutionary algorithms. In particular, we show for several classical problems in combinatorial optimization how drift analysis can be used in a uniform way to give bounds on the expected runtimes of evolutionary algorithms.Diese Dissertationsschrift beschäftigt sich mit der wahrscheinlichkeitstheoretischen Analyse von zufälligen kombinatorischen Strukturen und der Laufzeitanalyse randomisierter Suchheuristiken. Im Bereich der zufälligen Strukturen untersuchen wir zwei Klassen kombinatorischer Objekte. Dies sind zum einen die Klasse aller kombinatorischen Einbettungen planarer Graphen und zum anderen eine Klasse diskreter Funktionen mit bestimmten kombinatorischen Restriktionen (generalized parking functions). Für das Studium dieser Klassen entwickeln und verwenden wir zählkombinatorische Methoden die auf erzeugenden Funktionen basieren. Dies erlaubt uns, Konzentrationsresultate für die Gradsequenzen verschiedener Typen zufälliger kombinatorischer Einbettungen planarer Graphen zu erzielen. Darüber hinaus erhalten wir Konzentrationsresultate für den charakteristischen Parameter, den Defekt, zufälliger Instanzen der untersuchten diskreten Funktionen.
Im Bereich der randomisierten Suchheuristiken präsentieren und erweitern wir verschiedene wahrscheinlichkeitstheoretische Methoden der Analyse. Ein besonderer Fokus liegt dabei auf der Analyse der Drift stochastischer Prozesse. Wir wenden diese Methoden in der Laufzeitanalyse evolutionärer Algorithmen an. Insbesondere zeigen wir, wie mit Hilfe von Driftanalyse die erwarteten Laufzeiten evolutionärer Algorithmen auf verschiedenen klassischen Problemen der kombinatorischen Optimierung auf einheitliche Weise abgeschätzt werden können
Matroids, Complexity and Computation
The node deletion problem on graphs is: given a graph and integer k, can we
delete no more than k vertices to obtain a graph that satisfies some property π.
Yannakakis showed that this problem is NP-complete for an infinite family of well-
defined properties. The edge deletion problem and matroid deletion problem are
similar problems where given a graph or matroid respectively, we are asked if we
can delete no more than k edges/elements to obtain a graph/matroid that satisfies
a property π. We show that these problems are NP-hard for similar well-defined
infinite families of properties.
In 1991 Vertigan showed that it is #P-complete to count the number of bases
of a representable matroid over any fixed field. However no publication has been
produced. We consider this problem and show that it is #P-complete to count
the number of bases of matroids representable over any infinite fixed field or finite
fields of a fixed characteristic.
There are many different ways of describing a matroid. Not all of these are
polynomially equivalent. That is, given one description of a matroid, we cannot
create another description for the same matroid in time polynomial in the size of
the first description. Due to this, the complexity of matroid problems can vary
greatly depending on the method of description used. Given one description a
problem might be in P while another description gives an NP-complete problem.
Based on these interactions between descriptions, we create and study the hierarchy
of all matroid descriptions and generalize this to all descriptions of countable
objects
Computational complexity of evolutionary algorithms, hybridizations, and swarm intelligence
Bio-inspired randomized search heuristics such as evolutionary algorithms, hybridizations
with local search, and swarm intelligence are very popular among practitioners
as they can be applied in case the problem is not well understood or when there is
not enough knowledge, time, or expertise to design problem-specific algorithms. Evolutionary
algorithms simulate the natural evolution of species by iteratively applying
evolutionary operators such as mutation, recombination, and selection to a set of solutions
for a given problem. A recent trend is to hybridize evolutionary algorithms with
local search to refine newly constructed solutions by hill climbing. Swarm intelligence
comprises ant colony optimization as well as particle swarm optimization. These modern
search paradigms rely on the collective intelligence of many single agents to find good
solutions for the problem at hand. Many empirical studies demonstrate the usefulness
of these heuristics for a large variety of problems, but a thorough understanding is still
far away.
We regard these algorithms from the perspective of theoretical computer science and
analyze the random time these heuristics need to optimize pseudo-Boolean problems.
This is done in a mathematically rigorous sense, using tools known from the analysis of
randomized algorithms, and it leads to asymptotic bounds on their computational complexity.
This approach has been followed successfully for evolutionary algorithms, but
the theory of hybrid algorithms and swarm intelligence is still in its very infancy. Our
results shed light on the asymptotic performance of these heuristics, increase our understanding
of their dynamic behavior, and contribute to a rigorous theoretical foundation
of randomized search heuristics
Evolutionary algorithms and matroid optimization problems
We analyze the performance of evolutionary algorithms on various matroid optimization problems that encompass a vast number of efficiently solvable as well as NP-hard combinatorial optimization problems (including many well-known examples such as minimum spanning tree and maximum bipartite matching). We obtain very promising bounds on the expected running time and quality of the computed solution. Our results establish a better theoretical understanding of why randomized search heuristics yield empirically good results for many real-world optimization problems