6,757 research outputs found
A nonmonotone GRASP
A greedy randomized adaptive search procedure (GRASP) is an itera-
tive multistart metaheuristic for difficult combinatorial optimization problems. Each
GRASP iteration consists of two phases: a construction phase, in which a feasible
solution is produced, and a local search phase, in which a local optimum in the
neighborhood of the constructed solution is sought. Repeated applications of the con-
struction procedure yields different starting solutions for the local search and the
best overall solution is kept as the result. The GRASP local search applies iterative
improvement until a locally optimal solution is found. During this phase, starting from
the current solution an improving neighbor solution is accepted and considered as the
new current solution. In this paper, we propose a variant of the GRASP framework that
uses a new “nonmonotone” strategy to explore the neighborhood of the current solu-
tion. We formally state the convergence of the nonmonotone local search to a locally
optimal solution and illustrate the effectiveness of the resulting Nonmonotone GRASP
on three classical hard combinatorial optimization problems: the maximum cut prob-
lem (MAX-CUT), the weighted maximum satisfiability problem (MAX-SAT), and
the quadratic assignment problem (QAP)
Greedy MAXCUT Algorithms and their Information Content
MAXCUT defines a classical NP-hard problem for graph partitioning and it
serves as a typical case of the symmetric non-monotone Unconstrained Submodular
Maximization (USM) problem. Applications of MAXCUT are abundant in machine
learning, computer vision and statistical physics. Greedy algorithms to
approximately solve MAXCUT rely on greedy vertex labelling or on an edge
contraction strategy. These algorithms have been studied by measuring their
approximation ratios in the worst case setting but very little is known to
characterize their robustness to noise contaminations of the input data in the
average case. Adapting the framework of Approximation Set Coding, we present a
method to exactly measure the cardinality of the algorithmic approximation sets
of five greedy MAXCUT algorithms. Their information contents are explored for
graph instances generated by two different noise models: the edge reversal
model and Gaussian edge weights model. The results provide insights into the
robustness of different greedy heuristics and techniques for MAXCUT, which can
be used for algorithm design of general USM problems.Comment: This is a longer version of the paper published in 2015 IEEE
Information Theory Workshop (ITW
Cut Tree Construction from Massive Graphs
The construction of cut trees (also known as Gomory-Hu trees) for a given
graph enables the minimum-cut size of the original graph to be obtained for any
pair of vertices. Cut trees are a powerful back-end for graph management and
mining, as they support various procedures related to the minimum cut, maximum
flow, and connectivity. However, the crucial drawback with cut trees is the
computational cost of their construction. In theory, a cut tree is built by
applying a maximum flow algorithm for times, where is the number of
vertices. Therefore, naive implementations of this approach result in cubic
time complexity, which is obviously too slow for today's large-scale graphs. To
address this issue, in the present study, we propose a new cut-tree
construction algorithm tailored to real-world networks. Using a series of
experiments, we demonstrate that the proposed algorithm is several orders of
magnitude faster than previous algorithms and it can construct cut trees for
billion-scale graphs.Comment: Short version will appear at ICDM'1
Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics
Simple heuristics often show a remarkable performance in practice for
optimization problems. Worst-case analysis often falls short of explaining this
performance. Because of this, "beyond worst-case analysis" of algorithms has
recently gained a lot of attention, including probabilistic analysis of
algorithms.
The instances of many optimization problems are essentially a discrete metric
space. Probabilistic analysis for such metric optimization problems has
nevertheless mostly been conducted on instances drawn from Euclidean space,
which provides a structure that is usually heavily exploited in the analysis.
However, most instances from practice are not Euclidean. Little work has been
done on metric instances drawn from other, more realistic, distributions. Some
initial results have been obtained by Bringmann et al. (Algorithmica, 2013),
who have used random shortest path metrics on complete graphs to analyze
heuristics.
The goal of this paper is to generalize these findings to non-complete
graphs, especially Erd\H{o}s-R\'enyi random graphs. A random shortest path
metric is constructed by drawing independent random edge weights for each edge
in the graph and setting the distance between every pair of vertices to the
length of a shortest path between them with respect to the drawn weights. For
such instances, we prove that the greedy heuristic for the minimum distance
maximum matching problem, the nearest neighbor and insertion heuristics for the
traveling salesman problem, and a trivial heuristic for the -median problem
all achieve a constant expected approximation ratio. Additionally, we show a
polynomial upper bound for the expected number of iterations of the 2-opt
heuristic for the traveling salesman problem.Comment: An extended abstract appeared in the proceedings of WALCOM 201
Improving Optimization Bounds using Machine Learning: Decision Diagrams meet Deep Reinforcement Learning
Finding tight bounds on the optimal solution is a critical element of
practical solution methods for discrete optimization problems. In the last
decade, decision diagrams (DDs) have brought a new perspective on obtaining
upper and lower bounds that can be significantly better than classical bounding
mechanisms, such as linear relaxations. It is well known that the quality of
the bounds achieved through this flexible bounding method is highly reliant on
the ordering of variables chosen for building the diagram, and finding an
ordering that optimizes standard metrics is an NP-hard problem. In this paper,
we propose an innovative and generic approach based on deep reinforcement
learning for obtaining an ordering for tightening the bounds obtained with
relaxed and restricted DDs. We apply the approach to both the Maximum
Independent Set Problem and the Maximum Cut Problem. Experimental results on
synthetic instances show that the deep reinforcement learning approach, by
achieving tighter objective function bounds, generally outperforms ordering
methods commonly used in the literature when the distribution of instances is
known. To the best knowledge of the authors, this is the first paper to apply
machine learning to directly improve relaxation bounds obtained by
general-purpose bounding mechanisms for combinatorial optimization problems.Comment: Accepted and presented at AAAI'1
Exact Localisations of Feedback Sets
The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform
a given multi digraph into an acyclic graph by deleting as few arcs
(vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one
of the classic NP-complete problems. An important contribution of this paper is
that the subgraphs , of all elementary
cycles or simple cycles running through some arc , can be computed in
and , respectively. We use
this fact and introduce the notion of the essential minor and isolated cycles,
which yield a priori problem size reductions and in the special case of so
called resolvable graphs an exact solution in . We show
that weighted versions of the FASP and FVSP possess a Bellman decomposition,
which yields exact solutions using a dynamic programming technique in times
and
, where , , respectively. The parameters can
be computed in , ,
respectively and denote the maximal dimension of the cycle space of all
appearing meta graphs, decoding the intersection behavior of the cycles.
Consequently, equal zero if all meta graphs are trees. Moreover, we
deliver several heuristics and discuss how to control their variation from the
optimum. Summarizing, the presented results allow us to suggest a strategy for
an implementation of a fast and accurate FASP/FVSP-SOLVER
How to make a greedy heuristic for the asymmetric traveling salesman problem competitive
It is widely confirmed by many computational experiments that a greedy type heuristics for the Traveling Salesman Problem (TSP) produces rather poor solutions except for the Euclidean TSP. The selection of arcs to be included by a greedy heuristic is usually done on the base of cost values. We propose to use upper tolerances of an optimal solution to one of the relaxed Asymmetric TSP (ATSP) to guide the selection of an arc to be included in the final greedy solution. Even though it needs time to calculate tolerances, our computational experiments for the wide range of ATSP instances show that tolerance based greedy heuristics is much more accurate an faster than previously reported greedy type algorithms
- …