2,149 research outputs found
A Memetic Algorithm for the Generalized Traveling Salesman Problem
The generalized traveling salesman problem (GTSP) is an extension of the
well-known traveling salesman problem. In GTSP, we are given a partition of
cities into groups and we are required to find a minimum length tour that
includes exactly one city from each group. The recent studies on this subject
consider different variations of a memetic algorithm approach to the GTSP. The
aim of this paper is to present a new memetic algorithm for GTSP with a
powerful local search procedure. The experiments show that the proposed
algorithm clearly outperforms all of the known heuristics with respect to both
solution quality and running time. While the other memetic algorithms were
designed only for the symmetric GTSP, our algorithm can solve both symmetric
and asymmetric instances.Comment: 15 pages, to appear in Natural Computing, Springer, available online:
http://www.springerlink.com/content/5v4568l492272865/?p=e1779dd02e4d4cbfa49d0d27b19b929f&pi=1
Fine-Grained Complexity Analysis of Two Classic TSP Variants
We analyze two classic variants of the Traveling Salesman Problem using the
toolkit of fine-grained complexity. Our first set of results is motivated by
the Bitonic TSP problem: given a set of points in the plane, compute a
shortest tour consisting of two monotone chains. It is a classic
dynamic-programming exercise to solve this problem in time. While the
near-quadratic dependency of similar dynamic programs for Longest Common
Subsequence and Discrete Frechet Distance has recently been proven to be
essentially optimal under the Strong Exponential Time Hypothesis, we show that
bitonic tours can be found in subquadratic time. More precisely, we present an
algorithm that solves bitonic TSP in time and its bottleneck
version in time. Our second set of results concerns the popular
-OPT heuristic for TSP in the graph setting. More precisely, we study the
-OPT decision problem, which asks whether a given tour can be improved by a
-OPT move that replaces edges in the tour by new edges. A simple
algorithm solves -OPT in time for fixed . For 2-OPT, this is
easily seen to be optimal. For we prove that an algorithm with a runtime
of the form exists if and only if All-Pairs
Shortest Paths in weighted digraphs has such an algorithm. The results for
may suggest that the actual time complexity of -OPT is
. We show that this is not the case, by presenting an algorithm
that finds the best -move in time for
fixed . This implies that 4-OPT can be solved in time,
matching the best-known algorithm for 3-OPT. Finally, we show how to beat the
quadratic barrier for in two important settings, namely for points in the
plane and when we want to solve 2-OPT repeatedly.Comment: Extended abstract appears in the Proceedings of the 43rd
International Colloquium on Automata, Languages, and Programming (ICALP 2016
An interacting replica approach applied to the traveling salesman problem
We present a physics inspired heuristic method for solving combinatorial
optimization problems. Our approach is specifically motivated by the desire to
avoid trapping in metastable local minima- a common occurrence in hard problems
with multiple extrema. Our method involves (i) coupling otherwise independent
simulations of a system ("replicas") via geometrical distances as well as (ii)
probabilistic inference applied to the solutions found by individual replicas.
The {\it ensemble} of replicas evolves as to maximize the inter-replica
correlation while simultaneously minimize the local intra-replica cost function
(e.g., the total path length in the Traveling Salesman Problem within each
replica). We demonstrate how our method improves the performance of rudimentary
local optimization schemes long applied to the NP hard Traveling Salesman
Problem. In particular, we apply our method to the well-known "-opt"
algorithm and examine two particular cases- and . With the aid of
geometrical coupling alone, we are able to determine for the optimum tour
length on systems up to cities (an order of magnitude larger than the
largest systems typically solved by the bare opt). The probabilistic
replica-based inference approach improves even further and determines
the optimal solution of a problem with cities and find tours whose total
length is close to that of the optimal solutions for other systems with a
larger number of cities.Comment: To appear in SAI 2016 conference proceedings 12 pages,17 figure
Lin-Kernighan Heuristic Adaptations for the Generalized Traveling Salesman Problem
The Lin-Kernighan heuristic is known to be one of the most successful
heuristics for the Traveling Salesman Problem (TSP). It has also proven its
efficiency in application to some other problems. In this paper we discuss
possible adaptations of TSP heuristics for the Generalized Traveling Salesman
Problem (GTSP) and focus on the case of the Lin-Kernighan algorithm. At first,
we provide an easy-to-understand description of the original Lin-Kernighan
heuristic. Then we propose several adaptations, both trivial and complicated.
Finally, we conduct a fair competition between all the variations of the
Lin-Kernighan adaptation and some other GTSP heuristics. It appears that our
adaptation of the Lin-Kernighan algorithm for the GTSP reproduces the success
of the original heuristic. Different variations of our adaptation outperform
all other heuristics in a wide range of trade-offs between solution quality and
running time, making Lin-Kernighan the state-of-the-art GTSP local search.Comment: 25 page
The Unreasonable Success of Local Search: Geometric Optimization
What is the effectiveness of local search algorithms for geometric problems
in the plane? We prove that local search with neighborhoods of magnitude
is an approximation scheme for the following problems in the
Euclidian plane: TSP with random inputs, Steiner tree with random inputs,
facility location (with worst case inputs), and bicriteria -median (also
with worst case inputs). The randomness assumption is necessary for TSP
Applications of Bee Colony Optimization
Many computationally difficult problems are attacked using non-exact algorithms, such as approximation algorithms and heuristics. This thesis investigates an ex- ample of the latter, Bee Colony Optimization, on both an established optimization problem in the form of the Quadratic Assignment Problem and the FireFighting problem, which has not been studied before as an optimization problem. Bee Colony Optimization is a swarm intelligence algorithm, a paradigm that has increased in popularity in recent years, and many of these algorithms are based on natural pro- cesses.
We tested the Bee Colony Optimization algorithm on the QAPLIB library of Quadratic Assignment Problem instances, which have either optimal or best known solutions readily available, and enabled us to compare the quality of solutions found by the algorithm. In addition, we implemented a couple of other well known algorithms for the Quadratic Assignment Problem and consequently we could analyse the runtime of our algorithm.
We introduce the Bee Colony Optimization algorithm for the FireFighting problem. We also implement some greedy algorithms and an Ant Colony Optimization al- gorithm for the FireFighting problem, and compare the results obtained on some randomly generated instances.
We conclude that Bee Colony Optimization finds good solutions for the Quadratic Assignment Problem, however further investigation on speedup methods is needed to improve its performance to that of other algorithms. In addition, Bee Colony Optimization is effective on small instances of the FireFighting problem, however as instance size increases the results worsen in comparison to the greedy algorithms, and more work is needed to improve the decisions made on these instances
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