1,325 research outputs found
Exact value for the average optimal cost of bipartite traveling-salesman and 2-factor problems in two dimensions
We show that the average cost for the traveling-salesman problem in two
dimensions, which is the archetypal problem in combinatorial optimization, in
the bipartite case, is simply related to the average cost of the assignment
problem with the same Euclidean, increasing, convex weights. In this way we
extend a result already known in one dimension where exact solutions are
avalaible. The recently determined average cost for the assignment when the
cost function is the square of the distance between the points provides
therefore an exact prediction for
large number of points . As a byproduct of our analysis also the loop
covering problem has the same optimal average cost. We also explain why this
result cannot be extended at higher dimensions. We numerically check the exact
predictions.Comment: 5 pages, 3 figure
Ising formulations of many NP problems
We provide Ising formulations for many NP-complete and NP-hard problems,
including all of Karp's 21 NP-complete problems. This collects and extends
mappings to the Ising model from partitioning, covering and satisfiability. In
each case, the required number of spins is at most cubic in the size of the
problem. This work may be useful in designing adiabatic quantum optimization
algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more
references; v3: substantial revision and extension, to-be-published versio
Defragmenting the Module Layout of a Partially Reconfigurable Device
Modern generations of field-programmable gate arrays (FPGAs) allow for
partial reconfiguration. In an online context, where the sequence of modules to
be loaded on the FPGA is unknown beforehand, repeated insertion and deletion of
modules leads to progressive fragmentation of the available space, making
defragmentation an important issue. We address this problem by propose an
online and an offline component for the defragmentation of the available space.
We consider defragmenting the module layout on a reconfigurable device. This
corresponds to solving a two-dimensional strip packing problem. Problems of
this type are NP-hard in the strong sense, and previous algorithmic results are
rather limited. Based on a graph-theoretic characterization of feasible
packings, we develop a method that can solve two-dimensional defragmentation
instances of practical size to optimality. Our approach is validated for a set
of benchmark instances.Comment: 10 pages, 11 figures, 1 table, Latex, to appear in "Engineering of
Reconfigurable Systems and Algorithms" as a "Distinguished Paper
Una comparación de algoritmos basados en trayectoria granular para el problema de localización y ruteo con flota heterogénea (LRPH)
Indexación: Scopus.We consider the Location-Routing Problem with Heterogeneous Fleet (LRPH) in which the goal is to determine the depots to be opened, the customers to be assigned to each open depot, and the corresponding routes fulfilling the demand of the customers and by considering a heterogeneous fleet. We propose a comparison of granular approaches of Simulated Annealing (GSA), of Variable Neighborhood Search (GVNS) and of a probabilistic Tabu Search (pGTS) for the LRPH. Thus, the proposed approaches consider a subset of the search space in which non-favorable movements are discarded regarding a granularity factor. The proposed algorithms are experimentally compared for the solution of the LRPH, by taking into account the CPU time and the quality of the solutions obtained on the instances adapted from the literature. The computational results show that algorithm GSA is able to obtain high quality solutions within short CPU times, improving the results obtained by the other proposed approaches.https://revistas.unal.edu.co/index.php/dyna/article/view/55533/5896
The Traveling Salesman Problem
This paper presents a self-contained introduction into algorithmic and computational aspects of the traveling salesman problem and of related problems, along with their theoretical prerequisites as seen from the point of view of an operations researcher who wants to solve practical problem instances. Extensive computational results are reported on most of the algorithms described. Optimal solutions are reported for instances with sizes up to several thousand nodes as well as heuristic solutions with provably very high quality for larger instances
Application and Assessment of Divide-and-Conquer-based Heuristic Algorithms for some Integer Optimization Problems
In this paper three heuristic algorithms using the Divide-and-Conquer
paradigm are developed and assessed for three integer optimizations problems:
Multidimensional Knapsack Problem (d-KP), Bin Packing Problem (BPP) and
Travelling Salesman Problem (TSP). For each case, the algorithm is introduced,
together with the design of numerical experiments, in order to empirically
establish its performance from both points of view: its computational time and
its numerical accuracy.Comment: 16 pages, 6 figures, 8 table
Computational Complexity for Physicists
These lecture notes are an informal introduction to the theory of
computational complexity and its links to quantum computing and statistical
mechanics.Comment: references updated, reprint available from
http://itp.nat.uni-magdeburg.de/~mertens/papers/complexity.shtm
The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme
The Traveling Salesman Problem (TSP) is among the most famous NP-hard
optimization problems. We design for this problem a randomized polynomial-time
algorithm that computes a (1+eps)-approximation to the optimal tour, for any
fixed eps>0, in TSP instances that form an arbitrary metric space with bounded
intrinsic dimension.
The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the
above result holds in the special case of TSP in a fixed-dimensional Euclidean
space. Thus, our algorithm demonstrates that the algorithmic tractability of
metric TSP depends on the dimensionality of the space and not on its specific
geometry. This result resolves a problem that has been open since the
quasi-polynomial time algorithm of Talwar (T-04)
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
Comparative study of variations in quantum approximate optimization algorithms for the Traveling Salesman Problem
The Traveling Salesman Problem (TSP) is one of the most often-used NP-Hard
problems in computer science to study the effectiveness of computing models and
hardware platforms. In this regard, it is also heavily used as a vehicle to
study the feasibility of the quantum computing paradigm for this class of
problems. In this paper, we tackle the TSP using the quantum approximate
optimization algorithm (QAOA) approach by formulating it as an optimization
problem. By adopting an improved qubit encoding strategy and a layerwise
learning optimization protocol, we present numerical results obtained from the
gate-based digital quantum simulator, specifically targeting TSP instances with
3, 4, and 5 cities. We focus on the evaluations of three distinctive QAOA mixer
designs, considering their performances in terms of numerical accuracy and
optimization cost. Notably, we find a well-balanced QAOA mixer design exhibits
more promising potential for gate-based simulators and realistic quantum
devices in the long run, an observation further supported by our noise model
simulations. Furthermore, we investigate the sensitivity of the simulations to
the TSP graph. Overall, our simulation results show the digital quantum
simulation of problem-inspired ansatz is a successful candidate for finding
optimal TSP solutions.Comment: 18 pages, 6 figures, 3 table
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