76 research outputs found
An ETH-Tight Exact Algorithm for Euclidean TSP
We study exact algorithms for {\sc Euclidean TSP} in . In the
early 1990s algorithms with running time were presented for
the planar case, and some years later an algorithm with
running time was presented for any . Despite significant interest in
subexponential exact algorithms over the past decade, there has been no
progress on {\sc Euclidean TSP}, except for a lower bound stating that the
problem admits no algorithm unless ETH fails. Up to
constant factors in the exponent, we settle the complexity of {\sc Euclidean
TSP} by giving a algorithm and by showing that a
algorithm does not exist unless ETH fails.Comment: To appear in FOCS 201
Euclidean TSP with few inner points in linear space
Given a set of points in the Euclidean plane, such that just points
are strictly inside the convex hull of the whole set, we want to find the
shortest tour visiting every point. The fastest known algorithm for the version
when is significantly smaller than , i.e., when there are just few inner
points, works in time [Knauer and Spillner,
WG 2006], but also requires space of order . The best
linear space algorithm takes time [Deineko, Hoffmann, Okamoto,
Woeginer, Oper. Res. Lett. 34(1), 106-110]. We construct a linear space
time algorithm. The new insight is extending the
known divide-and-conquer method based on planar separators with a
matching-based argument to shrink the instance in every recursive call. This
argument also shows that the problem admits a quadratic bikernel.Comment: under submissio
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of points with their pairwise distances, the traveling
salesman problem (TSP) asks for a shortest tour that visits each point exactly
once. A TSP instance is rectilinear when the points lie in the plane and the
distance considered between two points is the distance. In this paper, a
fixed-parameter algorithm for the Rectilinear TSP is presented and relies on
techniques for solving TSP on bounded-treewidth graphs. It proves that the
problem can be solved in where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
strip
We investigate how the complexity of Euclidean TSP for point sets inside the strip depends on the strip width . We obtain two main results. First, for the case where the points have distinct integer -coordinates, we prove that a shortest bitonic tour (which can be computed in time using an existing algorithm) is guaranteed to be a shortest tour overall when , a bound which is best possible. Second, we present an algorithm that is fixed-parameter tractable with respect to . More precisely, our algorithm has running time for sparse point sets, where each rectangle inside the strip contains points. For random point sets, where the points are chosen uniformly at random from the rectangle~, it has an expected running time of
Production Scheduling with Capacity-Lot Size and Sequence Consideration
The General Lot-sizing and Scheduling Problem (GLSP) is a common problem found in continuous production planning. This problem involves many constraints and decisions including machine capacity, production lot-size, and production sequence. This study proposes a two-phase algorithm for solving large-scale GLSP models. In Phase 1, we generate patterns with a specific batch size and capacity and in Phase 2, based on the patterns selected in Phase 1, we optimize the production allocation. Additionally, the external supplies are included in the formulation to reflect the real situation in business with limited resources. In this work, the justification of the formulation is based on the ability of solving and calculation time. The proposed formulation was tested on eight scenarios. The results show that the proposed formulation is more tractable and is easier to solve than the GLSP.The General Lot-sizing and Scheduling Problem (GLSP) is a common problem found in continuous production planning. This problem involves many constraints and decisions including machine capacity, production lot-size, and production sequence. This study proposes a two-phase algorithm for solving large-scale GLSP models. In Phase 1, we generate patterns with a specific batch size and capacity and in Phase 2, based on the patterns selected in Phase 1, we optimize the production allocation. Additionally, the external supplies are included in the formulation to reflect the real situation in business with limited resources. In this work, the justification of the formulation is based on the ability of solving and calculation time. The proposed formulation was tested on eight scenarios. The results show that the proposed formulation is more tractable and is easier to solve than the GLSP
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