6,557 research outputs found
Bidirected minimum Manhattan network problem
In the bidirected minimum Manhattan network problem, given a set T of n
terminals in the plane, we need to construct a network N(T) of minimum total
length with the property that the edges of N(T) are axis-parallel and oriented
in a such a way that every ordered pair of terminals is connected in N(T) by a
directed Manhattan path. In this paper, we present a polynomial factor 2
approximation algorithm for the bidirected minimum Manhattan network problem.Comment: 14 pages, 16 figure
Quantifying the benefits of vehicle pooling with shareability networks
Taxi services are a vital part of urban transportation, and a considerable
contributor to traffic congestion and air pollution causing substantial adverse
effects on human health. Sharing taxi trips is a possible way of reducing the
negative impact of taxi services on cities, but this comes at the expense of
passenger discomfort quantifiable in terms of a longer travel time. Due to
computational challenges, taxi sharing has traditionally been approached on
small scales, such as within airport perimeters, or with dynamical ad-hoc
heuristics. However, a mathematical framework for the systematic understanding
of the tradeoff between collective benefits of sharing and individual passenger
discomfort is lacking. Here we introduce the notion of shareability network
which allows us to model the collective benefits of sharing as a function of
passenger inconvenience, and to efficiently compute optimal sharing strategies
on massive datasets. We apply this framework to a dataset of millions of taxi
trips taken in New York City, showing that with increasing but still relatively
low passenger discomfort, cumulative trip length can be cut by 40% or more.
This benefit comes with reductions in service cost, emissions, and with split
fares, hinting towards a wide passenger acceptance of such a shared service.
Simulation of a realistic online system demonstrates the feasibility of a
shareable taxi service in New York City. Shareability as a function of trip
density saturates fast, suggesting effectiveness of the taxi sharing system
also in cities with much sparser taxi fleets or when willingness to share is
low.Comment: Main text: 6 pages, 3 figures, SI: 24 page
Searching for Realizations of Finite Metric Spaces in Tight Spans
An important problem that commonly arises in areas such as internet
traffic-flow analysis, phylogenetics and electrical circuit design, is to find
a representation of any given metric on a finite set by an edge-weighted
graph, such that the total edge length of the graph is minimum over all such
graphs. Such a graph is called an optimal realization and finding such
realizations is known to be NP-hard. Recently Varone presented a heuristic
greedy algorithm for computing optimal realizations. Here we present an
alternative heuristic that exploits the relationship between realizations of
the metric and its so-called tight span . The tight span is a
canonical polytopal complex that can be associated to , and our approach
explores parts of for realizations in a way that is similar to the
classical simplex algorithm. We also provide computational results illustrating
the performance of our approach for different types of metrics, including
-distances and two-decomposable metrics for which it is provably possible
to find optimal realizations in their tight spans.Comment: 20 pages, 3 figure
Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems
We consider geometric instances of the Maximum Weighted Matching Problem
(MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000
vertices. Making use of a geometric duality relationship between MWMP, MTSP,
and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields
in near-linear time solutions as well as upper bounds. Using various
computational tools, we get solutions within considerably less than 1% of the
optimum.
An interesting feature of our approach is that, even though an FWP is hard to
compute in theory and Edmonds' algorithm for maximum weighted matching yields a
polynomial solution for the MWMP, the practical behavior is just the opposite,
and we can solve the FWP with high accuracy in order to find a good heuristic
solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental
Algorithms, 200
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
Integer Point Sets Minimizing Average Pairwise L1-Distance: What is the Optimal Shape of a Town?
An n-town, for a natural number n, is a group of n buildings, each occupying
a distinct position on a 2-dimensional integer grid. If we measure the distance
between two buildings along the axis-parallel street grid, then an n-town has
optimal shape if the sum of all pairwise Manhattan distances is minimized. This
problem has been studied for cities, i.e., the limiting case of very large n.
For cities, it is known that the optimal shape can be described by a
differential equation, for which no closed-form is known. We show that optimal
n-towns can be computed in O(n^7.5) time. This is also practically useful, as
it allows us to compute optimal solutions up to n=80.Comment: 26 pages, 6 figures, to appear in Computational Geometry: Theory and
Application
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