3,197 research outputs found
Shorter tours and longer detours: Uniform covers and a bit beyond
Motivated by the well known four-thirds conjecture for the traveling salesman
problem (TSP), we study the problem of {\em uniform covers}. A graph
has an -uniform cover for TSP (2EC, respectively) if the everywhere
vector (i.e. ) dominates a convex combination of
incidence vectors of tours (2-edge-connected spanning multigraphs,
respectively). The polyhedral analysis of Christofides' algorithm directly
implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP.
Seb\H{o} asked if such graphs have -uniform covers for TSP for
some . Indeed, the four-thirds conjecture implies that such
graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform
covers for TSP. We also study uniform covers for 2EC and show that the
everywhere 15/17 vector can be efficiently written as a convex combination of
2-edge-connected spanning multigraphs.
For a weighted, 3-edge-connected, cubic graph, our results show that if the
everywhere 2/3 vector is an optimal solution for the subtour linear programming
relaxation, then a tour with weight at most 27/19 times that of an optimal tour
can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall
into this category. In this special case, we can apply our tools to obtain an
even better approximation guarantee.
To extend our approach to input graphs that are 2-edge-connected, we present
a procedure to decompose an optimal solution for the subtour relaxation for TSP
into spanning, connected multigraphs that cover each 2-edge cut an even number
of times. Using this decomposition, we obtain a 17/12-approximation algorithm
for minimum weight 2-edge-connected spanning subgraphs on subcubic,
node-weighted graphs
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
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Probabilistic Analysis of Euclidean Capacitated Vehicle Routing
We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of n unit-demand customers modeled as independent, identically distributed uniform random points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most k customers, where k is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely.
The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when k is either o(?n) or ?(?n), and they asked whether the ITP algorithm was "also effective in the intermediate range". In this work, we show that when k = ?n, the ITP algorithm is at best a (1+c?)-approximation for some positive constant c?.
On the other hand, the approximation ratio of the ITP algorithm was known to be at most 0.995+? due to Bompadre, Dror, and Orlin, where ? is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to 0.915+?. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest
Layers and Matroids for the Traveling Salesman's Paths
Gottschalk and Vygen proved that every solution of the subtour elimination
linear program for traveling salesman paths is a convex combination of more and
more restrictive "generalized Gao-trees". We give a short proof of this fact,
as a layered convex combination of bases of a sequence of increasingly
restrictive matroids. A strongly polynomial, combinatorial algorithm follows
for finding this convex combination, which is a new tool offering polyhedral
insight, already instrumental in recent results for the path TSP
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