1,914 research outputs found
Diffusive behavior of a greedy traveling salesman
Using Monte Carlo simulations we examine the diffusive properties of the
greedy algorithm in the d-dimensional traveling salesman problem. Our results
show that for d=3 and 4 the average squared distance from the origin is
proportional to the number of steps t. In the d=2 case such a scaling is
modified with some logarithmic corrections, which might suggest that d=2 is the
critical dimension of the problem. The distribution of lengths also shows
marked differences between d=2 and d>2 versions. A simple strategy adopted by
the salesman might resemble strategies chosen by some foraging and hunting
animals, for which anomalous diffusive behavior has recently been reported and
interpreted in terms of Levy flights. Our results suggest that broad and
Levy-like distributions in such systems might appear due to dimension-dependent
properties of a search space.Comment: accepted in Phys. Rev.
Uncapacitated Flow-based Extended Formulations
An extended formulation of a polytope is a linear description of this
polytope using extra variables besides the variables in which the polytope is
defined. The interest of extended formulations is due to the fact that many
interesting polytopes have extended formulations with a lot fewer inequalities
than any linear description in the original space. This motivates the
development of methods for, on the one hand, constructing extended formulations
and, on the other hand, proving lower bounds on the sizes of extended
formulations.
Network flows are a central paradigm in discrete optimization, and are widely
used to design extended formulations. We prove exponential lower bounds on the
sizes of uncapacitated flow-based extended formulations of several polytopes,
such as the (bipartite and non-bipartite) perfect matching polytope and TSP
polytope. We also give new examples of flow-based extended formulations, e.g.,
for 0/1-polytopes defined from regular languages. Finally, we state a few open
problems
Domination Analysis of Greedy Heuristics For The Frequency Assignment Problem
We introduce the greedy expectation algorithm for the
fixed spectrum version of the frequency assignment problem. This
algorithm was previously studied for the travelling salesman
problem. We show that the domination number of this algorithm is
at least where is the
available span and the number of vertices in the constraint
graph. In contrast to this we show that the standard greedy
algorithm has domination number strictly less than
for large n and fixed
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
Generalization of the convex-hull-and-line traveling salesman problem
Two instances of the traveling salesman problem, on the same node set (1,2 n} but with different cost matrices C and C, are equivalent iff there exist {a, hi: -1, n} such that for any 1 _i, j _n, j, C(i, j) C(i,j) q-a -t-bj [7]. One ofthe well-solved special cases of the traveling salesman problem (TSP) is the convex-hull-and-line TSP. We extend the solution scheme for this class of TSP given in [9] to a more general class which is closed with respect to the above equivalence relation. The cost matrix in our general class is a certain composition of Kalmanson matrices. This gives a new, non-trivial solvable case of TSP
On Approximating Multi-Criteria TSP
We present approximation algorithms for almost all variants of the
multi-criteria traveling salesman problem (TSP).
First, we devise randomized approximation algorithms for multi-criteria
maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP,
where the edge weights have to be symmetric, we devise an algorithm with an
approximation ratio of 2/3 - eps. For multi-criteria Max-ATSP, where the edge
weights may be asymmetric, we present an algorithm with a ratio of 1/2 - eps.
Our algorithms work for any fixed number k of objectives. Furthermore, we
present a deterministic algorithm for bi-criteria Max-STSP that achieves an
approximation ratio of 7/27.
Finally, we present a randomized approximation algorithm for the asymmetric
multi-criteria minimum TSP with triangle inequality Min-ATSP. This algorithm
achieves a ratio of log n + eps.Comment: Preliminary version at STACS 2009. This paper is a revised full
version, where some proofs are simplifie
A Positive Semidefinite Approximation of the Symmetric Traveling Salesman Polytope
For a convex body B in a vector space V, we construct its approximation P_k,
k=1, 2, . . . using an intersection of a cone of positive semidefinite
quadratic forms with an affine subspace. We show that P_k is contained in B for
each k. When B is the Symmetric Traveling Salesman Polytope on n cities T_n, we
show that the scaling of P_k by n/k+ O(1/n) contains T_n for k no more than
n/2. Membership for P_k is computable in time polynomial in n (of degree linear
in k).
We discuss facets of T_n that lie on the boundary of P_k. We introduce a new
measure on each facet defining inequality for T_n in terms of the eigenvalues
of a quadratic form. Using these eigenvalues of facets, we show that the
scaling of P_1 by n^(1/2) has all of the facets of T_n defined by the subtour
elimination constraints either in its interior or lying on its boundary.Comment: 25 page
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