431 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
Robust Branch-Cut-and-Price for the Capacitated Minimum Spanning Tree Problem over a Large Extended Formulation
This paper presents a robust branch-cut-and-price algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to q-arbs, a structure that arises from a relaxation of the capacitated prize-collecting arbores- cence problem in order to make it solvable in pseudo-polynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the OR-Library show very signi¯cant improvements over previous algorithms. Several open instances could be solved to optimalityNo keywords;
On vertex adjacencies in the polytope of pyramidal tours with step-backs
We consider the traveling salesperson problem in a directed graph. The
pyramidal tours with step-backs are a special class of Hamiltonian cycles for
which the traveling salesperson problem is solved by dynamic programming in
polynomial time. The polytope of pyramidal tours with step-backs is
defined as the convex hull of the characteristic vectors of all possible
pyramidal tours with step-backs in a complete directed graph. The skeleton of
is the graph whose vertex set is the vertex set of and the
edge set is the set of geometric edges or one-dimensional faces of .
The main result of the paper is a necessary and sufficient condition for vertex
adjacencies in the skeleton of the polytope that can be verified in
polynomial time.Comment: in Englis
Survivable Networks, Linear Programming Relaxations and the Parsimonious Property
We consider the survivable network design problem - the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the k-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worstcase analyses of two heuristics for the survivable network design problem
Degree-bounded generalized polymatroids and approximating the metric many-visits TSP
In the Bounded Degree Matroid Basis Problem, we are given a matroid and a
hypergraph on the same ground set, together with costs for the elements of that
set as well as lower and upper bounds and for
each hyperedge . The objective is to find a minimum-cost basis
such that for
each hyperedge . Kir\'aly et al. (Combinatorica, 2012) provided an
algorithm that finds a basis of cost at most the optimum value which violates
the lower and upper bounds by at most , where is the
maximum degree of the hypergraph. When only lower or only upper bounds are
present for each hyperedge, this additive error is decreased to .
We consider an extension of the matroid basis problem to generalized
polymatroids, or g-polymatroids, and additionally allow element multiplicities.
The Bounded Degree g-polymatroid Element Problem with Multiplicities takes as
input a g-polymatroid instead of a matroid, and besides the lower and
upper bounds, each hyperedge has element multiplicities
. Building on the approach of Kir\'aly et al., we provide an
algorithm for finding a solution of cost at most the optimum value, having the
same additive approximation guarantee.
As an application, we develop a -approximation for the metric
Many-Visits TSP, where the goal is to find a minimum-cost tour that visits each
city a positive number of times. Our approach combines our algorithm
for the Bounded Degree g-polymatroid Element Problem with Multiplicities with
the principle of Christofides' algorithm from 1976 for the (single-visit)
metric TSP, whose approximation guarantee it matches.Comment: 17 page
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