23,185 research outputs found
Approximation Algorithms for Route Planning with Nonlinear Objectives
We consider optimal route planning when the objective function is a general
nonlinear and non-monotonic function. Such an objective models user behavior
more accurately, for example, when a user is risk-averse, or the utility
function needs to capture a penalty for early arrival. It is known that as
nonlinearity arises, the problem becomes NP-hard and little is known about
computing optimal solutions when in addition there is no monotonicity
guarantee. We show that an approximately optimal non-simple path can be
efficiently computed under some natural constraints. In particular, we provide
a fully polynomial approximation scheme under hop constraints. Our
approximation algorithm can extend to run in pseudo-polynomial time under a
more general linear constraint that sometimes is useful. As a by-product, we
show that our algorithm can be applied to the problem of finding a path that is
most likely to be on time for a given deadline.Comment: 9 pages, 2 figures, main part of this paper is to be appear in
AAAI'1
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
On the complexity of the Whitehead minimization problem
The Whitehead minimization problem consists in finding a minimum size element
in the automorphic orbit of a word, a cyclic word or a finitely generated
subgroup in a finite rank free group. We give the first fully polynomial
algorithm to solve this problem, that is, an algorithm that is polynomial both
in the length of the input word and in the rank of the free group. Earlier
algorithms had an exponential dependency in the rank of the free group. It
follows that the primitivity problem -- to decide whether a word is an element
of some basis of the free group -- and the free factor problem can also be
solved in polynomial time.Comment: v.2: Corrected minor typos and mistakes, improved the proof of the
main technical lemma (Statement 2.4); added a section of open problems. 30
page
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
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