973 research outputs found
Recommended from our members
On a linear program for minimum-weight triangulation
Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial-time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: For an LP formulation due to Dantzig, Hoffman, and Hu [Math. Programming, 31 (1985), pp. 1-14], (i) the integrality gap is constant, and (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP. © 2014 Society for Industrial and Applied Mathematics
On a Linear Program for Minimum-Weight Triangulation
Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time
constant-factor approximation algorithm, and a variety of effective polynomial-
time heuristics that, for many instances, can find the exact MWT. Linear
programs (LPs) for MWT are well-studied, but previously no connection was known
between any LP and any approximation algorithm or heuristic for MWT. Here we
show the first such connections: for an LP formulation due to Dantzig et al.
(1985): (i) the integrality gap is bounded by a constant; (ii) given any
instance, if the aforementioned heuristics find the MWT, then so does the LP.Comment: To appear in SICOMP. Extended abstract appeared in SODA 201
Hitting Diamonds and Growing Cacti
We consider the following NP-hard problem: in a weighted graph, find a
minimum cost set of vertices whose removal leaves a graph in which no two
cycles share an edge. We obtain a constant-factor approximation algorithm,
based on the primal-dual method. Moreover, we show that the integrality gap of
the natural LP relaxation of the problem is \Theta(\log n), where n denotes the
number of vertices in the graph.Comment: v2: several minor changes
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
Approximation algorithms for Capacitated Facility Location Problem with Penalties
In this paper, we address the problem of capacitated facility location
problem with penalties (CapFLPP) paid per unit of unserved demand. In case of
uncapacitated FLP with penalties demands of a client are either entirely met or
are entirely rejected and penalty is paid. In the uncapacitated case, there is
no reason to serve a client partially. Whereas, in case of CapFLPP, it may be
beneficial to serve a client partially instead of not serving at all and, pay
the penalty for the unmet demand. Charikar et. al.
\cite{charikar2001algorithms}, Jain et. al. \cite{jain2003greedy} and Xu- Xu
\cite{xu2009improved} gave , and approximation, respectively,
for the uncapacitated case . We present factor for the case
of uniform capacities and factor for non-uniform
capacities
A LP approximation for the Tree Augmentation Problem
In the Tree Augmentation Problem (TAP) the goal is to augment a tree by a
minimum size edge set from a given edge set such that is
-edge-connected. The best approximation ratio known for TAP is . In the
more general Weighted TAP problem, should be of minimum weight. Weighted
TAP admits several -approximation algorithms w.r.t. to the standard cut
LP-relaxation, but for all of them the performance ratio of is tight even
for TAP. The problem is equivalent to the problem of covering a laminar set
family. Laminar set families play an important role in the design of
approximation algorithms for connectivity network design problems. In fact,
Weighted TAP is the simplest connectivity network design problem for which a
ratio better than is not known. Improving this "natural" ratio is a major
open problem, which may have implications on many other network design
problems. It seems that achieving this goal requires finding an LP-relaxation
with integrality gap better than , which is a long time open problem even
for TAP. In this paper we introduce such an LP-relaxation and give an algorithm
that computes a feasible solution for TAP of size at most times the
optimal LP value. This gives some hope to break the ratio for the weighted
case. Our algorithm computes some initial edge set by solving a partial system
of constraints that form the integral edge-cover polytope, and then applies
local search on -leaf subtrees to exchange some of the edges and to add
additional edges. Thus we do not need to solve the LP, and the algorithm runs
roughly in time required to find a minimum weight edge-cover in a general
graph.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0279
- …