29 research outputs found
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
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
Algorithms and Adaptivity Gaps for Stochastic k-TSP
Given a metric and a , the classic
\textsf{k-TSP} problem is to find a tour originating at the
of minimum length that visits at least nodes in . In this work,
motivated by applications where the input to an optimization problem is
uncertain, we study two stochastic versions of \textsf{k-TSP}.
In Stoch-Reward -TSP, originally defined by Ene-Nagarajan-Saket [ENS17],
each vertex in the given metric contains a stochastic reward .
The goal is to adaptively find a tour of minimum expected length that collects
at least reward ; here "adaptively" means our next decision may depend on
previous outcomes. Ene et al. give an -approximation adaptive
algorithm for this problem, and left open if there is an -approximation
algorithm. We totally resolve their open question and even give an
-approximation \emph{non-adaptive} algorithm for this problem.
We also introduce and obtain similar results for the Stoch-Cost -TSP
problem. In this problem each vertex has a stochastic cost , and the
goal is to visit and select at least vertices to minimize the expected
\emph{sum} of tour length and cost of selected vertices. This problem
generalizes the Price of Information framework [Singla18] from deterministic
probing costs to metric probing costs.
Our techniques are based on two crucial ideas: "repetitions" and "critical
scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for
our problems, if we truncate the random variables at an ideal threshold and
repeat, then their expected values form a good surrogate. Unfortunately, this
ideal threshold is adaptive as it depends on how far we are from achieving our
target , so we truncate at various different scales and identify a
"critical" scale.Comment: ITCS 202
An approximate solution method based on tabu search for k-minimum spanning tree problems
This paper considers k-minimum spanning tree
problems. An existing solution algorithm based on tabu search, which was proposed by Katagiri et al., includes an iterative solving procedure of minimum spanning tree (MST) problems for subgraphs to obtain a local optimal solution of k-minimum spanning tree problems. This article provides a new tabu-searchbased approximate solution method that does not iteratively solve minimum spanning tree problems. Results of numerical experiments show that the proposed method provides a good performance in terms of accuracy over those of existing methods for relatively high cardinality k
Multi-source spanning trees: algorithms for minimizing source eccentricities
AbstractWe present two efficient algorithms constructing a spanning tree with minimum eccentricity of a source, for a given graph with weighted edges and a set of source vertices. The first algorithm is both simpler to implement and faster of the two. The second approach involves enumerating single-source shortest-path spanning trees for all points on a graph, a technique that may be useful in solving other problems
Constraint Programming for Flexible Service Function Chaining Deployment
Network Function Virtualization (NFV) and Software Defined Networking (SDN) are technologies that recently acquired a great momentum thanks to their promise of being a flexible and cost-effective solution for replacing hardware-based, vendor-dependent network middleboxes with software appliances running on general purpose hardware in the cloud. Delivering end-to-end networking services across multiple NFV/SDN network domains by implementing the so-called Service Function Chain (SFC) i.e., a sequence of Virtual Network Functions (VNF) that composes the service, is a challenging task. In this paper we address two crucial sub-problems of this task: i) the language to formalize the request of a given SFC to the network and ii) the solution of the SFC design problem, once the request is received. As for i) in our solution the request is built upon the intent-based approach, with a syntax that focuses on asking the user what she needs and not how it should be implemented, in a simple and high level language. Concerning ii) we define a formal model describing network architectures and VNF properties that is then used to solve the SFC design problem by means of Constraint Programming (CP), a programming paradigm which is often used in Artificial Intelligence applications. We argue that CP can be effectively used to address this kind of problems because it provides very expressive and flexible modeling languages which come with powerful solvers, thus providing efficient and scalable performance. We substantiate this claim by validating our tool on some typical and non trivial SFC design problems