46 research outputs found
Distributed Algorithms for Scheduling on Line and Tree Networks
We have a set of processors (or agents) and a set of graph networks defined
over some vertex set. Each processor can access a subset of the graph networks.
Each processor has a demand specified as a pair of vertices , along
with a profit; the processor wishes to send data between and . Towards
that goal, the processor needs to select a graph network accessible to it and a
path connecting and within the selected network. The processor requires
exclusive access to the chosen path, in order to route the data. Thus, the
processors are competing for routes/channels. A feasible solution selects a
subset of demands and schedules each selected demand on a graph network
accessible to the processor owning the demand; the solution also specifies the
paths to use for this purpose. The requirement is that for any two demands
scheduled on the same graph network, their chosen paths must be edge disjoint.
The goal is to output a solution having the maximum aggregate profit. Prior
work has addressed the above problem in a distibuted setting for the special
case where all the graph networks are simply paths (i.e, line-networks).
Distributed constant factor approximation algorithms are known for this case.
The main contributions of this paper are twofold. First we design a
distributed constant factor approximation algorithm for the more general case
of tree-networks. The core component of our algorithm is a tree-decomposition
technique, which may be of independent interest. Secondly, for the case of
line-networks, we improve the known approximation guarantees by a factor of 5.
Our algorithms can also handle the capacitated scenario, wherein the demands
and edges have bandwidth requirements and capacities, respectively.Comment: Accepted to PODC 2012, full versio
Finding Irrefutable Certificates for S_2^p via Arthur and Merlin
We show that , where is the
symmetric alternation class and refers to the promise
version of the Arthur-Merlin class . This is derived as a
consequence of our main result that presents an
algorithm for finding a small set of ``collectively irrefutable
certificates\u27\u27 of a given -type matrix. The main result also
yields some new consequences of the hypothesis that has
polynomial size circuits. It is known that the above hypothesis
implies a collapse of the polynomial time hierarchy () to
(Cai 2007, K"obler and Watanabe 1998).
Under the same hypothesis, we show that collapses to
. We also describe an algorithm for learning
polynomial size circuits for , assuming such circuits exist.
For the same problem, the previously best known result was a
algorithm (Bshouty et al. 1996)
Distributed and Parallel Algorithms for Set Cover Problems with Small Neighborhood Covers
In this paper, we study a class of set cover problems that satisfy a special
property which we call the {\em small neighborhood cover} property. This class
encompasses several well-studied problems including vertex cover, interval
cover, bag interval cover and tree cover. We design unified distributed and
parallel algorithms that can handle any set cover problem falling under the
above framework and yield constant factor approximations. These algorithms run
in polylogarithmic communication rounds in the distributed setting and are in
NC, in the parallel setting.Comment: Full version of FSTTCS'13 pape
A Near-linear Time Constant Factor Algorithm for Unsplittable Flow Problem on Line with Bag Constraints
Consider a scenario where we need to schedule a set of jobs on a system offering some resource (such as electrical power or communication bandwidth), which we shall refer to as bandwidth. Each job consists of a set (or bag) of job instances. For each job instance, the input specifies the start time, finish time, bandwidth requirement and profit. The bandwidth offered by the system varies at different points of time and is specified as part of the input. A feasible solution is to choose a subset of instances such that at
any point of time, the sum of bandwidth requirements of the chosen instances does not exceed the bandwidth available at that point of time, and furthermore, at most one instance is picked from each job.
The goal is to find a maximum profit feasible solution. We study this problem under a natural assumption called the no-bottleneck assumption (NBA), wherein the bandwidth requirement of any job instance is at most the minimum bandwidth available. We present a simple, near-linear time constant factor approximation algorithm for this problem, under NBA. When each job consists of only one job instance, the above problem is the same as the well-studied unsplittable flow problem (UFP) on lines. A constant factor approximation algorithm is known for the UFP on line, under NBA.
Our result leads to an alternative constant factor approximation algorithm for this problem. Though the approximation ratio achieved by our algorithm is inferior, it is much simpler, deterministic and faster in comparison to the existing algorithms. Our algorithm runs in near-linear time (), whereas the running time of the known algorithms is a high order polynomial. The core idea behind our algorithm is a reduction from the varying bandwidth case to the easier uniform bandwidth case, using a technique that we call slicing
Finding Independent Sets in Unions of Perfect Graphs
The maximum independent set problem (MaxIS) on general graphs is known to be NP-hard to approximate within a factor of , for any . However, there are many ``easy" classes of graphs on which the problem can be solved in polynomial time. In this context, an interesting question is that of computing the maximum independent set in a graph that can be expressed as the union of a small number of graphs from an easy class. The MaxIS problem has been studied on unions of interval graphs and chordal graphs. We study the MaxIS problem on unions of perfect graphs (which generalize the above two classes). We present an -approximation algorithm when the input graph is the
union of two perfect graphs. We also show that the MaxIS problem on unions of two comparability graphs (a subclass of perfect graphs)
cannot be approximated within any constant factor
Replica Placement via Capacitated Vertex Cover
In this paper, we study the replica placement problem on trees and present a constant factor approximation algorithm (with an additional additive constant factor). This improves the best known previous algorithm having an approximation ratio dependent on the maximum degree of the tree. Our techniques also extend to the partial cover version. Our algorithms are based on the LP rounding technique. The core component of our algorithm exploits a connection between the natural LP solutions of the replica placement problem and the capacitated vertex cover problem
Replica Placement on Directed Acyclic Graphs
The replica placement problem has been well studied on trees. In this paper, we study this problem on directed acyclic graphs. The replica placement problem on general DAGs generalizes the set cover problem. We present a constant factor approximation algorithm for the special case of DAGs having bounded degree and bounded tree-width (BDBT-DAGs). We also present a constant factor approximation algorithm for DAGs composed of local BDBT-DAGs connected in a tree like manner (TBDBT-DAGs). The latter class of DAGs generalizes trees as well; we improve upon the previously best known approximation ratio for the problem on trees. Our algorithms are based on the LP rounding technique; the core component of our algorithm exploits the structural properties of tree-decompositions to massage the LP solution into an integral solution
Knapsack Cover Subject to a Matroid Constraint
We consider the Knapsack Covering problem subject to a matroid constraint. In this problem, we are given an universe U of n items where item i has attributes: a cost c(i) and a size s(i). We also have a demand D. We are also given a matroid M = (U, I) on the set U. A feasible solution S to the problem is one such that (i) the cumulative size of the items chosen is at least D, and (ii) the set S is independent in the matroid M (i.e. S is in I). The objective is to minimize the total cost of the items selected, sum_{i in S}c(i).
Our main result proves a 2-factor approximation for this problem.
The problem described above falls in the realm of mixed packing covering problems. We also consider packing extensions of certain other covering problems and prove that in such cases it is not possible to derive any constant factor pproximations