16 research outputs found
Max-weight integral multicommodity flow in spiders and high-capacity trees
Abstract. We consider the max-weight integer multicommodity flow problem in trees. In this problem we are given an edge-capacitated tree and weighted pairs of terminals, and the objective is to find a max-weight integral flow between terminal pairs subject to the capacities. This problem was shown to be APX-hard by Garg, Vazirani and Yannakakis [Algorithmica, 1997], and a 4-approximation was given by Chekuri, Mydlarz and Shepherd [ACM Trans. Alg., 2007]. Some special cases are known to be solvable in polynomial time, including when the graph is a star (via b-matching) or a path. First, when every edge has capacity at least µ ≥ 2, we use iterated relaxation to obtain an improved approximation ratio of min{3, 1+4/µ+6/(µ 2 −µ)}. We show this ratio bounds the integrality gap of the natural LP relaxation. A complementary hardness result yields a 1+Θ(1/µ) threshold of approximability (if P ̸ = NP). Second, we extend the range of instances for which exact solutions can be found efficiently. When the tree is a spider (i.e. if only one vertex has degree greater than 2) we give a polynomial-time algorithm to find an optimal solution, as well as a polyhedral description of the integer hull of all In the max-weight integral multicommodity flow problem (WMCF), we are given an undirected supply graph G = (V,E), terminal pairs (s1,t1),...,(sk,tk) where si,ti ∈ V, non-negative weight
A Quasi-PTAS for Unsplittable Flow on Line Graphs
We study the Unsplittable Flow Problem (UFP) on a line graph, focusing on the long-standing open question of whether the problem is APX-hard. We describe a deterministic quasi-polynomial time approximation scheme for UFP on line graphs, thereby ruling out an APX-hardness result, unless NP is contained in DTIME(2^polylog(n)). Our result requires a quasi-polynomial bound on all edge capacities and demands in the input instance. Earlier results on this problem included a polynomial time (2+epsilon)-approximation under the assumption that no demand exceeds any edge capacity (the no-bottleneck assumption ) and a super-constant integrality gap if this assumption did not hold. Unlike most earlier work on UFP, our results do not require a no-bottleneck assumption
A Constant Factor Approximation Algorithm for Unsplittable Flow on Paths
In the unsplittable flow problem on a path, we are given a capacitated path
and tasks, each task having a demand, a profit, and start and end
vertices. The goal is to compute a maximum profit set of tasks, such that for
each edge of , the total demand of selected tasks that use does not
exceed the capacity of . This is a well-studied problem that has been
studied under alternative names, such as resource allocation, bandwidth
allocation, resource constrained scheduling, temporal knapsack and interval
packing.
We present a polynomial time constant-factor approximation algorithm for this
problem. This improves on the previous best known approximation ratio of
. The approximation ratio of our algorithm is for any
.
We introduce several novel algorithmic techniques, which might be of
independent interest: a framework which reduces the problem to instances with a
bounded range of capacities, and a new geometrically inspired dynamic program
which solves a special case of the maximum weight independent set of rectangles
problem to optimality. In the setting of resource augmentation, wherein the
capacities can be slightly violated, we give a -approximation
algorithm. In addition, we show that the problem is strongly NP-hard even if
all edge capacities are equal and all demands are either~1,~2, or~3.Comment: 37 pages, 5 figures Version 2 contains the same results as version 1,
but the presentation has been greatly revised and improved. References have
been adde
A Quasi-PTAS for Unsplittable Flow on Line Graphs
We study the Unsplittable Flow Problem (UFP) on a line graph, focusing on the long-standing open question of whether the problem is APX-hard. We describe a deterministic quasi-polynomial time approximation scheme for UFP on line graphs, thereby ruling out an APX-hardness result, unless NP is contained in DTIME(2^polylog(n)). Our result requires a quasi-polynomial bound on all edge capacities and demands in the input instance. Earlier results on this problem included a polynomial time (2+epsilon)-approximation under the assumption that no demand exceeds any edge capacity (the no-bottleneck assumption ) and a super-constant integrality gap if this assumption did not hold. Unlike most earlier work on UFP, our results do not require a no-bottleneck assumption
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