125 research outputs found
Approximability of the Unsplittable Flow Problem on Trees
We consider the approximability of the Unsplittable Flow Problem (UFP) on tree graphs, and give a deterministic quasi-polynomial time approximation scheme for the problem when the number of leaves in the tree graph is at most poly-logarithmic in (the number of demands), and when all edge capacities and resource requirements are suitably bounded. Our algorithm generalizes a recent technique that obtained the first such approximation scheme for line graphs. Our results show that the problem is not APX-hard for such graphs unless NP \subseteq DTIME(2^{polylog(n)}). Further, a reduction from the Demand Matching Problem shows that UFP is APX-hard when the number of leaves is Omega(n^\epsilon) for any constant \epsilon \u3e 0. Together, the two results give a nearly tight characterization of the approximability of the problem on tree graphs in terms of the number of leaves, and show the structure of the graph that results in hardness of approximation
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
From Electrical Power Flows to Unsplittabe Flows: A QPTAS for OPF with Discrete Demands in Line Distribution Networks
The {\it AC Optimal Power Flow} (OPF) problem is a fundamental problem in
power systems engineering which has been known for decades. It is a notoriously
hard problem due mainly to two reasons: (1) non-convexity of the power flow
constraints and (2) the (possible) existence of discrete power injection
constraints. Recently, sufficient conditions were provided for certain convex
relaxations of OPF to be exact in the continuous case, thus allowing one to
partially address the issue of non-convexity. In this paper we make a first
step towards addressing the combinatorial issue. Namely, by establishing a
connection to the well-known {\it unsplittable flow problem} (UFP), we are able
to generalize known techniques for the latter problem to provide approximation
algorithms for OPF with discrete demands. As an application, we give a
quasi-polynomial time approximation scheme for OPF in line networks under some
mild assumptions and a single generation source. We believe that this
connection can be further leveraged to obtain approximation algorithms for more
general settings, such as multiple generation sources and tree networks
Stochastic Unsplittable Flows
We consider the stochastic unsplittable flow problem: given a graph with edge-capacities, and source-sink pairs with each pair having a size and a value, the goal is to route the pairs unsplittably while respecting edge capacities to maximize the total value of the routed pairs. However, the size of each pair is a random variable and is revealed only after we decide to route that pair. Which pairs should we route, along which paths, and in what order so as to maximize the expected value?
We present results for several cases of the problem under the no-bottleneck assumption. We show a logarithmic approximation algorithm for the single-sink problem on general graphs, considerably improving on the prior results of Chawla and Roughgarden which worked for planar graphs. We present an approximation to the stochastic unsplittable flow problem on directed acyclic graphs, within less than a logarithmic factor of the best known approximation in the non-stochastic setting. We present a non-adaptive strategy on trees that is within a constant factor of the best adaptive strategy, asymptotically matching the best results for the non-stochastic unsplittable flow problem on trees. Finally, we give results for the stochastic unsplittable flow problem on general graphs.
Our techniques include using edge-confluent flows for the single-sink problem in order to control the interaction between flow-paths, and a reduction from general scheduling policies to "safe" ones (i.e., those guaranteeing no capacity violations), which may be of broader interest
Improved Algorithms for Scheduling Unsplittable Flows on Paths
In this paper, we investigate offline and online algorithms for Round-UFPP, the problem of minimizing the number of rounds required to schedule a set of unsplittable flows of non-uniform sizes
on a given path with non-uniform edge capacities. Round-UFPP is NP-hard and constant-factor approximation algorithms are known under the no bottleneck assumption (NBA), which stipulates that maximum size of a flow is at most the minimum edge capacity. We study Round-UFPP
without the NBA, and present improved online and offline algorithms. We first study offline Round-UFPP for a restricted class of instances called alpha-small, where the size of each flow is at
most alpha times the capacity of its bottleneck edge, and present an O(log(1/(1 - alpha)))-approximation algorithm. Our main result is an online O(log log cmax)-competitive algorithm for Round-UFPP
for general instances, where cmax is the largest edge capacities, improving upon the previous best bound of O(log cmax) due to [16]. Our result leads to an offline O(min(log n, log m, log log cmax))-
approximation algorithm and an online O(min(log m, log log cmax))-competitive algorithm for Round-UFPP, where n is the number of flows and m is the number of edges
Packing Cars into Narrow Roads: PTASs for Limited Supply Highway
In the Highway problem, we are given a path with n edges (the highway), and a set of m drivers, each one characterized by a subpath and a budget. For a given assignment of edge prices (the tolls), the highway owner collects from each driver the total price of the associated path when it does not exceed drivers\u27s budget, and zero otherwise. The goal is to choose the prices to maximize the total profit. A PTAS is known for this (strongly NP-hard) problem [Grandoni,Rothvoss-SODA\u2711, SICOMP\u2716].
In this paper we study the limited supply generalization of Highway, that incorporates capacity constraints. Here the input also includes a capacity u_e >= 0 for each edge e; we need to select, among drivers that can afford the required price, a subset such that the number of drivers that use each edge e is at most u_e (and we get profit only from selected drivers). To the best of our knowledge, the only approximation algorithm known for this problem is a folklore O(log m) approximation based on a reduction to the related Unsplittable Flow on a Path problem (UFP). The main result of this paper is a PTAS for limited supply highway.
As a second contribution, we study a natural generalization of the problem where each driver i demands a different amount d_i of capacity. Using known techniques, it is not hard to derive a QPTAS for this problem. Here we present a PTAS for the case that drivers have uniform budgets. Finding a PTAS for non-uniform-demand limited supply highway is left as a challenging open problem
Cluster Before You Hallucinate: Approximating Node-Capacitated Network Design and Energy Efficient Routing
We consider circuit routing with an objective of minimizing energy, in a
network of routers that are speed scalable and that may be shutdown when idle.
We consider both multicast routing and unicast routing. It is known that this
energy minimization problem can be reduced to a capacitated flow network design
problem, where vertices have a common capacity but arbitrary costs, and the
goal is to choose a minimum cost collection of vertices whose induced subgraph
will support the specified flow requirements. For the multicast (single-sink)
capacitated design problem we give a polynomial-time algorithm that is
O(log^3n)-approximate with O(log^4 n) congestion. This translates back to a
O(log ^(4{\alpha}+3) n)-approximation for the multicast energy-minimization
routing problem, where {\alpha} is the polynomial exponent in the dynamic power
used by a router. For the unicast (multicommodity) capacitated design problem
we give a polynomial-time algorithm that is O(log^5 n)-approximate with
O(log^12 n) congestion, which translates back to a O(log^(12{\alpha}+5)
n)-approximation for the unicast energy-minimization routing problem.Comment: 22 pages (full version of STOC 2014 paper
Prizing on Paths: A PTAS for the Highway Problem
In the highway problem, we are given an n-edge line graph (the highway), and
a set of paths (the drivers), each one with its own budget. For a given
assignment of edge weights (the tolls), the highway owner collects from each
driver the weight of the associated path, when it does not exceed the budget of
the driver, and zero otherwise. The goal is choosing weights so as to maximize
the profit.
A lot of research has been devoted to this apparently simple problem. The
highway problem was shown to be strongly NP-hard only recently
[Elbassioni,Raman,Ray-'09]. The best-known approximation is O(\log n/\log\log
n) [Gamzu,Segev-'10], which improves on the previous-best O(\log n)
approximation [Balcan,Blum-'06].
In this paper we present a PTAS for the highway problem, hence closing the
complexity status of the problem. Our result is based on a novel randomized
dissection approach, which has some points in common with Arora's quadtree
dissection for Euclidean network design [Arora-'98]. The basic idea is
enclosing the highway in a bounding path, such that both the size of the
bounding path and the position of the highway in it are random variables. Then
we consider a recursive O(1)-ary dissection of the bounding path, in subpaths
of uniform optimal weight. Since the optimal weights are unknown, we construct
the dissection in a bottom-up fashion via dynamic programming, while computing
the approximate solution at the same time. Our algorithm can be easily
derandomized. We demonstrate the versatility of our technique by presenting
PTASs for two variants of the highway problem: the tollbooth problem with a
constant number of leaves and the maximum-feasibility subsystem problem on
interval matrices. In both cases the previous best approximation factors are
polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09]
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