68 research outputs found
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 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
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
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]
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
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
How Unsplittable-Flow-Covering helps Scheduling with Job-Dependent Cost Functions
Generalizing many well-known and natural scheduling problems, scheduling with
job-specific cost functions has gained a lot of attention recently. In this
setting, each job incurs a cost depending on its completion time, given by a
private cost function, and one seeks to schedule the jobs to minimize the total
sum of these costs. The framework captures many important scheduling objectives
such as weighted flow time or weighted tardiness. Still, the general case as
well as the mentioned special cases are far from being very well understood
yet, even for only one machine. Aiming for better general understanding of this
problem, in this paper we focus on the case of uniform job release dates on one
machine for which the state of the art is a 4-approximation algorithm. This is
true even for a special case that is equivalent to the covering version of the
well-studied and prominent unsplittable flow on a path problem, which is
interesting in its own right. For that covering problem, we present a
quasi-polynomial time -approximation algorithm that yields an
-approximation for the above scheduling problem. Moreover, for
the latter we devise the best possible resource augmentation result regarding
speed: a polynomial time algorithm which computes a solution with \emph{optimal
}cost at speedup. Finally, we present an elegant QPTAS for the
special case where the cost functions of the jobs fall into at most
many classes. This algorithm allows the jobs even to have up to many
distinct release dates.Comment: 2 pages, 1 figur
A mazing 2+ε approximation for unsplittable flow on a path
We study the problem of unsplittable flow on a path (UFP), which arises naturally in many applications such as bandwidth allocation, job scheduling, and caching. Here we are given a path with nonnegative edge capacities and a set of tasks, which are characterized by a subpath, a demand, and a profit. The goal is to find the most profitable subset of tasks whose total demand does not violate the edge capacities. Not surprisingly, this problem has received a lot of attention in the research community. If the demand of each task is at most a small-enough fraction δ of the capacity along its subpath (δ-small tasks), then it has been known for a long time [Chekuri et al., ICALP 2003] how to compute a solution of value arbitrarily close to the optimum via LP rounding. However, much remains unknown for the complementary case, that is, when the demand of each task is at least some fraction δ > 0 of the smallest capacity of its subpath (δ-large tasks). For this setting, a constant factor approximation is known, improving on an earlier logarithmic approximation [Bonsma et al., FOCS 2011]. In this article, we present a polynomial-time approximation scheme (PTAS) for δ-large tasks, for any constant δ > 0. Key to this result is a complex geometrically inspired dynamic program. Each task is represented as a segment underneath the capacity curve, and we identify a proper maze-like structure so that each corridor of the maze is crossed by only O(1) tasks in the optimal solution. The maze has a tree topology, which guides our dynamic program. Our result implies a 2 + ε approximation for UFP, for any constant ε > 0, improving on the previously best 7 + ε approximation by Bonsma et al. We remark that our improved approximation algorithm matches the best known approximation ratio for the considerably easier special case of uniform edge capacities
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
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