70 research outputs found
Algorithms to Approximate Column-Sparse Packing Problems
Column-sparse packing problems arise in several contexts in both
deterministic and stochastic discrete optimization. We present two unifying
ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain
improved approximation algorithms for some well-known families of such
problems. As three main examples, we attain the integrality gap, up to
lower-order terms, for known LP relaxations for k-column sparse packing integer
programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set
packing (Bansal et al., Algorithmica, 2012), and go "half the remaining
distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and
Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM
Transactions of Algorithm
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
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
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
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
On k-Column Sparse Packing Programs
We consider the class of packing integer programs (PIPs) that are column
sparse, i.e. there is a specified upper bound k on the number of constraints
that each variable appears in. We give an (ek+o(k))-approximation algorithm for
k-column sparse PIPs, improving on recent results of and
. We also show that the integrality gap of our linear programming
relaxation is at least 2k-1; it is known that k-column sparse PIPs are
-hard to approximate. We also extend our result (at the loss
of a small constant factor) to the more general case of maximizing a submodular
objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail
- …