91 research outputs found
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
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
Randomized rounding algorithms for large scale unsplittable flow problems
Unsplittable flow problems cover a wide range of telecommunication and
transportation problems and their efficient resolution is key to a number of
applications. In this work, we study algorithms that can scale up to large
graphs and important numbers of commodities. We present and analyze in detail a
heuristic based on the linear relaxation of the problem and randomized
rounding. We provide empirical evidence that this approach is competitive with
state-of-the-art resolution methods either by its scaling performance or by the
quality of its solutions. We provide a variation of the heuristic which has the
same approximation factor as the state-of-the-art approximation algorithm. We
also derive a tighter analysis for the approximation factor of both the
variation and the state-of-the-art algorithm. We introduce a new objective
function for the unsplittable flow problem and discuss its differences with the
classical congestion objective function. Finally, we discuss the gap in
practical performance and theoretical guarantees between all the aforementioned
algorithms
Sherali-Adams gaps, flow-cover inequalities and generalized configurations for capacity-constrained Facility Location
Metric facility location is a well-studied problem for which linear
programming methods have been used with great success in deriving approximation
algorithms. The capacity-constrained generalizations, such as capacitated
facility location (CFL) and lower-bounded facility location (LBFL), have proved
notorious as far as LP-based approximation is concerned: while there are
local-search-based constant-factor approximations, there is no known linear
relaxation with constant integrality gap. According to Williamson and Shmoys
devising a relaxation-based approximation for \cfl\ is among the top 10 open
problems in approximation algorithms.
This paper advances significantly the state-of-the-art on the effectiveness
of linear programming for capacity-constrained facility location through a host
of impossibility results for both CFL and LBFL. We show that the relaxations
obtained from the natural LP at levels of the Sherali-Adams
hierarchy have an unbounded gap, partially answering an open question of
\cite{LiS13, AnBS13}. Here, denotes the number of facilities in the
instance. Building on the ideas for this result, we prove that the standard CFL
relaxation enriched with the generalized flow-cover valid inequalities
\cite{AardalPW95} has also an unbounded gap. This disproves a long-standing
conjecture of \cite{LeviSS12}. We finally introduce the family of proper
relaxations which generalizes to its logical extreme the classic star
relaxation and captures general configuration-style LPs. We characterize the
behavior of proper relaxations for CFL and LBFL through a sharp threshold
phenomenon.Comment: arXiv admin note: substantial text overlap with arXiv:1305.599
Models for the piecewise linear unsplittable multicommodity flow problems
International audienceIn this paper, we consider multicommodity flow problems, with unsplit-table flows and piecewise linear routing costs. We first focus on the case where the piecewise linear routing costs are convex. We show that this problem is N P-hard for the general case, but polynomially solvable when there is only one commodity. We then propose a strengthened mixed-integer programming formulation for the problem. We show that the linear relaxation of this formulation always gives the optimal solution of the problem for the single commodity case. We present a wide array of computational experiments, showing this formulation also produces very tight linear programming bounds for the multi-commodity case. Finally, we also adapt our formulation for the non-convex case. Our experimental results imply that the linear programming bounds for this case, are only slightly than the ones of state-of-the-art models for the splittable flow version of the problem
FPT and FPT-Approximation Algorithms for Unsplittable Flow on Trees
We study the unsplittable flow on trees (UFT) problem in which we are given a tree with capacities on its edges and a set of tasks, where each task is described by a path and a demand. Our goal is to select a subset of the given tasks of maximum size such that the demands of the selected tasks respect the edge capacities. The problem models throughput maximization in tree networks. The best known approximation ratio for (unweighted) UFT is O(log n). We study the problem under the angle of FPT and FPT-approximation algorithms. We prove that
- UFT is FPT if the parameters are the cardinality k of the desired solution and the number of different task demands in the input,
- UFT is FPT under (1+?)-resource augmentation of the edge capacities for parameters k and 1/?, and
- UFT admits an FPT-5-approximation algorithm for parameter k. One key to our results is to compute structured hitting sets of the input edges which partition the given tree into O(k) clean components. This allows us to guess important properties of the optimal solution. Also, in some settings we can compute core sets of subsets of tasks out of which at least one task i is contained in the optimal solution. These sets have bounded size, and hence we can guess this task i easily.
A consequence of our results is that the integral multicommodity flow problem on trees is FPT if the parameter is the desired amount of sent flow. Also, even under (1+?)-resource augmentation UFT is APX-hard, and hence our FPT-approximation algorithm for this setting breaks this boundary
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