88,180 research outputs found
The parallel approximability of a subclass of quadratic programming
In this paper we deal with the parallel approximability of a special class of Quadratic Programming (QP), called Smooth Positive Quadratic Programming. This subclass of QP is obtained by imposing restrictions on the coefficients of the QP instance. The Smoothness condition restricts the magnitudes of the coefficients while the positiveness requires that all the coefficients be non-negative. Interestingly, even with these restrictions several combinatorial problems can be modeled by Smooth QP. We show NC Approximation Schemes for the instances of Smooth Positive QP. This is done by reducing the instance of QP to an instance of Positive Linear Programming, finding in NC an approximate fractional solution to the obtained program, and then rounding the fractional solution to an integer approximate solution for the original problem. Then we show how to extend the result for positive instances of bounded degree to Smooth Integer Programming problems. Finally, we formulate several important combinatorial problems as Positive Quadratic Programs (or Positive Integer Programs) in packing/covering form and show that the techniques presented can be used to obtain NC Approximation Schemes for "dense" instances of such problems.Peer ReviewedPostprint (published version
A Parallelizable Acceleration Framework for Packing Linear Programs
This paper presents an acceleration framework for packing linear programming
problems where the amount of data available is limited, i.e., where the number
of constraints m is small compared to the variable dimension n. The framework
can be used as a black box to speed up linear programming solvers dramatically,
by two orders of magnitude in our experiments. We present worst-case guarantees
on the quality of the solution and the speedup provided by the algorithm,
showing that the framework provides an approximately optimal solution while
running the original solver on a much smaller problem. The framework can be
used to accelerate exact solvers, approximate solvers, and parallel/distributed
solvers. Further, it can be used for both linear programs and integer linear
programs
Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver
We study the design of polylogarithmic depth algorithms for approximately
solving packing and covering semidefinite programs (or positive SDPs for
short). This is a natural SDP generalization of the well-studied positive LP
problem.
Although positive LPs can be solved in polylogarithmic depth while using only
parallelizable iterations, the best known
positive SDP solvers due to Jain and Yao require parallelizable iterations. Several alternative solvers have
been proposed to reduce the exponents in the number of iterations. However, the
correctness of the convergence analyses in these works has been called into
question, as they both rely on algebraic monotonicity properties that do not
generalize to matrix algebra.
In this paper, we propose a very simple algorithm based on the optimization
framework proposed for LP solvers. Our algorithm only needs iterations, matching that of the best LP solver. To surmount
the obstacles encountered by previous approaches, our analysis requires a new
matrix inequality that extends Lieb-Thirring's inequality, and a
sign-consistent, randomized variant of the gradient truncation technique
proposed in
Nearly Linear-Work Algorithms for Mixed Packing/Covering and Facility-Location Linear Programs
We describe the first nearly linear-time approximation algorithms for
explicitly given mixed packing/covering linear programs, and for (non-metric)
fractional facility location. We also describe the first parallel algorithms
requiring only near-linear total work and finishing in polylog time. The
algorithms compute -approximate solutions in time (and work)
, where is the number of non-zeros in the constraint
matrix. For facility location, is the number of eligible client/facility
pairs
The Network Improvement Problem for Equilibrium Routing
In routing games, agents pick their routes through a network to minimize
their own delay. A primary concern for the network designer in routing games is
the average agent delay at equilibrium. A number of methods to control this
average delay have received substantial attention, including network tolls,
Stackelberg routing, and edge removal.
A related approach with arguably greater practical relevance is that of
making investments in improvements to the edges of the network, so that, for a
given investment budget, the average delay at equilibrium in the improved
network is minimized. This problem has received considerable attention in the
literature on transportation research and a number of different algorithms have
been studied. To our knowledge, none of this work gives guarantees on the
output quality of any polynomial-time algorithm. We study a model for this
problem introduced in transportation research literature, and present both
hardness results and algorithms that obtain nearly optimal performance
guarantees.
- We first show that a simple algorithm obtains good approximation guarantees
for the problem. Despite its simplicity, we show that for affine delays the
approximation ratio of 4/3 obtained by the algorithm cannot be improved.
- To obtain better results, we then consider restricted topologies. For
graphs consisting of parallel paths with affine delay functions we give an
optimal algorithm. However, for graphs that consist of a series of parallel
links, we show the problem is weakly NP-hard.
- Finally, we consider the problem in series-parallel graphs, and give an
FPTAS for this case.
Our work thus formalizes the intuition held by transportation researchers
that the network improvement problem is hard, and presents topology-dependent
algorithms that have provably tight approximation guarantees.Comment: 27 pages (including abstract), 3 figure
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