454 research outputs found
Separable Concave Optimization Approximately Equals Piecewise-Linear Optimization
We study the problem of minimizing a nonnegative separable concave function
over a compact feasible set. We approximate this problem to within a factor of
1+epsilon by a piecewise-linear minimization problem over the same feasible
set. Our main result is that when the feasible set is a polyhedron, the number
of resulting pieces is polynomial in the input size of the polyhedron and
linear in 1/epsilon. For many practical concave cost problems, the resulting
piecewise-linear cost problem can be formulated as a well-studied discrete
optimization problem. As a result, a variety of polynomial-time exact
algorithms, approximation algorithms, and polynomial-time heuristics for
discrete optimization problems immediately yield fully polynomial-time
approximation schemes, approximation algorithms, and polynomial-time heuristics
for the corresponding concave cost problems.
We illustrate our approach on two problems. For the concave cost
multicommodity flow problem, we devise a new heuristic and study its
performance using computational experiments. We are able to approximately solve
significantly larger test instances than previously possible, and obtain
solutions on average within 4.27% of optimality. For the concave cost facility
location problem, we obtain a new 1.4991+epsilon approximation algorithm.Comment: Full pape
Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows
The maximum multicommodity flow problem is a natural generalization of the
maximum flow problem to route multiple distinct flows. Obtaining a
approximation to the multicommodity flow problem on graphs is a well-studied
problem. In this paper we present an adaptation of recent advances in
single-commodity flow algorithms to this problem. As the underlying linear
systems in the electrical problems of multicommodity flow problems are no
longer Laplacians, our approach is tailored to generate specialized systems
which can be preconditioned and solved efficiently using Laplacians. Given an
undirected graph with m edges and k commodities, we give algorithms that find
approximate solutions to the maximum concurrent flow problem and
the maximum weighted multicommodity flow problem in time
\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))
Solving nonlinear multicommodity flow problems by the analytic center cutting plane method
The paper deals with nonlinear multicommodity flow problems with convex costs. A decomposition method is proposed to solve them. The approach applies a potential reduction algorithm to solve the master problem approximately and a column generation technique to define a sequence of primal linear programming problems. Each subproblem consists of finding a minimum cost flow between an origin and a destination node in an uncapacited network. It is thus formulated as a shortest path problem and solved with Dijkstra's d-heap algorithm. An implementation is described that takes full advantage of the supersparsity of the network in the linear algebra operations. Computational results show the efficiency of this approach on well-known nondifferentiable problems and also large scale randomly generated problems (up to 1000 arcs and 5000 commodities
Fast Algorithms for Separable Linear Programs
In numerical linear algebra, considerable effort has been devoted to
obtaining faster algorithms for linear systems whose underlying matrices
exhibit structural properties. A prominent success story is the method of
generalized nested dissection~[Lipton-Rose-Tarjan'79] for separable matrices.
On the other hand, the majority of recent developments in the design of
efficient linear program (LP) solves do not leverage the ideas underlying these
faster linear system solvers nor consider the separable structure of the
constraint matrix.
We give a faster algorithm for separable linear programs. Specifically, we
consider LPs of the form , where the
graphical support of the constraint matrix is -separable. These include flow problems on planar graphs
and low treewidth matrices among others. We present an time algorithm for these LPs, where is
the relative accuracy of the solution.
Our new solver has two important implications: for the -multicommodity
flow problem on planar graphs, we obtain an algorithm running in
time in the high accuracy regime; and when the
support of is -separable with , our
algorithm runs in time, which is nearly optimal. The latter
significantly improves upon the natural approach of combining interior point
methods and nested dissection, whose time complexity is lower bounded by
, where is the
matrix multiplication constant. Lastly, in the setting of low-treewidth LPs, we
recover the results of [DLY,STOC21] and [GS,22] with significantly simpler data
structure machinery.Comment: 55 pages. To appear at SODA 202
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