325 research outputs found
Minimum multicuts and Steiner forests for Okamura-Seymour graphs
We study the problem of finding minimum multicuts for an undirected planar
graph, where all the terminal vertices are on the boundary of the outer face.
This is known as an Okamura-Seymour instance. We show that for such an
instance, the minimum multicut problem can be reduced to the minimum-cost
Steiner forest problem on a suitably defined dual graph. The minimum-cost
Steiner forest problem has a 2-approximation algorithm. Hence, the minimum
multicut problem has a 2-approximation algorithm for an Okamura-Seymour
instance.Comment: 6 pages, 1 figur
ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems
This paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self-concordant augmented barrier. Our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component. The paper also proposes an approximation scheme for the original objective. An active set strategy can be applied to the transformed problem: it reduces the dimension of the dual space and accelerates computations. The new approach solves huge instances with high accuracy. The method is compared to alternative approaches proposed in the literatur
Network recovery after massive failures
This paper addresses the problem of efficiently restoring sufficient resources in a communications network to support the demand of mission critical services after a large scale disruption. We give a formulation of the problem as an MILP and show that it is NP-hard. We propose a polynomial time heuristic, called Iterative Split and Prune (ISP) that decomposes the original problem recursively into smaller problems, until it determines the set of network components to be restored. We performed extensive simulations by varying the topologies, the demand intensity, the number of critical services, and the disruption model. Compared to several greedy approaches ISP performs better in terms of number of repaired components, and does not result in any demand loss. It performs very close to the optimal when the demand is low with respect to the supply network capacities, thanks to the ability of the algorithm to maximize sharing of repaired resources
Algorithms for Multicommodity Flows in Planar Graphs
This paper gives efficient algorithms for the multicommodity flow problem for two classes Ct2 and Co~ of planar undirected graphs. Every graph in Ct2 has two face boundaries B t and B 2 such that each of the source-sink pairs lies on B 1 or B 2. On the other hand, every graph in Cot has a face boundary B t such that some of the source-sink pairs lie on B 1 and all the other pairs share a common sink lying on B t. The algorithms run in O(kn + nT(n)) time if a graph has n vertices and k source-sink pairs and T(n) is the time required for finding the single-source shortest paths in a planar graph of n vertices
Compact Oblivious Routing
Oblivious routing is an attractive paradigm for large distributed systems in which centralized control and frequent reconfigurations are infeasible or undesired (e.g., costly). Over the last almost 20 years, much progress has been made in devising oblivious routing schemes that guarantee close to optimal load and also algorithms for constructing such schemes efficiently have been designed. However, a common drawback of existing oblivious routing schemes is that they are not compact: they require large routing tables (of polynomial size), which does not scale.
This paper presents the first oblivious routing scheme which guarantees close to optimal load and is compact at the same time - requiring routing tables of polylogarithmic size. Our algorithm maintains the polylogarithmic competitive ratio of existing algorithms, and is hence particularly well-suited for emerging large-scale networks
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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