66,251 research outputs found
A Polynomial-time Bicriteria Approximation Scheme for Planar Bisection
Given an undirected graph with edge costs and node weights, the minimum
bisection problem asks for a partition of the nodes into two parts of equal
weight such that the sum of edge costs between the parts is minimized. We give
a polynomial time bicriteria approximation scheme for bisection on planar
graphs.
Specifically, let be the total weight of all nodes in a planar graph .
For any constant , our algorithm outputs a bipartition of the
nodes such that each part weighs at most and the total cost
of edges crossing the partition is at most times the total
cost of the optimal bisection. The previously best known approximation for
planar minimum bisection, even with unit node weights, was . Our
algorithm actually solves a more general problem where the input may include a
target weight for the smaller side of the bipartition.Comment: To appear in STOC 201
When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks
Let be a supply graph and a demand graph defined on the
same set of vertices. An assignment of capacities to the edges of and
demands to the edges of is said to satisfy the \emph{cut condition} if for
any cut in the graph, the total demand crossing the cut is no more than the
total capacity crossing it. The pair is called \emph{cut-sufficient} if
for any assignment of capacities and demands that satisfy the cut condition,
there is a multiflow routing the demands defined on within the network with
capacities defined on . We prove a previous conjecture, which states that
when the supply graph is series-parallel, the pair is
cut-sufficient if and only if does not contain an \emph{odd spindle} as
a minor; that is, if it is impossible to contract edges of and delete edges
of and so that becomes the complete bipartite graph , with
odd, and is composed of a cycle connecting the vertices of
degree 2, and an edge connecting the two vertices of degree . We further
prove that if the instance is \emph{Eulerian} --- that is, the demands and
capacities are integers and the total of demands and capacities incident to
each vertex is even --- then the multiflow problem has an integral solution. We
provide a polynomial-time algorithm to find an integral solution in this case.
In order to prove these results, we formulate properties of tight cuts (cuts
for which the cut condition inequality is tight) in cut-sufficient pairs. We
believe these properties might be useful in extending our results to planar
graphs.Comment: An extended abstract of this paper will be published at the 44th
Symposium on Theory of Computing (STOC 2012
Computing Socially-Efficient Cake Divisions
We consider a setting in which a single divisible good ("cake") needs to be
divided between n players, each with a possibly different valuation function
over pieces of the cake. For this setting, we address the problem of finding
divisions that maximize the social welfare, focusing on divisions where each
player needs to get one contiguous piece of the cake. We show that for both the
utilitarian and the egalitarian social welfare functions it is NP-hard to find
the optimal division. For the utilitarian welfare, we provide a constant factor
approximation algorithm, and prove that no FPTAS is possible unless P=NP. For
egalitarian welfare, we prove that it is NP-hard to approximate the optimum to
any factor smaller than 2. For the case where the number of players is small,
we provide an FPT (fixed parameter tractable) FPTAS for both the utilitarian
and the egalitarian welfare objectives
Almost-Tight Distributed Minimum Cut Algorithms
We study the problem of computing the minimum cut in a weighted distributed
message-passing networks (the CONGEST model). Let be the minimum cut,
be the number of nodes in the network, and be the network diameter. Our
algorithm can compute exactly in time. To the best of our knowledge, this is the first paper that
explicitly studies computing the exact minimum cut in the distributed setting.
Previously, non-trivial sublinear time algorithms for this problem are known
only for unweighted graphs when due to Pritchard and
Thurimella's -time and -time algorithms for
computing -edge-connected and -edge-connected components.
By using the edge sampling technique of Karger's, we can convert this
algorithm into a -approximation -time algorithm for any . This improves
over the previous -approximation -time algorithm and
-approximation -time algorithm of Ghaffari and Kuhn. Due to the lower
bound of by Das Sarma et al. which holds for any
approximation algorithm, this running time is tight up to a factor.
To get the stated running time, we developed an approximation algorithm which
combines the ideas of Thorup's algorithm and Matula's contraction algorithm. It
saves an factor as compared to applying Thorup's tree
packing theorem directly. Then, we combine Kutten and Peleg's tree partitioning
algorithm and Karger's dynamic programming to achieve an efficient distributed
algorithm that finds the minimum cut when we are given a spanning tree that
crosses the minimum cut exactly once
A Heuristic Strategy to Compute Ensemble of Trajectories for 3D Low Cost Earth-Moon Transfers
The problem of finding optimal trajectories is essential for modern space mission design. When considering multibody
gravitational dynamics and exploiting both low-thrust and high-thrust and alternative forms of propulsion such
as solar sailing, sets of good initial guesses are fundamental for the convergence to local or global optimal solutions,
using both direct or indirect methods available to solve the optimal control problem. This paper deals with obtaining
preliminary trajectories that are designed to be good initial guesses as input to search optimal low-energy short-time
Earth-Moon transfers with ballistic capture. A more realistic modelling is introduced, in which the restricted four-body
system Sun-Earth-Moon-Spacecraft is decoupled in two patched planar Circular Restricted Three-Body Problems,
taking into account the inclination of the orbital plane of the Moon with respect to the ecliptic. We present a heuristic
strategy based on the hyperbolic invariant manifolds of the Lyapunov orbits around the Lagrangian points of the Earth-
Moon system to obtain ballistic capture orbits around the Moon that fulfill specific mission requirements. Moreover,
quasi-periodic orbits of the Sun-Earth system are exploited using a genetic algorithm to find optimal solutions with
respect to total Dv, time of flight and altitude at departure. Finally, the procedure is illustrated and the full transfer
trajectories assessed in view of relevant properties. The proposed methodology provides sets of low-cost and shorttime
initial guesses to serve as inputs to compute fully optimized three-dimensional solutions considering different
propulsion technologies, such as low, high, and hybrid thrust, and/or using more realistic models
Efficient Linear Programming Decoding of HDPC Codes
We propose several improvements for Linear Programming (LP) decoding
algorithms for High Density Parity Check (HDPC) codes. First, we use the
automorphism groups of a code to create parity check matrix diversity and to
generate valid cuts from redundant parity checks. Second, we propose an
efficient mixed integer decoder utilizing the branch and bound method. We
further enhance the proposed decoders by removing inactive constraints and by
adapting the parity check matrix prior to decoding according to the channel
observations. Based on simulation results the proposed decoders achieve near-ML
performance with reasonable complexity.Comment: Submitted to the IEEE Transactions on Communications, November 200
Tree-based Coarsening and Partitioning of Complex Networks
Many applications produce massive complex networks whose analysis would
benefit from parallel processing. Parallel algorithms, in turn, often require a
suitable network partition. For solving optimization tasks such as graph
partitioning on large networks, multilevel methods are preferred in practice.
Yet, complex networks pose challenges to established multilevel algorithms, in
particular to their coarsening phase.
One way to specify a (recursive) coarsening of a graph is to rate its edges
and then contract the edges as prioritized by the rating. In this paper we (i)
define weights for the edges of a network that express the edges' importance
for connectivity, (ii) compute a minimum weight spanning tree with
respect to these weights, and (iii) rate the network edges based on the
conductance values of 's fundamental cuts. To this end, we also (iv)
develop the first optimal linear-time algorithm to compute the conductance
values of \emph{all} fundamental cuts of a given spanning tree. We integrate
the new edge rating into a leading multilevel graph partitioner and equip the
latter with a new greedy postprocessing for optimizing the maximum
communication volume (MCV). Experiments on bipartitioning frequently used
benchmark networks show that the postprocessing already reduces MCV by 11.3%.
Our new edge rating further reduces MCV by 10.3% compared to the previously
best rating with the postprocessing in place for both ratings. In total, with a
modest increase in running time, our new approach reduces the MCV of complex
network partitions by 20.4%
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