8,922 research outputs found
The generalized minimum spanning tree polytope and related polytopes
The Generalized Minimum Spanning Tree problem denoted by GMST is a variant of the classical Minimum Spanning Tree problem in which nodes are partitioned into clusters and the problem calls for a minimum cost tree spanning at least one node from each cluster. A different version of the problem, called E-GMST arises when exactly one node from each cluster has to be visited. Both GMST problem and E-GMST problem are NP-hard problems. In this paper, we model GMST problem and E-GMST problem as integer linear programs and study the facial structure of the corresponding polytopes
Clustering with shallow trees
We propose a new method for hierarchical clustering based on the optimisation
of a cost function over trees of limited depth, and we derive a
message--passing method that allows to solve it efficiently. The method and
algorithm can be interpreted as a natural interpolation between two well-known
approaches, namely single linkage and the recently presented Affinity
Propagation. We analyze with this general scheme three biological/medical
structured datasets (human population based on genetic information, proteins
based on sequences and verbal autopsies) and show that the interpolation
technique provides new insight.Comment: 11 pages, 7 figure
The cavity approach for Steiner trees packing problems
The Belief Propagation approximation, or cavity method, has been recently
applied to several combinatorial optimization problems in its zero-temperature
implementation, the max-sum algorithm. In particular, recent developments to
solve the edge-disjoint paths problem and the prize-collecting Steiner tree
problem on graphs have shown remarkable results for several classes of graphs
and for benchmark instances. Here we propose a generalization of these
techniques for two variants of the Steiner trees packing problem where multiple
"interacting" trees have to be sought within a given graph. Depending on the
interaction among trees we distinguish the vertex-disjoint Steiner trees
problem, where trees cannot share nodes, from the edge-disjoint Steiner trees
problem, where edges cannot be shared by trees but nodes can be members of
multiple trees. Several practical problems of huge interest in network design
can be mapped into these two variants, for instance, the physical design of
Very Large Scale Integration (VLSI) chips. The formalism described here relies
on two components edge-variables that allows us to formulate a massage-passing
algorithm for the V-DStP and two algorithms for the E-DStP differing in the
scaling of the computational time with respect to some relevant parameters. We
will show that one of the two formalisms used for the edge-disjoint variant
allow us to map the max-sum update equations into a weighted maximum matching
problem over proper bipartite graphs. We developed a heuristic procedure based
on the max-sum equations that shows excellent performance in synthetic networks
(in particular outperforming standard multi-step greedy procedures by large
margins) and on large benchmark instances of VLSI for which the optimal
solution is known, on which the algorithm found the optimum in two cases and
the gap to optimality was never larger than 4 %
Throughput-Optimal Multihop Broadcast on Directed Acyclic Wireless Networks
We study the problem of efficiently broadcasting packets in multi-hop
wireless networks. At each time slot the network controller activates a set of
non-interfering links and forwards selected copies of packets on each activated
link. A packet is considered jointly received only when all nodes in the
network have obtained a copy of it. The maximum rate of jointly received
packets is referred to as the broadcast capacity of the network. Existing
policies achieve the broadcast capacity by balancing traffic over a set of
spanning trees, which are difficult to maintain in a large and time-varying
wireless network. We propose a new dynamic algorithm that achieves the
broadcast capacity when the underlying network topology is a directed acyclic
graph (DAG). This algorithm is decentralized, utilizes local queue-length
information only and does not require the use of global topological structures
such as spanning trees. The principal technical challenge inherent in the
problem is the absence of work-conservation principle due to the duplication of
packets, which renders traditional queuing modelling inapplicable. We overcome
this difficulty by studying relative packet deficits and imposing in-order
delivery constraints to every node in the network. Although in-order packet
delivery, in general, leads to degraded throughput in graphs with cycles, we
show that it is throughput optimal in DAGs and can be exploited to simplify the
design and analysis of optimal algorithms. Our characterization leads to a
polynomial time algorithm for computing the broadcast capacity of any wireless
DAG under the primary interference constraints. Additionally, we propose an
extension of our algorithm which can be effectively used for broadcasting in
any network with arbitrary topology
Matroidal Degree-Bounded Minimum Spanning Trees
We consider the minimum spanning tree (MST) problem under the restriction
that for every vertex v, the edges of the tree that are adjacent to v satisfy a
given family of constraints. A famous example thereof is the classical
degree-constrained MST problem, where for every vertex v, a simple upper bound
on the degree is imposed. Iterative rounding/relaxation algorithms became the
tool of choice for degree-bounded network design problems. A cornerstone for
this development was the work of Singh and Lau, who showed for the
degree-bounded MST problem how to find a spanning tree violating each degree
bound by at most one unit and with cost at most the cost of an optimal solution
that respects the degree bounds.
However, current iterative rounding approaches face several limits when
dealing with more general degree constraints. In particular, when several
constraints are imposed on the edges adjacent to a vertex v, as for example
when a partition of the edges adjacent to v is given and only a fixed number of
elements can be chosen out of each set of the partition, current approaches
might violate each of the constraints by a constant, instead of violating all
constraints together by at most a constant number of edges. Furthermore, it is
also not clear how previous iterative rounding approaches can be used for
degree constraints where some edges are in a super-constant number of
constraints.
We extend iterative rounding/relaxation approaches both on a conceptual level
as well as aspects involving their analysis to address these limitations. This
leads to an efficient algorithm for the degree-constrained MST problem where
for every vertex v, the edges adjacent to v have to be independent in a given
matroid. The algorithm returns a spanning tree T of cost at most OPT, such that
for every vertex v, it suffices to remove at most 8 edges from T to satisfy the
matroidal degree constraint at v
A Framework for Algorithm Stability
We say that an algorithm is stable if small changes in the input result in
small changes in the output. This kind of algorithm stability is particularly
relevant when analyzing and visualizing time-varying data. Stability in general
plays an important role in a wide variety of areas, such as numerical analysis,
machine learning, and topology, but is poorly understood in the context of
(combinatorial) algorithms. In this paper we present a framework for analyzing
the stability of algorithms. We focus in particular on the tradeoff between the
stability of an algorithm and the quality of the solution it computes. Our
framework allows for three types of stability analysis with increasing degrees
of complexity: event stability, topological stability, and Lipschitz stability.
We demonstrate the use of our stability framework by applying it to kinetic
Euclidean minimum spanning trees
Throughput-Optimal Broadcast on Directed Acyclic Graphs
We study the problem of broadcasting packets in wireless networks. At each
time slot, a network controller activates non-interfering links and forwards
packets to all nodes at a common rate; the maximum rate is referred to as the
broadcast capacity of the wireless network. Existing policies achieve the
broadcast capacity by balancing traffic over a set of spanning trees, which are
difficult to maintain in a large and time-varying wireless network. We propose
a new dynamic algorithm that achieves the broadcast capacity when the
underlying network topology is a directed acyclic graph (DAG). This algorithm
utilizes local queue-length information, does not use any global topological
structures such as spanning trees, and uses the idea of in-order packet
delivery to all network nodes. Although the in-order packet delivery constraint
leads to degraded throughput in cyclic graphs, we show that it is throughput
optimal in DAGs and can be exploited to simplify the design and analysis of
optimal algorithms. Our simulation results show that the proposed algorithm has
superior delay performance as compared to tree-based approaches.Comment: To appear in the proceedings of INFOCOM, 201
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