1,824 research outputs found
Robust capacitated trees and networks with uniform demands
We are interested in the design of robust (or resilient) capacitated rooted
Steiner networks in case of terminals with uniform demands. Formally, we are
given a graph, capacity and cost functions on the edges, a root, a subset of
nodes called terminals, and a bound k on the number of edge failures. We first
study the problem where k = 1 and the network that we want to design must be a
tree covering the root and the terminals: we give complexity results and
propose models to optimize both the cost of the tree and the number of
terminals disconnected from the root in the worst case of an edge failure,
while respecting the capacity constraints on the edges. Second, we consider the
problem of computing a minimum-cost survivable network, i.e., a network that
covers the root and terminals even after the removal of any k edges, while
still respecting the capacity constraints on the edges. We also consider the
possibility of protecting a given number of edges. We propose three different
formulations: a cut-set based formulation, a flow based one, and a bilevel one
(with an attacker and a defender). We propose algorithms to solve each
formulation and compare their efficiency
Hypergraphic LP Relaxations for Steiner Trees
We investigate hypergraphic LP relaxations for the Steiner tree problem,
primarily the partition LP relaxation introduced by Koenemann et al. [Math.
Programming, 2009]. Specifically, we are interested in proving upper bounds on
the integrality gap of this LP, and studying its relation to other linear
relaxations. Our results are the following. Structural results: We extend the
technique of uncrossing, usually applied to families of sets, to families of
partitions. As a consequence we show that any basic feasible solution to the
partition LP formulation has sparse support. Although the number of variables
could be exponential, the number of positive variables is at most the number of
terminals. Relations with other relaxations: We show the equivalence of the
partition LP relaxation with other known hypergraphic relaxations. We also show
that these hypergraphic relaxations are equivalent to the well studied
bidirected cut relaxation, if the instance is quasibipartite. Integrality gap
upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap
of these hypergraph relaxations in general graphs. In the special case of
uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~
1.216. By our equivalence theorem, the latter result implies an improved upper
bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010
Parameterized Complexity Dichotomy for Steiner Multicut
The Steiner Multicut problem asks, given an undirected graph G, terminals
sets T1,...,Tt V(G) of size at most p, and an integer k, whether
there is a set S of at most k edges or nodes s.t. of each set Ti at least one
pair of terminals is in different connected components of G \ S. This problem
generalizes several graph cut problems, in particular the Multicut problem (the
case p = 2), which is fixed-parameter tractable for the parameter k [Marx and
Razgon, Bousquet et al., STOC 2011].
We provide a dichotomy of the parameterized complexity of Steiner Multicut.
That is, for any combination of k, t, p, and the treewidth tw(G) as constant,
parameter, or unbounded, and for all versions of the problem (edge deletion and
node deletion with and without deletable terminals), we prove either that the
problem is fixed-parameter tractable or that the problem is hard (W[1]-hard or
even (para-)NP-complete). We highlight that:
- The edge deletion version of Steiner Multicut is fixed-parameter tractable
for the parameter k+t on general graphs (but has no polynomial kernel, even on
trees). We present two proofs: one using the randomized contractions technique
of Chitnis et al, and one relying on new structural lemmas that decompose the
Steiner cut into important separators and minimal s-t cuts.
- In contrast, both node deletion versions of Steiner Multicut are W[1]-hard
for the parameter k+t on general graphs.
- All versions of Steiner Multicut are W[1]-hard for the parameter k, even
when p=3 and the graph is a tree plus one node. Hence, the results of Marx and
Razgon, and Bousquet et al. do not generalize to Steiner Multicut.
Since we allow k, t, p, and tw(G) to be any constants, our characterization
includes a dichotomy for Steiner Multicut on trees (for tw(G) = 1), and a
polynomial time versus NP-hardness dichotomy (by restricting k,t,p,tw(G) to
constant or unbounded).Comment: As submitted to journal. This version also adds a proof of
fixed-parameter tractability for parameter k+t using the technique of
randomized contraction
A reclaimer scheduling problem arising in coal stockyard management
We study a number of variants of an abstract scheduling problem inspired by
the scheduling of reclaimers in the stockyard of a coal export terminal. We
analyze the complexity of each of the variants, providing complexity proofs for
some and polynomial algorithms for others. For one, especially interesting
variant, we also develop a constant factor approximation algorithm.Comment: 26 page
Survivability in hierarchical telecommunications networks under dual homing
Cataloged from PDF version of article.The motivation behind this study is the essential need for survivability in the telecommunications networks.
An optical signal should find its destination even if the network experiences an occasional fiber cut. We consider
the design of a two-level survivable telecommunications network. Terminals compiling the access layer
communicate through hubs forming the backbone layer. To hedge against single link failures in the network,
we require the backbone subgraph to be two-edge connected and the terminal nodes to connect to the backbone
layer in a dual-homed fashion, i.e., at two distinct hubs. The underlying design problem partitions a given
set of nodes into hubs and terminals, chooses a set of connections between the hubs such that the resulting
backbone network is two-edge connected, and for each terminal chooses two hubs to provide the dual-homing
backbone access. All of these decisions are jointly made based on some cost considerations. We give alternative
formulations using cut inequalities, compare these formulations, provide a polyhedral analysis of the smallsized
formulation, describe valid inequalities, study the associated separation problems, and design variable
fixing rules. All of these findings are then utilized in devising an efficient branch-and-cut algorithm to solve
this network design problem
Secret Key Generation from Correlated Sources and Secure Link
In this paper, we study the problem of secret key generation from both
correlated sources and a secure channel. We obtain the optimal secret key rate
in this problem and show that the optimal scheme is to conduct secret key
generation and key distribution jointly, where every bit in the secret channel
will yield more than one bit of secret key rate. This joint scheme is better
than the separation-based scheme, where the secure channel is used for key
distribution, and as a result, every bit in the secure channel can only provide
one bit of secret key rate.Comment: 5 pages, 1 figur
Subset feedback vertex set is fixed parameter tractable
The classical Feedback Vertex Set problem asks, for a given undirected graph
G and an integer k, to find a set of at most k vertices that hits all the
cycles in the graph G. Feedback Vertex Set has attracted a large amount of
research in the parameterized setting, and subsequent kernelization and
fixed-parameter algorithms have been a rich source of ideas in the field.
In this paper we consider a more general and difficult version of the
problem, named Subset Feedback Vertex Set (SUBSET-FVS in short) where an
instance comes additionally with a set S ? V of vertices, and we ask for a set
of at most k vertices that hits all simple cycles passing through S. Because of
its applications in circuit testing and genetic linkage analysis SUBSET-FVS was
studied from the approximation algorithms perspective by Even et al.
[SICOMP'00, SIDMA'00].
The question whether the SUBSET-FVS problem is fixed-parameter tractable was
posed independently by Kawarabayashi and Saurabh in 2009. We answer this
question affirmatively. We begin by showing that this problem is
fixed-parameter tractable when parametrized by |S|. Next we present an
algorithm which reduces the given instance to 2^k n^O(1) instances with the
size of S bounded by O(k^3), using kernelization techniques such as the
2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow
us to give a 2^O(k log k) n^O(1) time algorithm solving the Subset Feedback
Vertex Set problem, proving that it is indeed fixed-parameter tractable.Comment: full version of a paper presented at ICALP'1
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