35,993 research outputs found
Approximating the minimum directed tree cover
Given a directed graph with non negative cost on the arcs, a directed
tree cover of is a rooted directed tree such that either head or tail (or
both of them) of every arc in is touched by . The minimum directed tree
cover problem (DTCP) is to find a directed tree cover of minimum cost. The
problem is known to be -hard. In this paper, we show that the weighted Set
Cover Problem (SCP) is a special case of DTCP. Hence, one can expect at best to
approximate DTCP with the same ratio as for SCP. We show that this expectation
can be satisfied in some way by designing a purely combinatorial approximation
algorithm for the DTCP and proving that the approximation ratio of the
algorithm is with is the maximum outgoing degree of
the nodes in .Comment: 13 page
Polylogarithmic Approximation Algorithm for k-Connected Directed Steiner Tree on Quasi-Bipartite Graphs
In the k-Connected Directed Steiner Tree problem (k-DST), we are given a directed graph G = (V,E) with edge (or vertex) costs, a root vertex r, a set of q terminals T, and a connectivity requirement k > 0; the goal is to find a minimum-cost subgraph H of G such that H has k edge-disjoint paths from the root r to each terminal in T. The k-DST problem is a natural generalization of the classical Directed Steiner Tree problem (DST) in the fault-tolerant setting in which the solution subgraph is required to have an r,t-path, for every terminal t, even after removing k-1 vertices or edges. Despite being a classical problem, there are not many positive results on the problem, especially for the case k ? 3. In this paper, we present an O(log k log q)-approximation algorithm for k-DST when an input graph is quasi-bipartite, i.e., when there is no edge joining two non-terminal vertices. To the best of our knowledge, our algorithm is the only known non-trivial approximation algorithm for k-DST, for k ? 3, that runs in polynomial-time Our algorithm is tight for every constant k, due to the hardness result inherited from the Set Cover problem
On Approximating Degree-Bounded Network Design Problems
Directed Steiner Tree (DST) is a central problem in combinatorial
optimization and theoretical computer science: Given a directed graph with edge costs , a root and
terminals , we need to output the minimum-cost arborescence in
that contains an \textrightarrow path for every . Recently,
Grandoni, Laekhanukit and Li, and independently Ghuge and Nagarajan, gave
quasi-polynomial time -approximation algorithms for the
problem, which are tight under popular complexity assumptions.
In this paper, we consider the more general Degree-Bounded Directed Steiner
Tree (DB-DST) problem, where we are additionally given a degree bound on
each vertex , and we require that every vertex in the output tree
has at most children. We give a quasi-polynomial time -bicriteria approximation: The algorithm produces a solution with
cost at most times the cost of the optimum solution that
violates the degree constraints by at most a factor of . This is
the first non-trivial result for the problem.
While our cost-guarantee is nearly optimal, the degree violation factor of
is an -factor away from the approximation lower bound
of from the set-cover hardness. The hardness result holds even
on the special case of the {\em Degree-Bounded Group Steiner Tree} problem on
trees (DB-GST-T). With the hope of closing the gap, we study the question of
whether the degree violation factor can be made tight for this special case. We
answer the question in the affirmative by giving an -bicriteria approximation algorithm for DB-GST-T
-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm
In the Directed Steiner Tree (DST) problem we are given an -vertex
directed edge-weighted graph, a root , and a collection of terminal
nodes. Our goal is to find a minimum-cost arborescence that contains a directed
path from to every terminal. We present an -approximation algorithm for DST that runs in
quasi-polynomial-time. By adjusting the parameters in the hardness result of
Halperin and Krauthgamer, we show the matching lower bound of
for the class of quasi-polynomial-time
algorithms. This is the first improvement on the DST problem since the
classical quasi-polynomial-time approximation algorithm by
Charikar et al. (The paper erroneously claims an approximation due
to a mistake in prior work.)
Our approach is based on two main ingredients. First, we derive an
approximation preserving reduction to the Label-Consistent Subtree (LCST)
problem. The LCST instance has quasi-polynomial size and logarithmic height. We
remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all
prior work on DST achieves a reduction to a tree instance of the related Group
Steiner Tree (GST) problem of similar height, however losing a logarithmic
factor in the approximation ratio. Our second ingredient is an LP-rounding
algorithm to approximately solve LCST instances, which is inspired by the
framework developed by Rothvo{\ss}. We consider a Sherali-Adams lifting of a
proper LP relaxation of LCST. Our rounding algorithm proceeds level by level
from the root to the leaves, rounding and conditioning each time on a proper
subset of label variables. A small enough (namely, polylogarithmic) number of
Sherali-Adams lifting levels is sufficient to condition up to the leaves
Parameterized Approximation Schemes for Steiner Trees with Small Number of Steiner Vertices
We study the Steiner Tree problem, in which a set of terminal vertices needs
to be connected in the cheapest possible way in an edge-weighted graph. This
problem has been extensively studied from the viewpoint of approximation and
also parametrization. In particular, on one hand Steiner Tree is known to be
APX-hard, and W[2]-hard on the other, if parameterized by the number of
non-terminals (Steiner vertices) in the optimum solution. In contrast to this
we give an efficient parameterized approximation scheme (EPAS), which
circumvents both hardness results. Moreover, our methods imply the existence of
a polynomial size approximate kernelization scheme (PSAKS) for the considered
parameter.
We further study the parameterized approximability of other variants of
Steiner Tree, such as Directed Steiner Tree and Steiner Forest. For neither of
these an EPAS is likely to exist for the studied parameter: for Steiner Forest
an easy observation shows that the problem is APX-hard, even if the input graph
contains no Steiner vertices. For Directed Steiner Tree we prove that
approximating within any function of the studied parameter is W[1]-hard.
Nevertheless, we show that an EPAS exists for Unweighted Directed Steiner Tree,
but a PSAKS does not. We also prove that there is an EPAS and a PSAKS for
Steiner Forest if in addition to the number of Steiner vertices, the number of
connected components of an optimal solution is considered to be a parameter.Comment: 23 pages, 6 figures An extended abstract appeared in proceedings of
STACS 201
Algorithmic Aspects of Energy-Delay Tradeoff in Multihop Cooperative Wireless Networks
We consider the problem of energy-efficient transmission in delay constrained
cooperative multihop wireless networks. The combinatorial nature of cooperative
multihop schemes makes it difficult to design efficient polynomial-time
algorithms for deciding which nodes should take part in cooperation, and when
and with what power they should transmit. In this work, we tackle this problem
in memoryless networks with or without delay constraints, i.e., quality of
service guarantee. We analyze a wide class of setups, including unicast,
multicast, and broadcast, and two main cooperative approaches, namely: energy
accumulation (EA) and mutual information accumulation (MIA). We provide a
generalized algorithmic formulation of the problem that encompasses all those
cases. We investigate the similarities and differences of EA and MIA in our
generalized formulation. We prove that the broadcast and multicast problems
are, in general, not only NP hard but also o(log(n)) inapproximable. We break
these problems into three parts: ordering, scheduling and power control, and
propose a novel algorithm that, given an ordering, can optimally solve the
joint power allocation and scheduling problems simultaneously in polynomial
time. We further show empirically that this algorithm used in conjunction with
an ordering derived heuristically using the Dijkstra's shortest path algorithm
yields near-optimal performance in typical settings. For the unicast case, we
prove that although the problem remains NP hard with MIA, it can be solved
optimally and in polynomial time when EA is used. We further use our algorithm
to study numerically the trade-off between delay and power-efficiency in
cooperative broadcast and compare the performance of EA vs MIA as well as the
performance of our cooperative algorithm with a smart noncooperative algorithm
in a broadcast setting.Comment: 12 pages, 9 figure
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