35 research outputs found
On the Capacity Bounds of Undirected Networks
In this work we improve on the bounds presented by Li&Li for network coding
gain in the undirected case. A tightened bound for the undirected multicast
problem with three terminals is derived. An interesting result shows that with
fractional routing, routing throughput can achieve at least 75% of the coding
throughput. A tighter bound for the general multicast problem with any number
of terminals shows that coding gain is strictly less than 2. Our derived bound
depends on the number of terminals in the multicast network and approaches 2
for arbitrarily large number of terminals.Comment: 5 pages, 5 figures, ISIT 2007 conferenc
A greedy approximation algorithm for the group Steiner problem
AbstractIn the group Steiner problem we are given an edge-weighted graph G=(V,E,w) and m subsets of vertices {gi}i=1m. Each subset gi is called a group and the vertices in ⋃igi are called terminals. It is required to find a minimum weight tree that contains at least one terminal from every group.We present a poly-logarithmic ratio approximation for this problem when the input graph is a tree. Our algorithm is a recursive greedy algorithm adapted from the greedy algorithm for the directed Steiner tree problem [Approximating the weight of shallow Steiner trees, Discrete Appl. Math. 93 (1999) 265–285, Approximation algorithms for directed Steiner problems, J. Algorithms 33 (1999) 73–91]. This is in contrast to earlier algorithms that are based on rounding a linear programming based relaxation for the problem [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66–84, preliminary version in Proceedings of SODA, 1998 pp. 253–259, On directed Steiner trees, Proceedings of SODA, 2002, pp. 59–63]. We answer in positive a question posed in [A polylogarithmic approximation algorithm for the Group Steiner tree problem, J. Algorithms 37 (2000) 66–84, preliminary version in Proceedings of SODA, 1998 pp. 253–259] on whether there exist good approximation algorithms for the group Steiner problem that are not based on rounding linear programs. For every fixed constant ε>0, our algorithm gives an O((log∑i|gi|)1+ε·logm) approximation in polynomial time. Approximation algorithms for trees can be extended to arbitrary undirected graphs by probabilistically approximating the graph by a tree. This results in an additional multiplicative factor of O(log|V|) in the approximation ratio, where |V| is the number of vertices in the graph. The approximation ratio of our algorithm on trees is slightly worse than the ratio of O(log(maxi|gi|)·logm) provided by the LP based approaches
Minimum-cost multicast over coded packet networks
We consider the problem of establishing minimum-cost multicast connections over coded packet networks, i.e., packet networks where the contents of outgoing packets are arbitrary, causal functions of the contents of received packets. We consider both wireline and wireless packet networks as well as both static multicast (where membership of the multicast group remains constant for the duration of the connection) and dynamic multicast (where membership of the multicast group changes in time, with nodes joining and leaving the group). For static multicast, we reduce the problem to a polynomial-time solvable optimization problem, and we present decentralized algorithms for solving it. These algorithms, when coupled with existing decentralized schemes for constructing network codes, yield a fully decentralized approach for achieving minimum-cost multicast. By contrast, establishing minimum-cost static multicast connections over routed packet networks is a very difficult problem even using centralized computation, except in the special cases of unicast and broadcast connections. For dynamic multicast, we reduce the problem to a dynamic programming problem and apply the theory of dynamic programming to suggest how it may be solved
Complexity of the Steiner Network Problem with Respect to the Number of Terminals
In the Directed Steiner Network problem we are given an arc-weighted digraph
, a set of terminals , and an (unweighted) directed
request graph with . Our task is to output a subgraph of the minimum cost such that there is a directed path from to in
for all .
It is known that the problem can be solved in time
[Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time
even if is planar, unless Exponential-Time Hypothesis
(ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other
reductions showing hardness of the problem) only shows that the problem cannot
be solved in time unless ETH fails, there is a significant
gap in the complexity with respect to in the exponent.
We show that Directed Steiner Network is solvable in time , where is a constant depending solely on the
genus of and is a computable function. We complement this result by
showing that there is no algorithm for
any function for the problem on general graphs, unless ETH fails
A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs
We demonstrate that the integrality gap of the natural cut-based LP relaxation for the directed Steiner tree problem is O(log k) in quasi-bipartite graphs with k terminals. Such instances can be seen to generalize set cover, so the integrality gap analysis is tight up to a constant factor. A novel aspect of our approach is that we use the primal-dual method; a technique that is rarely used in designing approximation algorithms for network design problems in directed graphs
Content Distribution by Multiple Multicast Trees and Intersession Cooperation: Optimal Algorithms and Approximations
In traditional massive content distribution with multiple sessions, the
sessions form separate overlay networks and operate independently, where some
sessions may suffer from insufficient resources even though other sessions have
excessive resources. To cope with this problem, we consider the universal
swarming approach, which allows multiple sessions to cooperate with each other.
We formulate the problem of finding the optimal resource allocation to maximize
the sum of the session utilities and present a subgradient algorithm which
converges to the optimal solution in the time-average sense. The solution
involves an NP-hard subproblem of finding a minimum-cost Steiner tree. We cope
with this difficulty by using a column generation method, which reduces the
number of Steiner-tree computations. Furthermore, we allow the use of
approximate solutions to the Steiner-tree subproblem. We show that the
approximation ratio to the overall problem turns out to be no less than the
reciprocal of the approximation ratio to the Steiner-tree subproblem.
Simulation results demonstrate that universal swarming improves the performance
of resource-poor sessions with negligible impact to resource-rich sessions. The
proposed approach and algorithm are expected to be useful for
infrastructure-based content distribution networks with long-lasting sessions
and relatively stable network environment
-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