22 research outputs found
A note on forbidding clique immersions
Robertson and Seymour proved that the relation of graph immersion is
well-quasi-ordered for finite graphs. Their proof uses the results of graph
minors theory. Surprisingly, there is a very short proof of the corresponding
rough structure theorem for graphs without -immersions; it is based on the
Gomory-Hu theorem. The same proof also works to establish a rough structure
theorem for Eulerian digraphs without -immersions, where
denotes the bidirected complete digraph of order
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
The generalized 3-edge-connectivity of lexicographic product graphs
The generalized -edge-connectivity of a graph is a
generalization of the concept of edge-connectivity. The lexicographic product
of two graphs and , denoted by , is an important graph
product. In this paper, we mainly study the generalized 3-edge-connectivity of
, and get upper and lower bounds of .
Moreover, all bounds are sharp.Comment: 14 page
Steiner connectivity problems in hypergraphs
We say that a tree is an -Steiner tree if and a
hypergraph is an -Steiner hypertree if it can be trimmed to an -Steiner
tree. We prove that it is NP-hard to decide, given a hypergraph
and some , whether there is a subhypergraph of
which is an -Steiner hypertree. As corollaries, we give two
negative results for two Steiner orientation problems in hypergraphs. Firstly,
we show that it is NP-hard to decide, given a hypergraph , some and some , whether this
hypergraph has an orientation in which every vertex of is reachable from
. Secondly, we show that it is NP-hard to decide, given a hypergraph
and some , whether this hypergraph
has an orientation in which any two vertices in are mutually reachable from
each other. This answers a longstanding open question of the Egerv\'ary
Research group. On the positive side, we show that the problem of finding a
Steiner hypertree and the first orientation problem can be solved in polynomial
time if the number of terminals is fixed
Greedy Algorithms for Online Survivable Network Design
In an instance of the network design problem, we are given a graph G=(V,E), an edge-cost function c:E -> R^{>= 0}, and a connectivity criterion. The goal is to find a minimum-cost subgraph H of G that meets the connectivity requirements. An important family of this class is the survivable network design problem (SNDP): given non-negative integers r_{u v} for each pair u,v in V, the solution subgraph H should contain r_{u v} edge-disjoint paths for each pair u and v.
While this problem is known to admit good approximation algorithms in the offline case, the problem is much harder in the online setting. Gupta, Krishnaswamy, and Ravi [Gupta et al., 2012] (STOC\u2709) are the first to consider the online survivable network design problem. They demonstrate an algorithm with competitive ratio of O(k log^3 n), where k=max_{u,v} r_{u v}. Note that the competitive ratio of the algorithm by Gupta et al. grows linearly in k. Since then, an important open problem in the online community [Naor et al., 2011; Gupta et al., 2012] is whether the linear dependence on k can be reduced to a logarithmic dependency.
Consider an online greedy algorithm that connects every demand by adding a minimum cost set of edges to H. Surprisingly, we show that this greedy algorithm significantly improves the competitive ratio when a congestion of 2 is allowed on the edges or when the model is stochastic. While our algorithm is fairly simple, our analysis requires a deep understanding of k-connected graphs. In particular, we prove that the greedy algorithm is O(log^2 n log k)-competitive if one satisfies every demand between u and v by r_{uv}/2 edge-disjoint paths. The spirit of our result is similar to the work of Chuzhoy and Li [Chuzhoy and Li, 2012] (FOCS\u2712), in which the authors give a polylogarithmic approximation algorithm for edge-disjoint paths with congestion 2.
Moreover, we study the greedy algorithm in the online stochastic setting. We consider the i.i.d. model, where each online demand is drawn from a single probability distribution, the unknown i.i.d. model, where every demand is drawn from a single but unknown probability distribution, and the prophet model in which online demands are drawn from (possibly) different probability distributions. Through a different analysis, we prove that a similar greedy algorithm is constant competitive for the i.i.d. and the prophet models. Also, the greedy algorithm is O(log n)-competitive for the unknown i.i.d. model, which is almost tight due to the lower bound of [Garg et al., 2008] for single connectivity
Graphs with large generalized (edge-)connectivity
The generalized -connectivity of a graph , introduced by
Hager in 1985, is a nice generalization of the classical connectivity.
Recently, as a natural counterpart, we proposed the concept of generalized
-edge-connectivity . In this paper, graphs of order such
that and for even
are characterized.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1207.183