1,157 research outputs found
Minimum congestion spanning trees in planar graphs
The main purpose of the paper is to develop an approach to evaluation or
estimation of the spanning tree congestion of planar graphs. This approach is
used to evaluate the spanning tree congestion of triangular grids
Low-Congestion Shortcut and Graph Parameters
Distributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of Omega~(sqrt{n} + D) rounds for many global problems, where n is the number of nodes and D is the diameter of the input graph. Since such a lower bound is derived from special "hard-core" instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts is initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. Specifically, given a specific graph class X, an f-round algorithm of constructing shortcuts of quality q for any instance in X results in O~(q + f)-round algorithms of solving several fundamental graph problems such as minimum spanning tree and minimum cut, for X. The main interest on this line is to identify the graph classes allowing the shortcuts which are efficient in the sense of breaking O~(sqrt{n}+D)-round general lower bounds.
In this paper, we consider the relationship between the quality of low-congestion shortcuts and three major graph parameters, chordality, diameter, and clique-width. The main contribution of the paper is threefold: (1) We show an O(1)-round algorithm which constructs a low-congestion shortcut with quality O(kD) for any k-chordal graph, and prove that the quality and running time of this construction is nearly optimal up to polylogarithmic factors. (2) We present two algorithms, each of which constructs a low-congestion shortcut with quality O~(n^{1/4}) in O~(n^{1/4}) rounds for graphs of D=3, and that with quality O~(n^{1/3}) in O~(n^{1/3}) rounds for graphs of D=4 respectively. These results obviously deduce two MST algorithms running in O~(n^{1/4}) and O~(n^{1/3}) rounds for D=3 and 4 respectively, which almost close the long-standing complexity gap of the MST construction in small-diameter graphs originally posed by Lotker et al. [Distributed Computing 2006]. (3) We show that bounding clique-width does not help the construction of good shortcuts by presenting a network topology of clique-width six where the construction of MST is as expensive as the general case
Fully dynamic all-pairs shortest paths with worst-case update-time revisited
We revisit the classic problem of dynamically maintaining shortest paths
between all pairs of nodes of a directed weighted graph. The allowed updates
are insertions and deletions of nodes and their incident edges. We give
worst-case guarantees on the time needed to process a single update (in
contrast to related results, the update time is not amortized over a sequence
of updates).
Our main result is a simple randomized algorithm that for any parameter
has a worst-case update time of and answers
distance queries correctly with probability , against an adaptive
online adversary if the graph contains no negative cycle. The best
deterministic algorithm is by Thorup [STOC 2005] with a worst-case update time
of and assumes non-negative weights. This is the first
improvement for this problem for more than a decade. Conceptually, our
algorithm shows that randomization along with a more direct approach can
provide better bounds.Comment: To be presented at the Symposium on Discrete Algorithms (SODA) 201
Communication tree problems
In this paper, we consider random communication
requirements and several cost
measures for a particular model of tree routing on a
complete network. First
we show that a random tree does not give any approximation.
Then give
approximation algorithms for the case for two random models
of requirements.Postprint (published version
Transitions in spatial networks
Networks embedded in space can display all sorts of transitions when their
structure is modified. The nature of these transitions (and in some cases
crossovers) can differ from the usual appearance of a giant component as
observed for the Erdos-Renyi graph, and spatial networks display a large
variety of behaviors. We will discuss here some (mostly recent) results about
topological transitions, `localization' transitions seen in the shortest paths
pattern, and also about the effect of congestion and fluctuations on the
structure of optimal networks. The importance of spatial networks in real-world
applications makes these transitions very relevant and this review is meant as
a step towards a deeper understanding of the effect of space on network
structures.Comment: Corrected version and updated list of reference
Parameterized complexity of the spanning tree congestion problem
We study the problem of determining the spanning tree congestion of a graph. We present some sharp contrasts in the parameterized complexity of this problem. First, we show that on apex-minor-free graphs, a general class of graphs containing planar graphs, graphs of bounded treewidth, and graphs of bounded genus, the problem to determine whether a given graph has spanning tree congestion at most k can be solved in linear time for every fixed k. We also show that for every fixed k and d the problem is solvable in linear time for graphs of degree at most d. In contrast, if we allow only one vertex of unbounded degree, the problem immediately becomes NP-complete for any fixed k≥8. Moreover, the hardness result holds for graphs excluding the complete graph on 6 vertices as a minor. We also observe that for k≤3 the problem becomes polynomially time solvable.publishedVersio
Maximum Edge-Disjoint Paths in -sums of Graphs
We consider the approximability of the maximum edge-disjoint paths problem
(MEDP) in undirected graphs, and in particular, the integrality gap of the
natural multicommodity flow based relaxation for it. The integrality gap is
known to be even for planar graphs due to a simple
topological obstruction and a major focus, following earlier work, has been
understanding the gap if some constant congestion is allowed.
In this context, it is natural to ask for which classes of graphs does a
constant-factor constant-congestion property hold. It is easy to deduce that
for given constant bounds on the approximation and congestion, the class of
"nice" graphs is nor-closed. Is the converse true? Does every proper
minor-closed family of graphs exhibit a constant factor, constant congestion
bound relative to the LP relaxation? We conjecture that the answer is yes.
One stumbling block has been that such bounds were not known for bounded
treewidth graphs (or even treewidth 3). In this paper we give a polytime
algorithm which takes a fractional routing solution in a graph of bounded
treewidth and is able to integrally route a constant fraction of the LP
solution's value. Note that we do not incur any edge congestion. Previously
this was not known even for series parallel graphs which have treewidth 2. The
algorithm is based on a more general argument that applies to -sums of
graphs in some graph family, as long as the graph family has a constant factor,
constant congestion bound. We then use this to show that such bounds hold for
the class of -sums of bounded genus graphs
- …