19,277 research outputs found
Algorithmic Aspects of Switch Cographs
This paper introduces the notion of involution module, the first
generalization of the modular decomposition of 2-structure which has a unique
linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm
and we take advantage of the involution modular decomposition tree to state
several algorithmic results. Cographs are the graphs that are totally
decomposable w.r.t modular decomposition. In a similar way, we introduce the
class of switch cographs, the class of graphs that are totally decomposable
w.r.t involution modular decomposition. This class generalizes the class of
cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We
use our new decomposition tool to design three practical algorithms for the
maximum cut, vertex cover and vertex separator problems. The complexity of
these problems was still unknown for this class of graphs. This paper also
improves the complexity of the maximum clique, the maximum independant set, the
chromatic number and the maximum clique cover problems by giving efficient
algorithms, thanks to the decomposition tree. Eventually, we show that this
class of graphs has Clique-Width at most 4 and that a Clique-Width expression
can be computed in linear time
Motion Planning for Unlabeled Discs with Optimality Guarantees
We study the problem of path planning for unlabeled (indistinguishable)
unit-disc robots in a planar environment cluttered with polygonal obstacles. We
introduce an algorithm which minimizes the total path length, i.e., the sum of
lengths of the individual paths. Our algorithm is guaranteed to find a solution
if one exists, or report that none exists otherwise. It runs in time
, where is the number of robots and is the total
complexity of the workspace. Moreover, the total length of the returned
solution is at most , where OPT is the optimal solution cost. To
the best of our knowledge this is the first algorithm for the problem that has
such guarantees. The algorithm has been implemented in an exact manner and we
present experimental results that attest to its efficiency
Multi-Embedding of Metric Spaces
Metric embedding has become a common technique in the design of algorithms.
Its applicability is often dependent on how high the embedding's distortion is.
For example, embedding finite metric space into trees may require linear
distortion as a function of its size. Using probabilistic metric embeddings,
the bound on the distortion reduces to logarithmic in the size.
We make a step in the direction of bypassing the lower bound on the
distortion in terms of the size of the metric. We define "multi-embeddings" of
metric spaces in which a point is mapped onto a set of points, while keeping
the target metric of polynomial size and preserving the distortion of paths.
The distortion obtained with such multi-embeddings into ultrametrics is at most
O(log Delta loglog Delta) where Delta is the aspect ratio of the metric. In
particular, for expander graphs, we are able to obtain constant distortion
embeddings into trees in contrast with the Omega(log n) lower bound for all
previous notions of embeddings.
We demonstrate the algorithmic application of the new embeddings for two
optimization problems: group Steiner tree and metrical task systems
Motifs in Temporal Networks
Networks are a fundamental tool for modeling complex systems in a variety of
domains including social and communication networks as well as biology and
neuroscience. Small subgraph patterns in networks, called network motifs, are
crucial to understanding the structure and function of these systems. However,
the role of network motifs in temporal networks, which contain many timestamped
links between the nodes, is not yet well understood.
Here we develop a notion of a temporal network motif as an elementary unit of
temporal networks and provide a general methodology for counting such motifs.
We define temporal network motifs as induced subgraphs on sequences of temporal
edges, design fast algorithms for counting temporal motifs, and prove their
runtime complexity. Our fast algorithms achieve up to 56.5x speedup compared to
a baseline method. Furthermore, we use our algorithms to count temporal motifs
in a variety of networks. Results show that networks from different domains
have significantly different motif counts, whereas networks from the same
domain tend to have similar motif counts. We also find that different motifs
occur at different time scales, which provides further insights into structure
and function of temporal networks
Efficient Local Search in Coordination Games on Graphs
We study strategic games on weighted directed graphs, where the payoff of a
player is defined as the sum of the weights on the edges from players who chose
the same strategy augmented by a fixed non-negative bonus for picking a given
strategy. These games capture the idea of coordination in the absence of
globally common strategies. Prior work shows that the problem of determining
the existence of a pure Nash equilibrium for these games is NP-complete already
for graphs with all weights equal to one and no bonuses. However, for several
classes of graphs (e.g. DAGs and cliques) pure Nash equilibria or even strong
equilibria always exist and can be found by simply following a particular
improvement or coalition-improvement path, respectively. In this paper we
identify several natural classes of graphs for which a finite improvement or
coalition-improvement path of polynomial length always exists, and, as a
consequence, a Nash equilibrium or strong equilibrium in them can be found in
polynomial time. We also argue that these results are optimal in the sense that
in natural generalisations of these classes of graphs, a pure Nash equilibrium
may not even exist.Comment: Extended version of a paper accepted to IJCAI1
Rerouting shortest paths in planar graphs
A rerouting sequence is a sequence of shortest st-paths such that consecutive
paths differ in one vertex. We study the the Shortest Path Rerouting Problem,
which asks, given two shortest st-paths P and Q in a graph G, whether a
rerouting sequence exists from P to Q. This problem is PSPACE-hard in general,
but we show that it can be solved in polynomial time if G is planar. To this
end, we introduce a dynamic programming method for reconfiguration problems.Comment: submitte
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