11,658 research outputs found
Search for an Immobile Hider on a Stochastic Network
Harry hides on an edge of a graph and does not move from there. Sally,
starting from a known origin, tries to find him as soon as she can. Harry's
goal is to be found as late as possible. At any given time, each edge of the
graph is either active or inactive, independently of the other edges, with a
known probability of being active. This situation can be modeled as a zero-sum
two-person stochastic game. We show that the game has a value and we provide
upper and lower bounds for this value. Finally, by generalizing optimal
strategies of the deterministic case, we provide more refined results for trees
and Eulerian graphs.Comment: 28 pages, 9 figure
Walking Through Waypoints
We initiate the study of a fundamental combinatorial problem: Given a
capacitated graph , find a shortest walk ("route") from a source to a destination that includes all vertices specified by a set
: the \emph{waypoints}. This waypoint routing problem
finds immediate applications in the context of modern networked distributed
systems. Our main contribution is an exact polynomial-time algorithm for graphs
of bounded treewidth. We also show that if the number of waypoints is
logarithmically bounded, exact polynomial-time algorithms exist even for
general graphs. Our two algorithms provide an almost complete characterization
of what can be solved exactly in polynomial-time: we show that more general
problems (e.g., on grid graphs of maximum degree 3, with slightly more
waypoints) are computationally intractable
Counting Euler Tours in Undirected Bounded Treewidth Graphs
We show that counting Euler tours in undirected bounded tree-width graphs is
tractable even in parallel - by proving a upper bound. This is in
stark contrast to #P-completeness of the same problem in general graphs.
Our main technical contribution is to show how (an instance of) dynamic
programming on bounded \emph{clique-width} graphs can be performed efficiently
in parallel. Thus we show that the sequential result of Espelage, Gurski and
Wanke for efficiently computing Hamiltonian paths in bounded clique-width
graphs can be adapted in the parallel setting to count the number of
Hamiltonian paths which in turn is a tool for counting the number of Euler
tours in bounded tree-width graphs. Our technique also yields parallel
algorithms for counting longest paths and bipartite perfect matchings in
bounded-clique width graphs.
While establishing that counting Euler tours in bounded tree-width graphs can
be computed by non-uniform monotone arithmetic circuits of polynomial degree
(which characterize ) is relatively easy, establishing a uniform
bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound
claimed in the previous version which has been subsequently remove
A parallel interaction potential approach coupled with the immersed boundary method for fully resolved simulations of deformable interfaces and membranes
In this paper we show and discuss the use of a versatile interaction
potential approach coupled with an immersed boundary method to simulate a
variety of flows involving deformable bodies. In particular, we focus on two
kinds of problems, namely (i) deformation of liquid-liquid interfaces and (ii)
flow in the left ventricle of the heart with either a mechanical or a natural
valve. Both examples have in common the two-way interaction of the flow with a
deformable interface or a membrane. The interaction potential approach (de
Tullio & Pascazio, Jou. Comp. Phys., 2016; Tanaka, Wada and Nakamura,
Computational Biomechanics, 2016) with minor modifications can be used to
capture the deformation dynamics in both classes of problems. We show that the
approach can be used to replicate the deformation dynamics of liquid-liquid
interfaces through the use of ad-hoc elastic constants. The results from our
simulations agree very well with previous studies on the deformation of drops
in standard flow configurations such as deforming drop in a shear flow or a
cross flow. We show that the same potential approach can also be used to study
the flow in the left ventricle of the heart. The flow imposed into the
ventricle interacts dynamically with the mitral valve (mechanical or natural)
and the ventricle which are simulated using the same model. Results from these
simulations are compared with ad- hoc in-house experimental measurements.
Finally, a parallelisation scheme is presented, as parallelisation is
unavoidable when studying large scale problems involving several thousands of
simultaneously deforming bodies on hundreds of distributed memory computing
processors
When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks
Let be a supply graph and a demand graph defined on the
same set of vertices. An assignment of capacities to the edges of and
demands to the edges of is said to satisfy the \emph{cut condition} if for
any cut in the graph, the total demand crossing the cut is no more than the
total capacity crossing it. The pair is called \emph{cut-sufficient} if
for any assignment of capacities and demands that satisfy the cut condition,
there is a multiflow routing the demands defined on within the network with
capacities defined on . We prove a previous conjecture, which states that
when the supply graph is series-parallel, the pair is
cut-sufficient if and only if does not contain an \emph{odd spindle} as
a minor; that is, if it is impossible to contract edges of and delete edges
of and so that becomes the complete bipartite graph , with
odd, and is composed of a cycle connecting the vertices of
degree 2, and an edge connecting the two vertices of degree . We further
prove that if the instance is \emph{Eulerian} --- that is, the demands and
capacities are integers and the total of demands and capacities incident to
each vertex is even --- then the multiflow problem has an integral solution. We
provide a polynomial-time algorithm to find an integral solution in this case.
In order to prove these results, we formulate properties of tight cuts (cuts
for which the cut condition inequality is tight) in cut-sufficient pairs. We
believe these properties might be useful in extending our results to planar
graphs.Comment: An extended abstract of this paper will be published at the 44th
Symposium on Theory of Computing (STOC 2012
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