959 research outputs found
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
A Fixed Parameter Tractable Approximation Scheme for the Optimal Cut Graph of a Surface
Given a graph cellularly embedded on a surface of genus , a
cut graph is a subgraph of such that cutting along yields a
topological disk. We provide a fixed parameter tractable approximation scheme
for the problem of computing the shortest cut graph, that is, for any
, we show how to compute a approximation of
the shortest cut graph in time .
Our techniques first rely on the computation of a spanner for the problem
using the technique of brick decompositions, to reduce the problem to the case
of bounded tree-width. Then, to solve the bounded tree-width case, we introduce
a variant of the surface-cut decomposition of Ru\'e, Sau and Thilikos, which
may be of independent interest
Low-Degree Spanning Trees of Small Weight
The degree-d spanning tree problem asks for a minimum-weight spanning tree in
which the degree of each vertex is at most d. When d=2 the problem is TSP, and
in this case, the well-known Christofides algorithm provides a
1.5-approximation algorithm (assuming the edge weights satisfy the triangle
inequality).
In 1984, Christos Papadimitriou and Umesh Vazirani posed the challenge of
finding an algorithm with performance guarantee less than 2 for Euclidean
graphs (points in R^n) and d > 2. This paper gives the first answer to that
challenge, presenting an algorithm to compute a degree-3 spanning tree of cost
at most 5/3 times the MST. For points in the plane, the ratio improves to 3/2
and the algorithm can also find a degree-4 spanning tree of cost at most 5/4
times the MST.Comment: conference version in Symposium on Theory of Computing (1994
On the complexity of the bilevel minimum spanning tree problem
We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. We show that this problem is NP-hard, even in the special case where the follower only controls a matching. Moreover, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a (|V|-1)-approximation algorithm for BMST, where |V| is the number of vertices. Moreover, we show that 2-approximating BMST is fixed-parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed-parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting
On the Complexity of the Bilevel Minimum Spanning Tree Problem
We consider the bilevel minimum spanning tree (BMST) problem where the leader
and the follower choose a spanning tree together, according to different
objective functions. By showing that this problem is NP-hard in general, we
answer an open question stated by Shi et al. We prove that BMST remains hard
even in the special case where the follower only controls a matching. Moreover,
by a polynomial reduction from the vertex-disjoint Steiner trees problem, we
give some evidence that BMST might even remain hard in case the follower
controls only few edges.
On the positive side, we present a polynomial-time -approximation
algorithm for BMST, where is the number of vertices in the input graph.
Moreover, considering the number of edges controlled by the follower as
parameter, we show that 2-approximating BMST is fixed-parameter tractable and
that, in case of uniform costs on leader's edges, even solving BMST exactly is
fixed-parameter tractable. We finally consider bottleneck variants of BMST and
settle the complexity landscape of all combinations of sum or bottleneck
objective functions for the leader and follower, for the optimistic as well as
the pessimistic setting
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
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