1,659 research outputs found
Sparse Fault-Tolerant BFS Trees
This paper addresses the problem of designing a sparse {\em fault-tolerant}
BFS tree, or {\em FT-BFS tree} for short, namely, a sparse subgraph of the
given network such that subsequent to the failure of a single edge or
vertex, the surviving part of still contains a BFS spanning tree for
(the surviving part of) . Our main results are as follows. We present an
algorithm that for every -vertex graph and source node constructs a
(single edge failure) FT-BFS tree rooted at with O(n \cdot
\min\{\Depth(s), \sqrt{n}\}) edges, where \Depth(s) is the depth of the BFS
tree rooted at . This result is complemented by a matching lower bound,
showing that there exist -vertex graphs with a source node for which any
edge (or vertex) FT-BFS tree rooted at has edges. We then
consider {\em fault-tolerant multi-source BFS trees}, or {\em FT-MBFS trees}
for short, aiming to provide (following a failure) a BFS tree rooted at each
source for some subset of sources . Again, tight bounds
are provided, showing that there exists a poly-time algorithm that for every
-vertex graph and source set of size constructs a
(single failure) FT-MBFS tree from each source , with
edges, and on the other hand there exist
-vertex graphs with source sets of cardinality , on
which any FT-MBFS tree from has edges.
Finally, we propose an approximation algorithm for constructing
FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result
stating that there exists no approximation algorithm for these
problems under standard complexity assumptions
Does Advice Help to Prove Propositional Tautologies?
One of the starting points of propositional proof complexity is the seminal paper by Cook and Reckhow [6], where they defined propositional proof systems as poly-time computable functions which have all propositional tautologies as their range. Motivated by provability consequences in bounded arithmetic, Cook and Krajíček [5] have recently started the investigation of proof systems which are computed by poly-time functions using advice. While this yields a more powerful model, it is also less directly applicable in practice. In this note we investigate the question whether the usage of advice in propositional proof systems can be simplified or even eliminated. While in principle, the advice can be very complex, we show that proof systems with logarithmic advice are also computable in poly-time with access to a sparse NP-oracle. In addition, we show that if advice is ”not very helpful” for proving tautologies, then there exists an optimal propositional proof system without advice. In our main result, we prove that advice can be transferred from the proof to the formula, leading to an easier computational model. We obtain this result by employing a recent technique by Buhrman and Hitchcock [4]
A Casual Tour Around a Circuit Complexity Bound
I will discuss the recent proof that the complexity class NEXP
(nondeterministic exponential time) lacks nonuniform ACC circuits of polynomial
size. The proof will be described from the perspective of someone trying to
discover it.Comment: 21 pages, 2 figures. An earlier version appeared in SIGACT News,
September 201
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