53 research outputs found
Hitting all Maximal Independent Sets of a Bipartite Graph
We prove that given a bipartite graph G with vertex set V and an integer k,
deciding whether there exists a subset of V of size k hitting all maximal
independent sets of G is complete for the class Sigma_2^P.Comment: v3: minor chang
Improving Strategies via SMT Solving
We consider the problem of computing numerical invariants of programs by
abstract interpretation. Our method eschews two traditional sources of
imprecision: (i) the use of widening operators for enforcing convergence within
a finite number of iterations (ii) the use of merge operations (often, convex
hulls) at the merge points of the control flow graph. It instead computes the
least inductive invariant expressible in the domain at a restricted set of
program points, and analyzes the rest of the code en bloc. We emphasize that we
compute this inductive invariant precisely. For that we extend the strategy
improvement algorithm of [Gawlitza and Seidl, 2007]. If we applied their method
directly, we would have to solve an exponentially sized system of abstract
semantic equations, resulting in memory exhaustion. Instead, we keep the system
implicit and discover strategy improvements using SAT modulo real linear
arithmetic (SMT). For evaluating strategies we use linear programming. Our
algorithm has low polynomial space complexity and performs for contrived
examples in the worst case exponentially many strategy improvement steps; this
is unsurprising, since we show that the associated abstract reachability
problem is Pi-p-2-complete
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Opacity Issues in Games with Imperfect Information
We study in depth the class of games with opacity condition, which are
two-player games with imperfect information in which one of the players only
has imperfect information, and where the winning condition relies on the
information he has along the play. Those games are relevant for security
aspects of computing systems: a play is opaque whenever the player who has
imperfect information never "knows" for sure that the current position is one
of the distinguished "secret" positions. We study the problems of deciding the
existence of a winning strategy for each player, and we call them the
opacity-violate problem and the opacity-guarantee problem. Focusing on the
player with perfect information is new in the field of games with
imperfect-information because when considering classical winning conditions it
amounts to solving the underlying perfect-information game. We establish the
EXPTIME-completeness of both above-mentioned problems, showing that our winning
condition brings a gap of complexity for the player with perfect information,
and we exhibit the relevant opacity-verify problem, which noticeably
generalizes approaches considered in the literature for opacity analysis in
discrete-event systems. In the case of blindfold games, this problem relates to
the two initial ones, yielding the determinacy of blindfold games with opacity
condition and the PSPACE-completeness of the three problems.Comment: In Proceedings GandALF 2011, arXiv:1106.081
- …