9,645 research outputs found
On Algorithms and Complexity for Sets with Cardinality Constraints
Typestate systems ensure many desirable properties of imperative programs,
including initialization of object fields and correct use of stateful library
interfaces. Abstract sets with cardinality constraints naturally generalize
typestate properties: relationships between the typestates of objects can be
expressed as subset and disjointness relations on sets, and elements of sets
can be represented as sets of cardinality one. Motivated by these applications,
this paper presents new algorithms and new complexity results for constraints
on sets and their cardinalities. We study several classes of constraints and
demonstrate a trade-off between their expressive power and their complexity.
Our first result concerns a quantifier-free fragment of Boolean Algebra with
Presburger Arithmetic. We give a nondeterministic polynomial-time algorithm for
reducing the satisfiability of sets with symbolic cardinalities to constraints
on constant cardinalities, and give a polynomial-space algorithm for the
resulting problem.
In a quest for more efficient fragments, we identify several subclasses of
sets with cardinality constraints whose satisfiability is NP-hard. Finally, we
identify a class of constraints that has polynomial-time satisfiability and
entailment problems and can serve as a foundation for efficient program
analysis.Comment: 20 pages. 12 figure
The Satisfiability Threshold for k-XORSAT
We consider "unconstrained" random -XORSAT, which is a uniformly random
system of linear non-homogeneous equations in over
variables, each equation containing variables, and also consider a
"constrained" model where every variable appears in at least two equations.
Dubois and Mandler proved that is a sharp threshold for satisfiability
of constrained 3-XORSAT, and analyzed the 2-core of a random 3-uniform
hypergraph to extend this result to find the threshold for unconstrained
3-XORSAT.
We show that remains a sharp threshold for satisfiability of
constrained -XORSAT for every , and we use standard results on the
2-core of a random -uniform hypergraph to extend this result to find the
threshold for unconstrained -XORSAT. For constrained -XORSAT we narrow
the phase transition window, showing that implies almost-sure
satisfiability, while implies almost-sure unsatisfiability.Comment: Version 2 adds sharper phase transition result, new citation in
literature survey, and improvements in presentation; removes Appendix
treating k=
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