Pushing the frontier of minimality

The Minimal Constraint Satisfaction Problem, or Minimal CSP for short, arises in a number of real-world applications, most notably in constraint-based product configuration. It is composed of the set of CSP problems where every allowed tuple can be extended to a solution. Despite the very restrictive structure, computing a solution to a Minimal CSP instance is NP-hard in the general case. In this paper, we look at three independent ways to add further restrictions to the problem. First, we bound the size of the domains. Second, we define the arity as a function on the number of variables. Finally we study the complexity of computing a solution to a Minimal CSP instance when not just every allowed tuple, but every partial solution smaller than a given size, can be extended to a solution. In all three cases, we show that finding a solution remains NP-hard. All these results reveal that the hardness of minimality is very robust

k-Step Relative Inductive Generalization

We introduce a new form of SAT-based symbolic model checking. One common idea in SAT-based symbolic model checking is to generate new clauses from states that can lead to property violations. Our previous work suggests applying induction to generalize from such states. While effective on some benchmarks, the main problem with inductive generalization is that not all such states can be inductively generalized at a given time in the analysis, resulting in long searches for generalizable states on some benchmarks. This paper introduces the idea of inductively generalizing states relative to $k$-step over-approximations: a given state is inductively generalized relative to the latest $k$-step over-approximation relative to which the negation of the state is itself inductive. This idea motivates an algorithm that inductively generalizes a given state at the highest level $k$ so far examined, possibly by generating more than one mutually $k$-step relative inductive clause. We present experimental evidence that the algorithm is effective in practice.Comment: 14 page

Explicit rank bounds for cyclic covers

Let $M$ be a closed, orientable hyperbolic 3-manifold and $\phi$ a homomorphism of its fundamental group onto $\mathbb{Z}$ that is not induced by a fibration over the circle. For each natural number $n$ we give an explicit lower bound, linear in $n$, on rank of the fundamental group of the cover of $M$ corresponding to $\phi^{-1}(n\mathbb{Z})$. The key new ingredient is the following result: for such a manifold $M$ and a connected, two-sided incompressible surface of genus $g$ in $M$ that is not a fiber or semi-fiber, a reduced homotopy in $(M,S)$ has length at most $14g-12$.Comment: 21 pages; changes suggested by a referee. Most are minor, but the previous Lemma 3.5 has been removed and all dependence on it has been written ou

Sweepouts of amalgamated 3-manifolds

We show that if two 3-manifolds with toroidal boundary are glued via a `sufficiently complicated' map then every Heegaard splitting of the resulting 3-manifold is weakly reducible. Additionally, if Z is a manifold obtained by gluing X and Y, two connected small manifolds with incompressible boundary, along a closed surface F. Then the genus g(Z) of Z is greater than or equal to 1/2(g(X)+g(Y)-2g(F)). Both results follow from a new technique to simplify the intersection between an incompressible surface and a strongly irreducible Heegaard splitting.Comment: This is the version published by Algebraic & Geometric Topology on 24 February 200