On the Path-Width of Integer Linear Programming

We consider the feasibility problem of integer linear programming (ILP). We show that solutions of any ILP instance can be naturally represented by an FO-definable class of graphs. For each solution there may be many graphs representing it. However, one of these graphs is of path-width at most 2n, where n is the number of variables in the instance. Since FO is decidable on graphs of bounded path- width, we obtain an alternative decidability result for ILP. The technique we use underlines a common principle to prove decidability which has previously been employed for automata with auxiliary storage. We also show how this new result links to automata theory and program verification.Comment: In Proceedings GandALF 2014, arXiv:1408.556

A Decidable Class of Nested Iterated Schemata (extended version)

Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called "iterated schemata", allow to express such patterns. Schemata extend propositional logic with indexed propositions, e.g. P_i, P_i+1, P_1, and with generalized connectives, e.g. /\i=1..n or i=1..n (called "iterations") where n is an (unbound) integer variable called a "parameter". The expressive power of iterated schemata is strictly greater than propositional logic: it is even out of the scope of first-order logic. We define a proof procedure, called DPLL*, that can prove that a schema is satisfiable for at least one value of its parameter, in the spirit of the DPLL procedure. However the converse problem, i.e. proving that a schema is unsatisfiable for every value of the parameter, is undecidable so DPLL* does not terminate in general. Still, we prove that it terminates for schemata of a syntactic subclass called "regularly nested". This is the first non trivial class for which DPLL* is proved to terminate. Furthermore the class of regularly nested schemata is the first decidable class to allow nesting of iterations, i.e. to allow schemata of the form /\i=1..n (/\j=1..n ...).Comment: 43 pages, extended version of "A Decidable Class of Nested Iterated Schemata", submitted to IJCAR 200

Grad and Classes with Bounded Expansion II. Algorithmic Aspects

Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇r(G). These classes are also characterized by the existence of several partition results such as the existence of low tree-width and low tree-depth colorings. These results lead to several new linear time algorithms, such as an algorithm for counting all the isomorphs of a fixed graph in an input graph or an algorithm for checking whether there exists a subset of vertices of a priori bounded size such that the subgraph induced by this subset satisfies some arbirtrary but fixed first order sentence. We also show that for fixed p, computing the distances between two vertices up to distance p may be performed in constant time per query after a linear time preprocessing. We also show, extending several earlier results, that a class of graphs has sublinear separators if it has sub-exponential expansion. This result result is best possible in general