The complexity of finite-valued CSPs

We study the computational complexity of exact minimisation of rational-valued discrete functions. Let $\Gamma$ be a set of rational-valued functions on a fixed finite domain; such a set is called a finite-valued constraint language. The valued constraint satisfaction problem, $\operatorname{VCSP}(\Gamma)$, is the problem of minimising a function given as a sum of functions from $\Gamma$. We establish a dichotomy theorem with respect to exact solvability for all finite-valued constraint languages defined on domains of arbitrary finite size. We show that every constraint language $\Gamma$ either admits a binary symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of $\operatorname{VCSP}(\Gamma)$ exactly, or $\Gamma$ satisfies a simple hardness condition that allows for a polynomial-time reduction from Max-Cut to $\operatorname{VCSP}(\Gamma)$

Lot sizing with piecewise concave production costs

Cataloged from PDF version of article.We study the lot-sizing problem with piecewise concave production costs and concave holding costs. This problem is a generalization of the lot-sizing problem with quantity discounts, minimum order quantities, capacities, overloading, subcontracting or a combination of these. We develop a dynamic programming algorithm to solve this problem and answer an open question in the literature: we show that the problem is polynomially solvable when the breakpoints of the production cost function are time invariant and the number of breakpoints is fixed. For the special cases with capacities and subcontracting, the time complexity of our algorithm is as good as the complexity of algorithms available in the literature. We report the results of a computational experiment where the dynamic programming is able to solve instances that are hard for a mixed-integer programming solver. We enhance the mixed-integer programming formulation with valid inequalities based on mixing sets and use a cut-and-branch algorithm to compute better bounds. We propose a state space reduction–based heuristic algorithm for large instances and show that the solutions are of good quality by comparing them with the bounds obtained from the cut-and-branch

Is the Optimal Implementation Inefficient? Elementarily Not

Sharing graphs are a local and asynchronous implementation of lambda-calculus beta-reduction (or linear logic proof-net cut-elimination) that avoids useless duplications. Empirical benchmarks suggest that they are one of the most efficient machineries, when one wants to fully exploit the higher-order features of lambda-calculus. However, we still lack confirming grounds with theoretical solidity to dispel uncertainties about the adoption of sharing graphs. Aiming at analysing in detail the worst-case overhead cost of sharing operators, we restrict to the case of elementary and light linear logic, two subsystems with bounded computational complexity of multiplicative exponential linear logic. In these two cases, the bookkeeping component is unnecessary, and sharing graphs are simplified to the so-called "abstract algorithm". By a modular cost comparison over a syntactical simulation, we prove that the overhead of shared reductions is quadratically bounded to cost of the naive implementation, i.e. proof-net reduction. This result generalises and strengthens a previous complexity result, and implies that the price of sharing is negligible, if compared to the obtainable benefits on reductions requiring a large amount of duplication

Soft Session Types

We show how systems of session types can enforce interactions to be bounded for all typable processes. The type system we propose is based on Lafont's soft linear logic and is strongly inspired by recent works about session types as intuitionistic linear logic formulas. Our main result is the existence, for every typable process, of a polynomial bound on the length of any reduction sequence starting from it and on the size of any of its reducts.Comment: In Proceedings EXPRESS 2011, arXiv:1108.407