[1] Corresponding author e-mail: [email protected]   [1] Corresponding author e-mail: [email protected]   Lot-sizing problems (LSPs) belong to the class of production planning problems in which the availability quantities of the production plan are always considered as a decision variable. This paper aims at developing a new mathematical model for the multi-level ca...

Let P = {C1, C2, . . . , Cn} be a set of color classes, where each color class Ci consists of a set of points. In this paper, we address a family of covering problems, in which one is allowed to cover at most one point from each color class. We prove that the problems in this family are NP-complete (or NP-hard) and offer several constant-factor approximation algorithms.

Computing the clique number and chromatic number of a general graph are well-known to be NP-Hard problems. Codenotti et al. (Bruno Codenotti, Ivan Gerace, and Sebastiano Vigna. Hardness results and spectral techniques for combinatorial problems on circulant graphs. Linear Algebra Appl., 285(1-3): 123–142, 1998) showed that computing the clique number and chromatic number are still NP-Hard probl...

A generally-accepted minimum requirement for an algorithm to be considered ‘efficient’ is that its running time is polynomial: O(nc) for some constant c, where n is the size of the input.1 Researchers recognized early on that not all problems can be solved this quickly, but we had a hard time figuring out exactly which ones could and which ones couldn’t. there are several so-called NP-hard prob...

We survey a collective achievement of a group of researchers: the PCP Theorems. They give new definitions of the class NP, and imply that computing approximate solutions to many NP-hard problems is itself NP-hard. Techniques developed to prove them have had many other consequences. 2000 Mathematics Subject Classification: 68Q10, 68Q15, 68Q17, 68Q25.

Trust region method for a class of large-scale minimization problems, the unconstrained discrete-time optimal control (DTOC) problems, is considered. Although the trust region algorithms developed in 4] and 13] are very economical they lack the ability to handle the so-called hard case. In this paper, We show that the trust region subproblem can be solved within an acceptable accuracy without f...

We study classic cake-cutting problems, but in discrete models rather than using infiniteprecision real values, specifically, focusing on their communication complexity. Using general discrete simulations of classical infinite-precision protocols (Robertson-Webb and movingknife), we roughly partition the various fair-allocation problems into 3 classes: “easy” (constant number of rounds of logar...

Lagrangian relaxation is a powerful bounding technique that has been applied successfully to many / /9-hard combinatorial optimization problems. The basic idea is to see an ./K~-hard problem as an "easy-to-solve" problem complicated by a number of "nasty" side constraints. We show that reformulating nasty inequality constraints as equalities by using slack variables leads to stronger lower boun...

Many combinatorial optimisation problems can be modelled as integer linear programs. We consider a class of generalised integer programs where the constraints are allowed to be taken from a broader set of relations (instead of just being linear inequalities). The set of allowed relations is defined using a many-valued logic and the resulting class of relations have provably strong modelling pro...