We present a work in progress describing attribute grammar approaches to Grammatical Evolution, which allow us to encode context-sensitive and semantic information. Performance of the different grammars adopted are directly compared with a more traditional GA representation on five instances of an NP-hard knapsack problem. The results presented are encouraging, demonstrating that Grammatical Ev...

This paper develops an algorithm based on the Modular Approach to solve singly constrained separable discrete optimization problems (Nonlinear Knapsack Problems). The Modular Approach uses fathoming and integration techniques repeatedly. The fathoming reduces the decision space of variables. The integration reduces the number of variables in the problem by combining several variables into one v...

The program is written in Java, hence it should in principle run in any browser having Java enabled. The present version has mainly been tested using the Windows browser Explorer, hence if you have problems, please use the Windows computers in Multimedie Lab, 1st floor, south. You may also download all the relevant files index.html, AppletTest.jar, data.1, data.2, data.3 to your home directory ...

This paper deals with a general mixed integer knapsack polyhedron for which we introduce and analyze a new family of inequalities. We discuss the value of this family both from a theoretic and a computational point of view.

Consider the problem of partitioning a group of b indistinguishable objects into subgroups, each of size at leastl and at most u. The objective is to minimize the additive separable cost of the partition, where the cost associated with a subgroup of size j is c(j). In the case that c(.) is convex, we show how to solve the problem in O(log u-/ + 1) steps. In the case that c(.) is concave, we sol...

The multiple knapsack problem (MKP) is a classical combinatorial optimization problem. A recent algorithm for some classes of the MKP is bincompletion, a bin-oriented, branch-and-bound algorithm. We investigate mechanisms for detecting and breaking symmetry in the bin-completion search space. We propose path-symmetry and pathdominance, two new symmetry relations which are more powerful generali...

In the last lecture, we studied the KNAPSACK problem. The KNAPSACK problem is an NP-hard problem but does admit a pseudo-polynomial time algorithm and can be solved efficiently if the input size is small. We used this pseudo-polynomial time algorithm to obtain an FPTAS for KNAPSACK. In this lecture, we study another class of problems, known as strongly NP-hard problems. Definition 1 (Strongly N...

The most efficient algorithms for solving the single-criterion {0, 1}-knapsack problem are based on the core concept (i.e., based on a small number of relevant variables). But this concept is not used in problems with more than one criterion. The main purpose of this paper is to validate the existence of such a set of variables in bi-criteria {0.1}-knapsack instances. Numerical experiments were...