Cardinality of relations and relational approximation algorithms

Cardinality of relations and relational approximation algorithms

First, we discuss three specific classes of relations, which allow to model the essential constituents of graphs, such as vertices and (directed or undirected) edges. Based on Kawahara’s characterisation of the cardinality of relations we then derive fundamental properties on their cardinalities. As main applications of these results, we formally verify four relational programs, which implement...

Cardinality and Equivalence Relations

Cardinality of Equivalence Relations

This entry provides formulae for counting the number of equivalence relations and partial equivalence relations over a finite carrier set with given cardinality. To count the number of equivalence relations, we provide bijections between equivalence relations and set partitions [4], and then transfer the main results of the two AFP entries, Cardinality of Set Partitions [1] and Spivey’s General...

Approximation Algorithms for Knapsack Problems with Cardinality Constraintsy

We address a variant of the classical knapsack problem in which an upper bound is imposed on the number of items that can be selected. This problem arises in the solution of real-life cutting stock problems by column generation, and may be used to separate cover inequalities with small support within cutting plane approaches to integer linear programs. We focus our attention on approximation al...

Improved Approximation Algorithms for Max-2SAT with Cardinality Constraint

The optimization problem Max-2SAT-CC is Max-2SAT with the additional cardinality constraint that the value one may be assigned to at most K variables. We present an approximation algorithm with polynomial running time for Max-2SAT-CC. This algorithm achieves, for any > 0, approximation ratio 6+3·e 16+2·e − ≈ 0.6603. Furthermore, we present a greedy algorithm with running time O(N logN) and appr...