Lower bounds and heuristic algorithms for the ki-partitioning problem

Abstract

We consider the problem of partitioning a set of positive integers values into a given number of subsets, each having an associated cardinality limit, so that the maximum sum of values in a subset is minimized, and the number of values in each subset does not exceed the corresponding limit. The problem is related to scheduling and bin packing problems. We give combinatorial lower bounds, reduction criteria, constructive heuristics, a scatter search approach, and a lower bound based on column generation. The outcome of extensive computational experiments is presented.

EPTAS for the dual of splittable bin packing with cardinality constraint

“…Since each bin in B c has a total size of small items in the interval [γ c ε 2 , (γ c + 1)ε 2 ), so over the y c bins in B c we have a total size of small items that is not larger than y c (γ c + 1)ε 2 , and Constraint (8) is satisfied. Constraint ( 9), ( 10), (11), and ( 12) are satisfied trivially based on the definition of the v, x, y, and z.…”

EPTAS for parallel identical machine scheduling with time restrictions

“…Similarly, a large job will also be placed in only one configuration. Thus constraints ( 6), ( 7), (8), and (10) are satisfied. Recall that T S is the set of short containers.…”

Approximation Schemes for the 3-Partitioning Problems

“…For the min-max objective, Babel, et al [2] showed the relationship between the scheduling problems and the k-partitioning problem, and devised a -approximation algorithm. Upper (lower) bounds 4 / 3 and heuristic algorithms for the min-max k-partitioning problem can be found in [7][8][9]. He et al [11] investigated the max-min k-partitioning problem and presented an algorithm with performance ratio .…”