The maximum saving partition problem

Abstract

The input to the MAXIMUM SAVING PARTITION PROBLEM consists of a set V = {1, . . . , n}, weights w i , a function f, and a family S of feasible subsets of V. The output is a partition (S 1 , . . . , S l ) such that S i ∈ S, and j ∈V w j − l i=1 f (S i ) is maximized. We present a general 1 2 -approximation algorithm, and improved algorithms for special cases of the function f.

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

“…Using the results of Kratochvil [13] on the NP-completeness of PrExt node coloring in bipartite planar graphs for k = 3 and P 13 -free bipartite graphs for k = 5, we deduce: Corollary 1. In bipartite planar graphs, min weighted node coloring is strongly NP-complete and it is not 8 7 − ε-approximable unless P=NP.…”

“…This problem is easily shown NP-hard; it suffices to consider w(v) = 1, ∀v ∈ V and min weighted node coloring becomes the classical node coloring problem. Other versions of weighted colorings have been studied in Hassin and Monnot [8].…”

A better differential approximation ratio for symmetric TSP

“…Introduced in [2,3], this ratio has been first used for studying mathematical programming problems, where the standard ratio is not suitable when very common operations such as "removing a constant" are performed, see for instance [31]. Afterwards, this approach has been considered for the main combinatorial optimization problems, leading to the development of new techniques and interesting results (see for instance [5] for vehicle routing, [20] for several results on graph problems, [10,15,17] for MinColoring, [21] for several weighted versions of graph partitioning, [12,13] for Bin Packing, [7,16] for satisfiability, [11,6] for Set Cover, and very recently [18] for weighted Set Cover, etc.). A survey of many results about differential approximation can be found in the book chapter [4].…”

A survey on the structure of approximation classes

“…After a first paper operationally and mathematically justifying the use of the differential ratio ( [38]), a systematic study of differential approximation of NPO problems has started and still continues. Several results (positive or negative) for classical combinatorial problems have appeared (min coloring [34,42,55], min tsp, max tsp and vehicle routing problems [74,73,18,54], bin packing [35,36], min set cover [19], optimal satisfiability problems [20,45], etc.). Several structural and computational aspects are also investigated in [33,75,72,87].…”