Partitioning a weighted partial order

Abstract

The problem of partitioning a partially ordered set into a minimum number of chains is a well-known problem. In this paper we study a generalization of this problem, where we not only assume that the chains have bounded size, but also that a weight w i is given for each element i in the partial order such that w i ≤ w j if i ≺ j. The problem is then to partition the partial order into a minimum-weight set of chains of bounded size, where the weight of a chain equals the weight of the heaviest element in the chain. We prove that this problem is APX-hard, and we propose and analyze lower bounds for this problem. Based on these lower bounds, we exhibit a 2-approximation algorithm, and show that it is tight. We report computational results for a number of real-world and randomly generated problem instances.

Perfect Phylogenies via Branchings in Acyclic Digraphs and a Generalization of Dilworth’s Theorem

“…The min-max theorem has a constructive algorithmic proof, leading to a polynomial-time algorithm to compute a chain partition of a given partially ordered set equipped with a monotone weight function such that the of sum of the maximum weights in the chains is minimized. This result contrasts with known results in the literature implying that two natural variants of the problem are NP-hard: (i) the variant in which the chains used in the partition have to be of bounded size [25,29], and (ii) the variant in which the weight function is not necessarily monotone, which corresponds to a variant of the graph coloring problem known as Weighted Coloring (see, e.g., [3,6,8,15]), in the class of cocomparability graphs. We refer to the remarks following Corollary 3.5 in Section 3.1 for more details.…”

“…Two remarks are in order here, showing that the result of Corollary 3.5 is sharp in two ways. First, let us note that the variant of the MinimumPriceChainPartition problem in which the chains used in the partition have to be of bounded size was studied by Moonen and Spieksma in [25], who described a practical application encountered at Bruynzeel Storage Systems, a manufacturing company in the Netherlands, to a problem of optimally loading pallets on a truck. 3 Moonen and Spieksma referred to the problem as "Minimum Weight Partition into B-chains" (where B is the upper bound on the size of the chains) and showed that the problem is APX-hard even in the case of unit weights, strengthening the previous NP-hardness result from [29].…”

Some results on triangle partitions

“…The following problem has been investigated in great detail due to its applications in various scheduling problems. Suppose we wish to find a vertex coloring with a minimal number of colors, such that each color class contains at most q vertices [30,32,35,42,49,50]. This problem is NP-complete on permutation graphs for each q 6 [34].…”