THE PROBLEM OF PARTITIONING WITH DUPLICATIONS AND ITS APPLICATIONS

1994 ◽  
Vol 03 (03) ◽  
pp. 395-405
Author(s):  
J. HARALAMBIDES ◽  
S. TRAGOUDAS
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Vlsi Layout ◽  
Optimal Polynomial ◽  
Partitioning Problem ◽  
Polynomially Solvable ◽  
The Cost ◽  
Parallel Graph ◽  
Special Case

The problem of partitioning the elements of a graph G=(V, E) into two equal size sets A and B that share at most d elements such that the total number of edges (u, v), u∈A−B, v∈B−A is minimized, arises in the areas of Hypermedia Organization, Network Integrity, and VLSI Layout. We formulate the problem in terms of element duplication, where each element c∈A∩B is substituted by two copies c′∈A and c″∈B As a result, edges incident to c′ or c″ need not count in the cost of the partition. We show that this partitioning problem is NP-hard in general, and we present a solution which utilizes an optimal polynomial time algorithm for the special case where G is a series-parallel graph. We also discuss special other cases where the partitioning problem or variations are polynomially solvable.

A Pseudo-Polynomial Time Algorithm for a Special Multiobjective Order Picking Problem

2015 ◽  
Vol 14 (05) ◽  
pp. 1111-1128 ◽  
Author(s):  
Özgür Özpeynirci ◽  
Cansu Kandemir
Keyword(s):  
Polynomial Time ◽  
Exact Algorithm ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Order Picking ◽  
Special Design ◽  
Nondominated Points ◽  
Polynomially Solvable ◽  
Nondominated Point ◽  
Special Case

In this study, we work on the order picking problem (OPP) in a specially designed warehouse with a single picker. Ratliff and Rosenthal [Operations Research31(3) (1983) 507–521] show that the special design of the warehouse and use of one picker lead to a polynomially solvable case. We address the multiobjective version of this special case and investigate the properties of the nondominated points. We develop an exact algorithm that finds any nondominated point and present an illustrative example. Finally we conduct a computational test and report the results.

scholarly journals Tree Reconstruction from Triplet Cover Distances

10.37236/3388 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Katharina T. Huber ◽  
Mike Steel
Keyword(s):  
Polynomial Time ◽  
Classical Result ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Interior Vertex ◽  
Evolutionary Genomics ◽  
Tree Reconstruction ◽  
Finite Tree ◽  
Path Distance ◽  
Special Case

It is a classical result that any finite tree with positively weighted edges, and without vertices of degree 2, is uniquely determined by the weighted path distance between each pair of leaves. Moreover, it is possible for a (small) strict subset $\mathcal{L}$ of leaf pairs to suffice for reconstructing the tree and its edge weights, given just the distances between the leaf pairs in $\mathcal{L}$. It is known that any set ${\mathcal L}$ with this property for a tree in which all interior vertices have degree 3 must form a cover  for $T$ - that is, for each interior vertex $v$ of $T$, ${\mathcal L}$ must contain a pair of leaves from each pair of the three components of  $T-v$.  Here we provide a partial converse of this result by showing that if a set ${\mathcal L}$ of leaf pairs forms a cover  of a certain type for such a tree $T$ then $T$ and its edge weights can be uniquely determined from the distances between the pairs of leaves in ${\mathcal L}$. Moreover,  there is a polynomial-time algorithm for achieving this reconstruction. The result establishes a special case of a recent question concerning 'triplet covers', and is relevant to a problem arising in evolutionary genomics.

scholarly journals Scheduling Jobs and a Variable Maintenance on a Single Machine with Common Due-Date Assignment

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Long Wan
Keyword(s):  
Single Machine ◽  
Optimal Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Due Date ◽  
Due Date Assignment ◽  
Common Due Date ◽  
Optimal Polynomial ◽  
Special Case ◽  
Identical Jobs

We investigate a common due-date assignment scheduling problem with a variable maintenance on a single machine. The goal is to minimize the total earliness, tardiness, and due-date cost. We derive some properties on an optimal solution for our problem. For a special case with identical jobs we propose an optimal polynomial time algorithm followed by a numerical example.

Optimal Polynomial Time Algorithm for Restoring Multicast Cloud Services

2016 ◽  
Vol 20 (8) ◽  
pp. 1543-1546 ◽  
Author(s):  
Sara Ayoubi ◽  
Chadi Assi ◽  
Lata Narayanan ◽  
Khaled Shaban
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Cloud Services ◽  
Optimal Polynomial

The cost of a cycle is a square

2002 ◽  
Vol 67 (1) ◽  
pp. 35-60 ◽  
Author(s):  
A. Carbone
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Heuristic Algorithms ◽  
Sequent Calculus ◽  
Propositional Calculus ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Proof Search ◽  
The Cost ◽  
Flow Graphs

AbstractThe logical flow graphs of sequent calculus proofs might contain oriented cycles. For the predicate calculus the elimination of cycles might be non-elementary and this was shown in [Car96]. For the propositional calculus, we prove that if a proof of k lines contains n cycles then there exists an acyclic proof with (kn+1) lines. In particular, there is a polynomial time algorithm which eliminates cycles from a proof. These results are motivated by the search for general methods on proving lower bounds on proof size and by the design of more efficient heuristic algorithms for proof search.

scholarly journals Economic lot-sizing problem with remanufacturing option: complexity and algorithms

2021 ◽  
Author(s):  
Ashwin Arulselvan ◽  
Kerem Akartunalı ◽  
Wilco van den Heuvel
Keyword(s):  
Polynomial Time ◽  
Lot Sizing ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Single Item ◽  
Inventory Costs ◽  
Time Period ◽  
Lot Sizing Problem ◽  
Dynamic Lot Sizing ◽  
The Cost

AbstractIn a single item dynamic lot-sizing problem, we are given a time horizon and demand for a single item in every time period. The problem seeks a solution that determines how much to produce and carry at each time period, so that we will incur the least amount of production and inventory cost. When the remanufacturing option is included, the input comprises of number of returned products at each time period that can be potentially remanufactured to satisfy the demands, where remanufacturing and inventory costs are applicable. For this problem, we first show that it cannot have a fully polynomial time approximation scheme. We then provide a polynomial time algorithm, when we make certain realistic assumptions on the cost structure.

scholarly journals Arc-Completion of 2-Colored Best Match Graphs to Binary-Explainable Best Match Graphs

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 110
Author(s):  
David Schaller ◽  
Manuela Geiß ◽  
Marc Hellmuth ◽  
Peter F. Stadler
Keyword(s):  
Phylogenetic Tree ◽  
Polynomial Time ◽  
Binary Tree ◽  
A Priori ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Cardinality ◽  
Mathematical Phylogenetics ◽  
Special Case

Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.

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