scholarly journals An Overview of Algorithms for Network Survivability

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
F. A. Kuipers
Keyword(s):  
Polynomial Time ◽  
Network Connectivity ◽  
Network Survivability ◽  
Disjoint Paths ◽  
Polynomial Time Algorithms ◽  
Concise Overview ◽  
Main Components

Network survivability—the ability to maintain operation when one or a few network components fail—is indispensable for present-day networks. In this paper, we characterize three main components in establishing network survivability for an existing network, namely, (1) determining network connectivity, (2) augmenting the network, and (3) finding disjoint paths. We present a concise overview of network survivability algorithms, where we focus on presenting a few polynomial-time algorithms that could be implemented by practitioners and give references to more involved algorithms.

scholarly journals On Structural Parameterizations of the Edge Disjoint Paths Problem

Algorithmica ◽  
2021 ◽  
Author(s):  
Robert Ganian ◽  
Sebastian Ordyniak ◽  
M. S. Ramanujan
Keyword(s):  
Polynomial Time ◽  
Disjoint Paths ◽  
Structural Parameter ◽  
Input Graph ◽  
Feedback Vertex Set ◽  
Fpt Algorithms ◽  
Edge Disjoint ◽  
Fpt Algorithm ◽  
Vertex Set ◽  
Terminal Pair

AbstractIn this paper we revisit the classical edge disjoint paths (EDP) problem, where one is given an undirected graph G and a set of terminal pairs P and asks whether G contains a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our focus lies on structural parameterizations for the problem that allow for efficient (polynomial-time or FPT) algorithms. As our first result, we answer an open question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that the problem can be solved in polynomial time if the input graph has a feedback vertex set of size one. We also show that EDP parameterized by the treewidth and the maximum degree of the input graph is fixed-parameter tractable. Having developed two novel algorithms for EDP using structural restrictions on the input graph, we then turn our attention towards the augmented graph, i.e., the graph obtained from the input graph after adding one edge between every terminal pair. In constrast to the input graph, where EDP is known to remain -hard even for treewidth two, a result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved in non-uniform polynomial time if the augmented graph has constant treewidth; we note that the possible improvement of this result to an FPT-algorithm has remained open since then. We show that this is highly unlikely by establishing the [1]-hardness of the problem parameterized by the treewidth (and even feedback vertex set) of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting a novel structural parameter of the augmented graph.

scholarly journals Reconstruction of score sets

2014 ◽  
Vol 6 (2) ◽  
pp. 210-229
Author(s):  
Antal Iványi
Keyword(s):  
Polynomial Time ◽  
Constructive Proof ◽  
Construction Algorithm ◽  
Polynomial Time Algorithms ◽  
General Polynomial ◽  
Degree Set ◽  
New Algorithms

Abstract The score set of a tournament is defined as the set of its different outdegrees. In 1978 Reid [15] published the conjecture that for any set of nonnegative integers D there exists a tournament T whose degree set is D. Reid proved the conjecture for tournaments containing n = 1, 2, and 3 vertices. In 1986 Hager [4] published a constructive proof of the conjecture for n = 4 and 5 vertices. In 1989 Yao [18] presented an arithmetical proof of the conjecture, but general polynomial construction algorithm is not known. In [6] we described polynomial time algorithms which reconstruct the score sets containing only elements less than 7. In [5] we improved this bound to 9. In this paper we present and analyze new algorithms Hole-Map, Hole-Pairs, Hole-Max, Hole-Shift, Fill-All, Prefix-Deletion, and using them improve the above bound to 12, giving a constructive partial proof of Reid’s conjecture.

scholarly journals Polynomial-Time Algorithms for Stochastic Uncapacitated Lot-Sizing Problems

2008 ◽  
Vol 56 (5) ◽  
pp. 1172-1183 ◽  
Author(s):  
Yongpei Guan ◽  
Andrew J. Miller
Keyword(s):  
Polynomial Time ◽  
Lot Sizing ◽  
Polynomial Time Algorithms
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