MULTI-TERMINAL NETWORK CONNECTEDNESS ON SERIES-PARALLEL NETWORKS

2009 ◽  
Vol 01 (02) ◽  
pp. 253-265 ◽  
Author(s):  
TONI R. FARLEY ◽  
CHARLES J. COLBOURN
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Polynomial Time Algorithms ◽  
Parallel Networks ◽  
Network Connectedness ◽  
Network Operation ◽  
Terminal Reliability ◽  
Special Case ◽  
General Networks

Network operation may require that a specified number k of nodes be able to communicate via paths consisting of operating edges and nodes. In an environment of node and edge failure, this leads to associated reliability measures. When the k nodes are known in advance, this has been widely studied as k-terminal reliability; when the k nodes are chosen uniformly at random, this has been studied as k-resilience. A third notion, when it suffices to have anyk nodes communicate, is related to the expected size of the largest component in the network. We generalize these three measures to the probability that given h nodes chosen in advance and i nodes chosen at random, they appear in a component of size at least k = h + i + j. As expected, for general networks, for most choices of (h, i, j) the computation is #P-complete and hence unlikely to admit a polynomial time algorithm. We develop polynomial time algorithms in the special case that the network is series-parallel, which subsume and generalize earlier methods for k-terminal reliability and k-resilience.

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.

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 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.

scholarly journals Some results on stable sets for k-colorable P₆-free graphs and generalizations

2012 ◽  
Vol Vol. 14 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Raffaele Mosca
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Stable Set ◽  
Stable Sets ◽  
Polynomial Time Algorithms ◽  
Complexity Status ◽  
International Audience ◽  
Hard Problems ◽  
Structure Properties

Graphs and Algorithms International audience This article deals with the Maximum Weight Stable Set (MWS) problem (and some other related NP-hard problems) and the class of P-6-free graphs. The complexity status of MWS is open for P-6-free graphs and is open even for P-5-free graphs (as a long standing open problem). Several results are known for MWS on subclasses of P-5-free: in particular, MWS can be solved for k-colorable P-5-free graphs in polynomial time for every k (depending on k) and more generally for (P-5, K-p)-free graphs (depending on p), which is a useful result since for every graph G one can easily compute a k-coloring of G, with k not necessarily minimum. This article studies the MWS problem for k-colorable P-6-free graphs and more generally for (P-6, K-p)-free graphs. Though we were not able to define a polynomial time algorithm for this problem for every k, this article introduces: (i) some structure properties of P-6-free graphs with respect to stable sets, (ii) two reductions for MWS on (P-6; K-p)-free graphs for every p, (iii) three polynomial time algorithms to solve MWS respectively for 3-colorable P-6-free, for 4-colorable P-6-free, and for (P-6, K-4)-free graphs (the latter allows one to state, together with other known results, that MWS can be solved for (P-6, F)-free graphs in polynomial time where F is any four vertex graph).

scholarly journals On Efficiently Explaining Graph-Based Classifiers

2021 ◽  
Author(s):  
Xuanxiang Huang ◽  
Yacine Izza ◽  
Alexey Ignatiev ◽  
Joao Marques-Silva
Keyword(s):  
Decision Trees ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Graph Representation ◽  
Efficient Computation ◽  
Binary Decision ◽  
Polynomial Time Algorithms ◽  
Wide Range ◽  
Novel Algorithms

Recent work has shown that not only decision trees (DTs) may not be interpretable but also proposed a polynomial-time algorithm for computing one PI-explanation of a DT. This paper shows that for a wide range of classifiers, globally referred to as decision graphs, and which include decision trees and binary decision diagrams, but also their multi-valued variants, there exist polynomial-time algorithms for computing one PI-explanation. In addition, the paper also proposes a polynomial-time algorithm for computing one contrastive explanation. These novel algorithms build on explanation graphs (XpG's). XpG's denote a graph representation that enables both theoretical and practically efficient computation of explanations for decision graphs. Furthermore, the paper proposes a practically efficient solution for the enumeration of explanations, and studies the complexity of deciding whether a given feature is included in some explanation. For the concrete case of decision trees, the paper shows that the set of all contrastive explanations can be enumerated in polynomial time. Finally, the experimental results validate the practical applicability of the algorithms proposed in the paper on a wide range of publicly available benchmarks.

A polynomial time algorithm for big data in a special case of minimum constraint removal problem

2019 ◽  
Vol 13 (2) ◽  
pp. 247-254
Author(s):  
Bahram Sadeghi Bigham ◽  
Fariba Noorizadeh ◽  
Salman Khodayifar
Keyword(s):  
Big Data ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Special Case

M-Convex Function Minimization Under L1-Distance Constraint and Its Application to Dock Reallocation in Bike-Sharing System

2021 ◽  
Author(s):  
Akiyoshi Shioura
Keyword(s):  
Convex Function ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Steepest Descent ◽  
Nonlinear Integer Programming ◽  
Distance Constraint ◽  
Polynomial Time Algorithms ◽  
Bike Sharing ◽  
Pseudo Polynomial Time

In this paper, we consider a problem of minimizing an M-convex function under an L1-distance constraint (MML1); the constraint is given by an upper bound for L1-distance between a feasible solution and a given “center.” This is motivated by a nonlinear integer programming problem for reallocation of dock capacity in a bike-sharing system discussed by Freund et al. (2017). The main aim of this paper is to better understand the combinatorial structure of the dock reallocation problem through the connection with M-convexity and show its polynomial-time solvability using this connection. For this, we first show that the dock reallocation problem and its generalizations can be reformulated in the form of (MML1). We then present a pseudo-polynomial-time algorithm for (MML1) based on the steepest descent approach. We also propose two polynomial-time algorithms for (MML1) by replacing the L1-distance constraint with a simple linear constraint. Finally, we apply the results for (MML1) to the dock reallocation problem to obtain a pseudo-polynomial-time steepest descent algorithm and also polynomial-time algorithms for this problem. For this purpose, we develop a polynomial-time algorithm for a relaxation of the dock reallocation problem by using a proximity-scaling approach, which is of interest in its own right.

CONSTRUCTING A MINIMUM PHYLOGENETIC NETWORK FROM A DENSE TRIPLET SET

2012 ◽  
Vol 10 (05) ◽  
pp. 1250013 ◽  
Author(s):  
MICHEL HABIB ◽  
THU-HIEN TO
Keyword(s):  
Polynomial Time ◽  
Phylogenetic Network ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Complete Answer ◽  
Biconnected Components ◽  
Polynomial Time Algorithms ◽  
Minimum Number

For a given set [Formula: see text] of species and a set [Formula: see text] of triplets on [Formula: see text], we seek to construct a phylogenetic network which is consistent with [Formula: see text] i.e. which represents all triplets of [Formula: see text]. The level of a network is defined as the maximum number of hybrid vertices in its biconnected components. When [Formula: see text] is dense, there exist polynomial time algorithms to construct level-0,1 and 2 networks (Aho et al., 1981; Jansson, Nguyen and Sung, 2006; Jansson and Sung, 2006; Iersel et al., 2009). For higher levels, partial answers were obtained in the paper by Iersel and Kelk (2008), with a polynomial time algorithm for simple networks. In this paper, we detail the first complete answer for the general case, solving a problem proposed in Jansson and Sung (2006) and Iersel et al. (2009). For any k fixed, it is possible to construct a level-k network having the minimum number of hybrid vertices and consistent with [Formula: see text], if there is any, in time [Formula: see text].

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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