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].

A POLYNOMIAL-TIME ALGORITHM FOR COMPUTING THE RESILIENCE OF ARRANGEMENTS OF RAY SENSORS

2014 ◽  
Vol 24 (03) ◽  
pp. 225-236 ◽  
Author(s):  
DAVID KIRKPATRICK ◽  
BOTING YANG ◽  
SANDRA ZILLES
Keyword(s):  
Polynomial Time ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
The Other ◽  
Np Hard ◽  
Sensor Coverage ◽  
Entire Plane ◽  
Line Segments ◽  
Minimum Number

Given an arrangement A of n sensors and two points s and t in the plane, the barrier resilience of A with respect to s and t is the minimum number of sensors whose removal permits a path from s to t such that the path does not intersect the coverage region of any sensor in A. When the surveillance domain is the entire plane and sensor coverage regions are unit line segments, even with restricted orientations, the problem of determining the barrier resilience is known to be NP-hard. On the other hand, if sensor coverage regions are arbitrary lines, the problem has a trivial linear time solution. In this paper, we study the case where each sensor coverage region is an arbitrary ray, and give an O(n2m) time algorithm for computing the barrier resilience when there are m ⩾ 1 sensor intersections.

scholarly journals Obtaining a Proportional Allocation by Deleting Items

Algorithmica ◽  
2021 ◽  
Author(s):  
Britta Dorn ◽  
Ronald de Haan ◽  
Ildikó Schlotter
Keyword(s):  
Control Problem ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Lower And Upper Bounds ◽  
Proportional Allocation ◽  
Minimum Number

AbstractWe consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is $${\mathsf {W}}[3]$$ W [ 3 ] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation $$\pi $$ π in advance as part of the input, and our aim is to delete a minimum number of items such that $$\pi $$ π is proportional in the remainder; this variant turns out to be $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -hard for six agents, but polynomial-time solvable for two agents, and we show that it is $$\mathsf {W[2]}$$ W [ 2 ] -hard when parameterized by the number k of

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.

The Deletion-Insertion model applied to the genome rearrangement problem

2019 ◽  
Vol 28 (1) ◽  
pp. 1-13
Author(s):  
Abra Brisbin ◽  
Manda Riehl ◽  
Noah Williams
Keyword(s):  
Polynomial Time ◽  
Genome Rearrangement ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Small Scale ◽  
Permutation Statistics ◽  
Np Hard ◽  
Minimum Number ◽  
Linear Chromosomes

Abstract Permutations are frequently used in solving the genome rearrangement problem, whose goal is finding the shortest sequence of mutations transforming one genome into another. We introduce the Deletion-Insertion model (DI) to model small-scale mutations in species with linear chromosomes, such as humans. Applying one restriction to this model, we obtain the transposition model for genome rearrangement, which was shown to be NP-hard in [4]. We use combinatorial reasoning and permutation statistics to develop a polynomial-time algorithm to approximate the minimum number of transpositions required in the transposition model and to analyze the sharpness of several bounds on transpositions between genomes.

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.

A POLYNOMIAL-TIME ALGORITHM FOR FINDING TOTAL COLORINGS OF PARTIAL k-TREES

1999 ◽  
Vol 10 (02) ◽  
pp. 171-194 ◽  
Author(s):  
SHUJI ISOBE ◽  
XIAO ZHOU ◽  
TAKAO NISHIZEKI
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Combinatorial Problems ◽  
Total Coloring ◽  
Minimum Number

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Many combinatorial problems can be efficiently solved for partial k-trees, that is, graphs of treewidth bounded by a constant k. However, no polynomial-time algorithm has been known for the problem of finding a total coloring of a given partial k-tree with the minimum number of colors. This paper gives such a first polynomial-time algorithm.

OPTIMAL, EFFICIENT RECONSTRUCTION OF PHYLOGENETIC NETWORKS WITH CONSTRAINED RECOMBINATION

2004 ◽  
Vol 02 (01) ◽  
pp. 173-213 ◽  
Author(s):  
DAN GUSFIELD ◽  
SATISH EDDHU ◽  
CHARLES LANGLEY
Keyword(s):  
Phylogenetic Tree ◽  
Polynomial Time ◽  
Phylogenetic Network ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Phylogenetic Networks ◽  
Seminal Paper ◽  
Back Mutation ◽  
True Tree ◽  
The One

A phylogenetic network is a generalization of a phylogenetic tree, allowing structural properties that are not tree-like. In a seminal paper, Wang et al.1 studied the problem of constructing a phylogenetic network, allowing recombination between sequences, with the constraint that the resulting cycles must be disjoint. We call such a phylogenetic network a "galled-tree". They gave a polynomial-time algorithm that was intended to determine whether or not a set of sequences could be generated on galled-tree. Unfortunately, the algorithm by Wang et al.1 is incomplete and does not constitute a necessary test for the existence of a galled-tree for the data. In this paper, we completely solve the problem. Moreover, we prove that if there is a galled-tree, then the one produced by our algorithm minimizes the number of recombinations over all phylogenetic networks for the data, even allowing multiple-crossover recombinations. We also prove that when there is a galled-tree for the data, the galled-tree minimizing the number of recombinations is "essentially unique". We also note two additional results: first, any set of sequences that can be derived on a galled tree can be derived on a true tree (without recombination cycles), where at most one back mutation per site is allowed; second, the site compatibility problem (which is NP-hard in general) can be solved in polynomial time for any set of sequences that can be derived on a galled tree. Perhaps more important than the specific results about galled-trees, we introduce an approach that can be used to study recombination in general phylogenetic networks. This paper greatly extends the conference version that appears in an earlier work.8 PowerPoint slides of the conference talk can be found at our website.7

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.

Covering Orthogonal Polygons with Non-Piercing Rectangles

1997 ◽  
Vol 07 (05) ◽  
pp. 473-484 ◽  
Author(s):  
J. Mark Keil
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Visibility Graph ◽  
Orthogonal Polygon ◽  
Orthogonal Polygons ◽  
Minimum Number ◽  
Simply Connected

Given a simply connected orthogonal polygon P, a polynomial time algorithm is presented to cover the polygon with the minimum number of rectangles, under the restriction that if A and B are two overlapping rectangles in the cover then either A - B or B - A is connected. The algorithm runs in O(n log n + nm) time, where n is the number of vertices of P and m is the number of edges in the visibility graph of P that are either horizontal, vertical or form the diagonal of an empty rectangle.

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