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.

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

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.

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.

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].

scholarly journals The Minimum Tollbooth Problem in Atomic Network Congestion Games with Unsplittable Flows

2021 ◽  
Author(s):  
Julian Nickerl
Keyword(s):  
Nash Equilibrium ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Congestion Games ◽  
Social Optimum ◽  
Network Congestion ◽  
Pure Nash Equilibrium ◽  
Positive Side ◽  
Minimum Number

AbstractThis work analyzes the minimum tollbooth problem in atomic network congestion games with unsplittable flows. The goal is to place tolls on edges, such that there exists a pure Nash equilibrium in the tolled game that is a social optimum in the untolled one. Additionally, we require the number of tolled edges to be the minimum. This problem has been extensively studied in non-atomic games, however, to the best of our knowledge, it has not been considered for atomic games before. By a reduction from the weighted CNF SAT problem, we show both the NP-hardness of the problem and the W[2]-hardness when parameterizing the problem with the number of tolled edges. On the positive side, we present a polynomial time algorithm for networks on series-parallel graphs that turns any given state of the untolled game into a pure Nash equilibrium of the tolled game with the minimum number of tolled edges.

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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