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

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.

2014 ◽  
Vol 24 (03) ◽  
pp. 225-236 ◽  
Author(s):  
DAVID KIRKPATRICK ◽  
BOTING YANG ◽  
SANDRA ZILLES

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.


10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski

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.


Algorithmica ◽  
2021 ◽  
Author(s):  
Britta Dorn ◽  
Ronald de Haan ◽  
Ildikó Schlotter

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


2020 ◽  
Vol 34 (02) ◽  
pp. 2070-2078
Author(s):  
Yasushi Kawase ◽  
Hanna Sumita

We study the problem of fairly allocating a set of indivisible goods to risk-neutral agents in a stochastic setting. We propose an (approximation) algorithm to find a stochastic allocation that maximizes the minimum utility among the agents. The algorithm runs by repeatedly finding an (approximate) allocation to maximize the total virtual utility of the agents. This implies that the problem is solvable in polynomial time when the utilities are gross-substitutes (which is a subclass of submodular). When the utilities are submodular, we can find a (1 − 1/e)-approximate solution for the problem and this is best possible unless P=NP. We also extend the problem where a stochastic allocation must satisfy the (ex ante) envy-freeness. Under this condition, we demonstrate that the problem is NP-hard even when every agent has an additive utility with a matroid constraint (which is a subclass of gross-substitutes). Furthermore, we propose a polynomial-time algorithm for the setting with a restriction that the matroid constraint is common to all agents.


2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Jan Kára ◽  
Jan Kratochvil ◽  
David R. Wood

Graphs and Algorithms International audience We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.


2021 ◽  
Vol 13 (4) ◽  
pp. 1-24
Author(s):  
Jessica Chen ◽  
Henry Milner ◽  
Ion Stoica ◽  
Jibin Zhan

The HTTP adaptive streaming technique opened the door to cope with the fluctuating network conditions during the streaming process by dynamically adjusting the volume of the future chunks to be downloaded. The bitrate selection in this adjustment inevitably involves the task of predicting the future throughput of a video session, owing to which various heuristic solutions have been explored. The ultimate goal of the present work is to explore the theoretical upper bounds of the QoE that any ABR algorithm can possibly reach, therefore providing an essential step to benchmarking the performance evaluation of ABR algorithms. In our setting, the QoE is defined in terms of a linear combination of the average perceptual quality and the buffering ratio. The optimization problem is proven to be NP-hard when the perceptual quality is defined by chunk size and conditions are given under which the problem becomes polynomially solvable. Enriched by a global lower bound, a pseudo-polynomial time algorithm along the dynamic programming approach is presented. When the minimum buffering is given higher priority over higher perceptual quality, the problem is shown to be also NP-hard, and the above algorithm is simplified and enhanced by a sequence of lower bounds on the completion time of chunk downloading, which, according to our experiment, brings a 36.0% performance improvement in terms of computation time. To handle large amounts of data more efficiently, a polynomial-time algorithm is also introduced to approximate the optimal values when minimum buffering is prioritized. Besides its performance guarantee, this algorithm is shown to reach 99.938% close to the optimal results, while taking only 0.024% of the computation time compared to the exact algorithm in dynamic programming.


1996 ◽  
Vol 07 (01) ◽  
pp. 23-41
Author(s):  
MARTIN FÜRER ◽  
WEBB MILLER

An alignment of k given sequences is a k-rowed matrix frequently used by molecular biologists to display correspondences between entries from each sequence. Under one approach, an alignment is represented by a matrix of ‘x’ and ’-’ characters, where each x in row r indicates the position of an entry of sequence r. It is sometimes efficient to store only the run-length encoding of each row of this bit-matrix. A natural class of commands for editing one such row into another consists of operations of the form: “Move the d dashes that begin at position i of row r to position j of that row,” for relevant values of r, d, i and j. We show that the problem of determining a shortest sequence of such operations that converts one given alignment to another is NP-hard and give a polynomial-time algorithm that always comes within a factor 5/4 of optimality. An application of these ideas to alignments of long DNA sequences is discussed.


2003 ◽  
Vol 01 (01) ◽  
pp. 71-94 ◽  
Author(s):  
MICHAL OZERY-FLATO ◽  
RON SHAMIR

A central problem in genome rearrangement is finding a most parsimonious rearrangement scenario using certain rearrangement operations. An important problem of this type is sorting a signed genome by reversals and translocations (SBRT). Hannenhalli and Pevzner presented a duality theorem for SBRT which leads to a polynomial time algorithm for sorting a multi-chromosomal genome using a minimum number of reversals and translocations. However, there is one case for which their theorem and algorithm fail. We describe that case and suggest a correction to the theorem and the polynomial algorithm. The solution of SBRT uses a reduction to the problem of sorting a signed permutation by reversals (SBR). The best extant algorithms for SBR require quadratic time. The common approach to solve SBR is by finding a safe reversal using the overlap graph or the interleaving graph of a permutation. We describe a family of signed permutations which proves a quadratic lower bound on the number of affected vertices in the overlap/interleaving graph during any optimal sorting scenario. This implies, in particular, an Ω(n3) lower bound for Bergeron's algorithm.


1999 ◽  
Vol 10 (02) ◽  
pp. 171-194 ◽  
Author(s):  
SHUJI ISOBE ◽  
XIAO ZHOU ◽  
TAKAO NISHIZEKI

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.


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