Bounded-Length Smith-Waterman Alignment

Given a fixed alignment scoring scheme, the bounded length (respectively, bounded total length) Smith-Waterman alignment problem on a pair of strings of lengths m, n, asks for the maximum alignment score across all substring pairs, such that the first substring\u27s length (respectively, the sum of the two substrings\u27 lengths) is above the given threshold w. The latter problem was introduced by Arslan and Egecioglu under the name "local alignment with length threshold". They proposed a dynamic programming algorithm solving the problem in time O(mn^2), and also an approximation algorithm running in time O(rmn), where r is a parameter controlling the accuracy of approximation. We show that both these problems can be solved exactly in time O(mn), assuming a rational scoring scheme; furthermore, this solution can be used to obtain an exact algorithm for the normalised bounded total length Smith - Waterman alignment problem, running in time O(mn log n). Our algorithms rely on the techniques of fast window-substring alignment and implicit unit-Monge matrix searching, developed previously by the author and others

Bounded-length Smith-Waterman alignment

Given a fixed alignment scoring scheme, the bounded length (respectively, bounded total length) Smith-Waterman alignment problem on a pair of strings of lengths m, n, asks for the maximum alignment score across all substring pairs, such that the first substring's length (respectively, the sum of the two substrings' lengths) is above the given threshold w. The latter problem was introduced by Arslan and Egecioglu under the name "local alignment with length threshold". They proposed a dynamic programming algorithm solving the problem in time O(mn^2), and also an approximation algorithm running in time O(rmn), where r is a parameter controlling the accuracy of approximation. We show that both these problems can be solved exactly in time O(mn), assuming a rational scoring scheme; furthermore, this solution can be used to obtain an exact algorithm for the normalised bounded total length Smith - Waterman alignment problem, running in time O(mn log n). Our algorithms rely on the techniques of fast window-substring alignment and implicit unit-Monge matrix searching, developed previously by the author and others

Departure time choice equilibrium and optimal transport problems

This paper presents a systematic approach for analyzing the departure-time choice equilibrium (DTCE) problem of a single bottleneck with heterogeneous commuters. The approach is based on the fact that the DTCE is equivalently represented as a linear programming problem with a special structure, which can be analytically solved by exploiting the theory of optimal transport combined with a decomposition technique. By applying the proposed approach to several types of models with heterogeneous commuters, it is shown that the dynamic equilibrium distribution of departure times exhibits striking regularities under mild assumptions regarding schedule delay functions, in which commuters sort themselves according to their attributes, such as desired arrival times, schedule delay functions (value of times), and travel distances to a destination

A new look at departure time choice equilibrium models with heterogeneous users

This paper presents a systematic approach for analyzing the departure-time choice equilibrium (DTCE) problem of a single bottleneck with heterogeneous commuters. The approach is based on the fact that the DTCE is equivalently represented as a linear programming problem with a special structure, which can be analytically solved by exploiting the theory of optimal transport combined with a decomposition technique. By applying the proposed approach to several types of models with heterogeneous commuters, it is shown that (i) the essential condition for emerging equilibrium “sorting patterns,” which have been known in the literature, is that the schedule delay functions have the “Monge property,” (ii) the equilibrium problems with the Monge property can be solved analytically, and (iii) the proposed approach can be applied to a more general problem with more than two types of heterogeneities

Algorithmic and Linear Programming-Based Techniques for the Maximum Utility Problem

A common topic of study in the subfield of Operations Research known as Revenue Management is finding optimal prices for a line of products given customer preferences. While there exists a large number of ways to model optimal pricing problems, in this thesis we study a price-based Revenue Management model known as the Maximum Utility Problem (MUP). In this model, we are given a set of n customer segments and m products, as well as reservation prices Rij which reflect the amount that Segment i is willing to pay for Product j. Using a number of structural and behavioral assumptions, if we derive a vector of prices for our line of products, we can compute an assignment of customers to products. We wish to find the set of prices that leads to the optimal amount of revenue given our rules for assigning customers to products. Using this framework, we can formulate a Nonlinear Mixed Integer Programming formulation that, while difficult to solve, has a surprising amount of underlying structure. If we fix an assignment and simply ask for the optimal set of prices such that said assignment is feasible, we obtain a new linear program, the dual of which happens to be a set of shortest-paths problems. This fact lead to the development of the Dobson-Kalish Algorithm, which explores a large number of assignments and quickly computes their optimal prices. Since the introduction of the Dobson-Kalish Algorithm, there has been a rich variety of literature surrounding MUP and its relatives. These include the introduction of utility tolerances to increase the robustness of the model, as well as new approximation algorithms, hardness results, and insights into the underlying combinatorial structure of the problem. After detailing this history, this thesis discusses a range of settings under which MUP can be solved in polynomial time. Relating it to other equilibria and price-based optimization problems, we overview Stackelberg Network Pricing Games as well as the general formulations of Bilevel Mixed Integer Linear Programs and Bilinear Mixed Integer Programs, showing that our formulation of the latter is in fact a more general version of the former. We provide some new structured instances for which we can prove additional ap- proximation and runtime results for existing algorithms. We also contribute a generalized heuristic algorithm and show that MUP can be solved exactly when the matrix of reservation prices is rank 1. Finally, we discuss techniques for improving the upper bound to the overall problem, analyzing the primal and dual of the linear programming relaxation of MUP. To test the effectiveness of our approach, we analyze numerous examples that have been solved using Gurobi and present possible avenues for improving our ideas

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Tohoku University赤松隆課