Robustness of Mission Plans for Unmanned Aircraft.

This thesis studies the robustness of optimal mission plans for unmanned aircraft. Mission planning typically involves tactical planning and path planning. Tactical planning refers to task scheduling and in multi aircraft scenarios also includes establishing a communication topology. Path planning refers to computing a feasible and collision-free trajectory. For a prototypical mission planning problem, the traveling salesman problem on a weighted graph, the robustness of an optimal tour is analyzed with respect to changes to the edge costs. Specifically, the stability region of an optimal tour is obtained, i.e., the set of all edge cost perturbations for which that tour is optimal. The exact stability region of solutions to variants of the traveling salesman problems is obtained from a linear programming relaxation of an auxiliary problem. Edge cost tolerances and edge criticalities are derived from the stability region. For Euclidean traveling salesman problems, robustness with respect to perturbations to vertex locations is considered and safe radii and vertex criticalities are introduced. For weighted-sum multi-objective problems, stability regions with respect to changes in the objectives, weights, and simultaneous changes are given. Most critical weight perturbations are derived. Computing exact stability regions is intractable for large instances. Therefore, tractable approximations are desirable. The stability region of solutions to relaxations of the traveling salesman problem give under approximations and sets of tours give over approximations. The application of these results to the two-neighborhood and the minimum 1-tree relaxation are discussed. Bounds on edge cost tolerances and approximate criticalities are obtainable likewise. A minimum spanning tree is an optimal communication topology for minimizing the cumulative transmission power in multi aircraft missions. The stability region of a minimum spanning tree is given and tolerances, stability balls, and criticalities are derived. This analysis is extended to Euclidean minimum spanning trees. This thesis aims at enabling increased mission performance by providing means of assessing the robustness and optimality of a mission and methods for identifying critical elements. Examples of the application to mission planning in contested environments, cargo aircraft mission planning, multi-objective mission planning, and planning optimal communication topologies for teams of unmanned aircraft are given.PhDAerospace EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120837/1/mniendo_1.pd

The Traveling Salesman Problem Under Squared Euclidean Distances

Let $P$ be a set of points in $\mathbb{R}^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by TSP($d,\alpha$). We design a 5-approximation algorithm for TSP(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\alpha}/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-TSP of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-TSP$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-TSP$(d, \alpha)$ is APX-hard if $d\ge3$ and $\alpha>1$. The APX-hardness proof carries over to TSP$(d, \alpha)$ for the same parameter ranges.Comment: 12 pages, 4 figures. (v2) Minor linguistic change

On the Approximability of TSP on Local Modifications of Optimally Solved Instances

Given an instance of TSP together with an optimal solution, we consider the scenario in which this instance is modified locally, where a local modification consists in the alteration of the weight of a single edge. More generally, for a problem U, let LM-U (local-modification-U) denote the same problem as U, but in LM-U, we are also given an optimal solution to an instance from which the input instance can be derived by a local modification. The question is how to exploit this additional knowledge, i.e., how to devise better algorithms for LM-U than for U. Note that this need not be possible in all cases: The general problem of LM-TSP is as hard as TSP itself, i.e., unless P=NP, there is no polynomial-time p(n)-approximation algorithm for LM-TSP for any polynomial p. Moreover, LM-TSP where inputs must satisfy the β-triangle inequality (LM-Δβ-TSP) remains NP-hard for all β>½. However, for LM-Δ-TSP (i.e., metric LM-TSP), we will present an efficient 1.4-approximation algorithm. In other words, the additional information enables us to do better than if we simply used Christofides' algorithm for the modified input. Similarly, for all 1<β<3.34899, we achieve a better approximation ratio for LM-Δ-TSP than for Δβ-TSP. For ½≤β<1, we show how to obtain an approximation ratio arbitrarily close to 1, for sufficiently large input graphs

The Parameterized Approximability of TSP with Deadlines

Modern algorithm theory includes numerous techniques to attack hard problems, such as approximation algorithms on the one hand and parameterized complexity on the other hand. However, it is still uncommon to use these two techniques simultaneously, which is unfortunate, as there are natural problems that cannot be solved using either technique alone, but rather well if we combine them. The problem to be studied here is not only natural, but also practical: Consider TSP, generalized as follows. We impose deadlines on some of the vertices, effectively constraining them to be visited prior to a given point of time. The resulting problem DlTSP (a special case of the well-known TSP with time windows) inherits its hardness from classical TSP, which is both well known from practice and renowned to be one of the hardest problems with respect to approximability: Within polynomial time, not even a polynomial approximation ratio (let alone a constant one) can be achieved (unless P = NP). We will show that DlTSP is even harder than classical TSP in the following sense. Classical TSP, despite its hardness, admits good approximation algorithms if restricted to metric (or near-metric) inputs. Not so DlTSP (and hence, neither TSP with time windows): We will prove that even for metric inputs, no constant approximation ratio can ever be achieved (unless P = NP). This is where parameterization becomes crucial: By combining methods from the field of approximation algorithms with ideas from the theory of parameterized complexity, we apply the concept of parameterized approximation. Thereby, we obtain a 2.5-approximation algorithm for DlTSP with a running time of k! · poly(|G|), where k denotes the number of deadlines. Furthermore, we prove that there is no fpt-algorithm with an approximation guarantee of 2-ε for any ε > 0, unless P = NP. Finally, we show that, unlike TSP, DlTSP becomes much harder when relaxing the triangle inequality. More precisely, for an arbitrary small violation of the triangle inequality, DlTSP does not admit an fpt-algorithm with approximation guarantee ((1-ε)/2)|V| for any ε > 0, unless P = N

Approximability of the Minimum Steiner Cycle Problem

In this paper, we consider variants of a new problem that we call minimum Steiner cycle problem (SCP). The problem is defined as follows. Given is a weighted complete graph and a set of terminal vertices. In the SCP problem, we are looking for a minimum-cost cycle that passes through every terminal exactly once and through every other vertex of the graph at most once. We show that, if P<>NP, there is no approximation algorithm for SCP on directed graphs with an approximation ratio polynomial in the input size. Moreover, this result holds even in the case when the number of terminals is 4. On the contrary, we show that SCP on undirected graphs with constant number of terminals and edge costs satisfying the beta-relaxed triangle inequality is approximable with the ratio beta^2+beta. When the number of terminals k is a part of the input, the problem is obviously a generalization of TSP. For the metric case, we present a 3/2- and a 2/3 log_2 k-approximation algorithm for undirected and directed graphs G=(V,E), respectively. For the case with the beta-relaxed triangle inequality, we present a (beta^2+beta)-approximation algorithm

Reusing optimal TSP solutions for locally modified input instances : Extended abstract

Given an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e. g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification- U) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i. e., whether lm-U is computationally more tractable than U. Here, we give non-trivial examples both of problems where this is and problems where this is not the case4th IFIP International Conference on Theoretical Computer ScienceRed de Universidades con Carreras en Informática (RedUNCI