A Fast Approximation Algorithm for TSP with Neighborhoods and Red-Blue Separation

Resource assignment problems occur in a vast variety of applications, from scheduling problems over image recognition to communication networks. Often these problems can be modeled by a maximum weight matching problem in (bipartite) graphs or generalizations thereof, and efficient and practical algorithms are known for these problems. Although in some of the applications an assignment of the resources may be needed only once, in many of these applications, the assignment has to be computed more often for different scenarios. In that case it is often essential that the assignments can be computed very fast. Moreover, implementing different assignments in different scenarios may come with a certain cost for the reconfiguration of the system. In this paper, we consider the problem of determining optimal assignments sequentially over a given time horizon, where consecutive assignments are coupled by constraints that control the cost of reconfiguration. We develop fast approximation and online algorithms for this problem with provable approximation guarantees and competitive ratios. Moreover, we present an extensive computational study about the applicability of our model and our algorithms in the context of orthogonal frequency division multiple access (OFDMA) wireless networks, finding a significant performance improvement for the total bandwidth of the system using our algorithms. For this application (the downlink of an OFDMA wireless cell) , the run time of matching algorithms is extremely important, having an acceptable range of a few milliseconds only. For the considered realistic instances, our algorithms perform extremely well: the solution quality is, on average, within a factor of 0.8–0.9 of optimal off-line solutions, and the running times are at most 5 ms per phase even in the worst case. Thus, our algorithms are well suited to be applied in the context of OFDMA systems. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2221 . This paper was accepted by Teck-Hua Ho, optimization.

We consider the linear time-dependent Schrodinger equation with a time- dependent smooth potential on an unbounded domain. A Galerkin spectral method with a tensor-product Hermite basis is used as a discretization in space. Discretizing the resulting ODE for the Hermite expansion coecients involves the computation of the action of the Galerkin matrix on a vector in each time step. We propose a fast algorithm for the direct computation of this matrix-vector product without actually assembling the matrix itself. The costs scale linearly in the size of the basis. Together with the application of a hyperbolically reduced basis, this reduces the computational eort considerably and helps cope with the infamous curse of dimensionality. The ap- plication of the fast algorithm is limited to the case of the potential being significantly smoother than the solution. The error analysis is based on a binary tree representation of the three-term recurrence relation for the one-dimensional Hermite functions. The fast algorithm constitutes an ecient tool for schemes involving the action of a ma- trix due to spectral discretization on a vector, and it is also applicable in the context of spectral approximations for linear problems other than the Schrodinger equation. Keywords: linear Schrodinger equation, spectral Galerkin methods, reduced index sets, fast algorithm, direct computation, curse of dimensionality, binary trees

We present a hybrid approach to generate plausible motions for high-DOF human-like articulated figures in constrained environment with multiple obstacles. The method combines motion planning and motion blending techniques to efficiently produce natural and collision-free human motion for different tasks.

Traveling salesman problem (TSP) is a well-known NP-hard combinatorial optimization problem. It has been solved by a number of exact and approximate algorithms and serves as a testbed for new heuristic and metaheuristic optimization algorithms. However, it is often not easy to evaluate the hardness (complexity) of a TSP instance. Simple measures such as the number of cities or the minimum (maximum) route length do not capture the internal structure of a TSP instance sufficiently. In this work, we propose a new method for the assessment of TSP instance complexity based on clustering. The new approach is evaluated on a set of randomized TSP instances with different structure and its relation to the performance of a selected metaheuristic TSP solver is studied.

Parallel techniques for solving large scale travelling salesperson problems

Traveling salesman problem in which all the vertices are assumed to be on a spherical surface is a special case of the conventional travelling salesman problem. There are exact and approximate algorithms for the travelling salesman problem. As the solution time is a performance parameter in most real-time applications, approximate algorithms always have an important area of research for both researchers and engineers. In this chapter, approximate algorithms based on heuristic methods are considered for the travelling salesman problem on the sphere. Firstly, 28 test instances were newly generated on the unit sphere. Then, using various heuristic methods such as genetic algorithms, ant colony optimization, and fluid genetic algorithms, the initial solutions for solving test instances of the traveling salesman problem are obtained in Matlab®. Then, the initial heuristic solutions are used as input for the 2-opt algorithm. The performances and time complexities of the applied methods are analyzed as a conclusion.

The problem of partitioning a set of geographically distributed tasks among a team of UAVs that must travel to perform the tasks arises in many contexts, from surveillance to environmental monitoring. When UAVs have limits in their travel distances, it is important to select subsets of tasks for each UAV that achieve highest team values while remaining feasible given limits on UAV resources. This class of problems has been studied previously in computer science and operations research as orienteering problems. This problem is known to be NP-Hard. Previous algorithms for solving orienteering problems as integer programming problems are slow; faster algorithms have been proposed based on heuristic techniques used in traveling salesperson problems. In this paper, we develop a different class of fast algorithms, based on a decomposition of the orienteering problem into a knapsack assignment problem and a subsequent traveling salesperson problem. We combine greedy algorithms for knapsack problems with the use of spanning trees to estimate traveling salesperson tour lengths to obtain new, approximate algorithms for orienteering problems. We conduct experiments with single and multiple UAV algorithms on benchmark problems as well as problems derived from a LANDAT dataset, comparing our new algorithms with other algorithms in the literature. Our results indicate that our new algorithms are consistently fast, and perform better than competing algorithms.

Enumeration of all maximal bicliques in bipartite graphs is a well-studied fundamental problem. However, a wide range of applications need less overlapping bicliques with specific size constraints instead of all the maximal bicliques. In this paper, we study a new biclique problem, called the top-k t-biclique coverage problem. A t-biclique is a biclique with a size constraint <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$t$</tex> for one vertex set and the problem aims to find <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> t-bicliques maximising the coverage on the other vertex set. The top-k t-biclique coverage problem has novel applications such as finding top-k courses while maximising student engagement. We prove that this problem is NP-hard. A straightforward way to address the problem first needs to enumerate and store all t-bicliques and then greedily select <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k$</tex> promising t-bicliques, leading an approximate guarantee on the coverage. However, it takes exponential space, which is impractical. We then apply a fast approximation scheme to solve this problem, which shaves the exponential space consumption by progressively updating top-k results during the t-biclique enumeration. Observing that the fast approximation algorithm takes too much time on updating the results due to the coverage is computed from scratch for each update, an online index is devised to address the drawback. Due the hardness of the problem, even the fast approximation algorithm cannot scale to large dataset. To devise a scalable solution, we then propose a heuristic algorithm running in polynomial time. Thanks for four carefully designed heuristic rules, the heuristic algorithm can find large coverage top-k t-bicliques extremely fast for large datasets. Apart from that, the heuristic result with large coverage can effectively prune unpromising enumerations in the fast greedy algorithm, which improves the efficiency of the fast approximation algorithm without compromising the approximation ratio. Extensive experiments are conducted on real datasets to justify the effectiveness and efficiency of the proposed algorithms.

An improved approximation algorithm for the traveling salesman problem with relaxed triangle inequality

Various neural network models have been proposed to tackle combinatorial optimization problems such as the travelling salesman problem (TSP). Existing learning-based TSP methods adopt a simple setting that the training and testing data are independent and identically distributed. However, the existing literature fails to solve TSP instances when training and testing data have different distributions. Concretely, we find that different training and testing distribution will result in more difficult TSP instances, i.e., the solution obtained by the model has a large gap from the optimal solution. To tackle this problem, in this work, we study learning-based TSP methods when training and testing data have different distributions using adaptive-hardness, i.e., how difficult a TSP instance can be for a solver. This problem is challenging because it is non-trivial to (1) define hardness measurement quantitatively; (2) efficiently and continuously generate sufficiently hard TSP instances upon model training; (3) fully utilize instances with different levels of hardness to learn a more powerful TSP solver. To solve these challenges, we first propose a principled hardness measurement to quantify the hardness of TSP instances. Then, we propose a hardness-adaptive generator to generate instances with different hardness. We further propose a curriculum learner fully utilizing these instances to train the TSP solver. Experiments show that our hardness-adaptive generator can generate instances ten times harder than the existing methods, and our proposed method achieves significant improvement over state-of-the-art models in terms of the optimality gap. The codes are publicly available.

A hybrid GRASP/VND heuristic for the one-commodity pickup-and-delivery traveling salesman problem

This work investigates an unmanned aerial vehicle (UAV) assisted IoT system, where a UAV flies to each foothold to collect data from IoT devices, and then return to its start point. For such a system, we aim to minimize the energy consumption by jointly optimizing the deployment and flight trajectory of UAV. It is a mixed-integer non-convex and NP-hard problem. In order to address it, a bilevel optimization approach is proposed, where an upper-level method aims to optimize the deployment of UAV and a lower-level one aims to plan UAV flight trajectory. Specifically, the former optimizes the number and locations of footholds of UAV. This work proposes an improved dandelion algorithm with a novel encoding strategy, in which each dandelion represents a foothold of UAV and the entire dandelion population is seen as an entire deployment. Then, two mutation strategies are designed to adjust the number and locations of footholds. Based on the footholds of the UAV provided by the former, the latter transforms flight trajectory planning into a traveling salesman problem (TSP). This work proposes an iterated greedy algorithm to solve it efficiently. The effectiveness of the proposed bilevel optimization approach is verified on ten instances, and the experimental results show that it significantly outperforms other benchmark approaches.

We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a $(1+\epsilon)$ approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate $B$ such that $B \le B^* \le \lceil (1+\epsilon)B \rceil +1$ where $B^*$ is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization.

The problem of traversing a set of points in the order that minimizes the total distance traveled (traveling salesman problem) is one of the most famous and well-studied problems in combinatorial optimization. In this paper, we introduce the metric of minimizing the number of turns in the tour, given that the input points are in the Euclidean plane. We give approximation algorithms for several variants under this metric. For the general case we give a logarithmic approximation algorithm. For the case when the lines of the tour are restricted to being either horizontal or vertical, we give a 2-approximation algorithm. If we have the further restriction that no two points are allowed to have the same x- or y-coordinate, we give an algorithm that finds a tour which makes at most two turns more than the optimal tour. We also introduce several interesting algorithmic techniques for decomposing sets of points in the Euclidean plane that we believe to be of independent interest.

We build on the classical greedy sequential set cover algorithm, in the spirit of the primal-dual schema, to obtain simple parallel approximation algorithms for the set cover problem and its generalizations. Our algorithms use randomization, and our randomized voting lemmas may be of independent interest. Fast parallel approximation algorithms were known before for set cover, though not for any of its generalizations. >