Greedy algorithm

About: Greedy algorithm is a research topic. Over the lifetime, 15347 publications have been published within this topic receiving 393945 citations.

Minimal Controllability Problems

Abstract: Given a linear system, we consider the problem of finding a small set of variables to affect with an input so that the resulting system is controllable. We show that this problem is NP-hard; indeed, we show that even approximating the minimum number of variables that need to be affected within a multiplicative factor of clog n is NP-hard for some positive c. On the positive side, we show it is possible to find sets of variables matching this in approximability barrier in polynomial time. This can be done with a simple greedy heuristic which sequentially picks variables to maximize the rank increase of the controllability matrix. Experiments on Erdos-Renyi random graphs that demonstrate this heuristic almost always succeed at finding the minimum number of variables.

An Optimal Algorithm for the Traveling Salesman Problem with Time Windows

Abstract: This paper presents the development of new elimination tests which greatly enhance the performance of a relatively well established dynamic programming approach and its application to the minimization of the total traveling cost for the traveling salesman problem with time windows. The tests take advantage of the time window constraints to significantly reduce the state space and the number of state transitions. These reductions are performed both a priori and during the execution of the algorithm. The approach does not experience problems stemming from increasing problem size, wider or overlapping time windows, or an increasing number of states nearly as rapidly as other methods. Our computational results indicate that the algorithm was successful in solving problems with up to 200 nodes and fairly wide time windows. When the density of the nodes in the geographical region was kept constant as the problem size was increased, the algorithm was capable of solving problems with up to 800 nodes. For these problems, the CPU time increased linearly with problem size. These problem sizes are much larger than those of problems previously reported in the literature.

Learning with Structured Sparsity

Abstract: This paper investigates a learning formulation called structured sparsity, which is a natural extension of the standard sparsity concept in statistical learning and compressive sensing. By allowing arbitrary structures on the feature set, this concept generalizes the group sparsity idea that has become popular in recent years. A general theory is developed for learning with structured sparsity, based on the notion of coding complexity associated with the structure. It is shown that if the coding complexity of the target signal is small, then one can achieve improved performance by using coding complexity regularization methods, which generalize the standard sparse regularization. Moreover, a structured greedy algorithm is proposed to efficiently solve the structured sparsity problem. It is shown that the greedy algorithm approximately solves the coding complexity optimization problem under appropriate conditions. Experiments are included to demonstrate the advantage of structured sparsity over standard sparsity on some real applications.

Approximation algorithms for some routing problems

Abstract: Several polynomial time approximation algorithms for some NP-complete routing problems are presented, and the worst-case ratios of the cost of the obtained route to that of an optimal are determined. A mixed-strategy heuristic with a bound of 9/5 is presented for the Stacker-Crane problem (a modified Traveling Salesman problem). A tour-splitting heuristic is given for k-person variants of the Traveling Salesman problem, the Chinese Postman problem, and the Stacker-Crane problem, for which a minimax solution is sought. This heuristic has a bound of e + 1 - 1/k, where e is the bound for the corresponding 1-person algorithm.

Cellular UAV-to-X Communications: Design and Optimization for Multi-UAV Networks

Abstract: In this paper, we consider a single-cell cellular network with a number of cellular users (CUs) and unmanned aerial vehicles (UAVs), in which multiple UAVs upload their collected data to the base station (BS). Two transmission modes are considered to support the multi-UAV communications, i.e., UAV-to-network (U2N) and UAV-to-UAV (U2U) communications. Specifically, the UAV with a high signal-to-noise ratio (SNR) for the U2N link uploads its collected data directly to the BS through U2N communication, while the UAV with a low SNR for the U2N link can transmit data to a nearby UAV through underlaying U2U communication for the sake of quality of service. We first propose a cooperative UAV sense-and-send protocol to enable the UAV-to-X communications, and then formulate the subchannel allocation and UAV speed optimization problem to maximize the uplink sum-rate. To solve this NP-hard problem efficiently, we decouple it into three sub-problems: U2N and cellular user (CU) subchannel allocation, U2U subchannel allocation, and UAV speed optimization. An iterative subchannel allocation and speed optimization algorithm (ISASOA) is proposed to solve these sub-problems jointly. The simulation results show that the proposed ISASOA can upload 10% more data than the greedy algorithm.