The Travelling Thief Problem (TTP) is a challenging combinatorial optimization problem that attracts many scholars. The TTP interconnects two well-known NP-hard problems: the Travelling Salesman Problem (TSP) and the 0-1 Knapsack Problem (KP). Increasingly algorithms have been proposed for solving this novel problem that combines two interdependent sub-problems. In this paper, TTP is investigated theoretically and empirically. An algorithm based on the score value calculated by our proposed formulation in picking items and sorting items in the reverse order in the light of the scoring value is proposed to solve the problem. Different approaches for solving the TTP are compared and analyzed; the experimental investigations suggest that our proposed approach is very efficient in meeting or beating current state-of-the-art heuristic solutions on a comprehensive set of benchmark TTP instances.

In the travelling thief problem (TTP), a thief undertakes a cyclic tour through a set of cities, and according to a picking plan, picks a subset of available items into a rented knapsack with limited capacity. The overall aim is to maximise profit while minimising renting cost. TTP combines two interdependent components: the travelling salesman problem (TSP) and the knapsack problem (KP). Existing TTP approaches typically solve the TSP and KP components in an interleaved fashion: the solution of one component is fixed while the solution of the other is changed. This indicates poor coordination between solving the two components, which may lead to poor quality TTP solutions. The 2-OPT heuristic is often used for solving the TSP component, which reverses a segment in the tour. Within the TTP context, 2-OPT does not take into account the picking plan, which can result in a lower objective value. This in turn can result in the tour modification to be rejected by a solver. To address this, we propose an extended form of 2-OPT in order to change the picking plan in coordination with modifying the tour. Items deemed as less profitable and picked in cities earlier in the reversed segment are replaced by items that tend to be equally or more profitable and not picked in cities later in the reversed segment. The picking plan is further changed through a modified form of the bit-flip search, where changes in the picking state are only permitted for boundary items, which are defined as lowest profitable picked items or highest profitable unpicked items. This restriction reduces the amount of time spent on the KP component, allowing more tours to be evaluated by the TSP component within a given time budget. The two modified heuristics form the basis of a new cooperative coordination solver, which is shown to outperform several state-of-the-art TTP solvers on a broad range of benchmark TTP instances.

This paper presents a new method for estimating high dimensional covariance matrices. The method, permuted rank-penalized least-squares (PRLS), is based on a Kronecker product series expansion of the true covariance matrix. Assuming an i.i.d. Gaussian random sample, we establish high dimensional rates of convergence to the true covariance as both the number of samples and the number of variables go to infinity. For covariance matrices of low separation rank, our results establish that PRLS has significantly faster convergence than the standard sample covariance matrix (SCM) estimator. The convergence rate captures a fundamental tradeoff between estimation error and approximation error, thus providing a scalable covariance estimation framework in terms of separation rank, similar to low rank approximation of covariance matrices. The MSE convergence rates generalize the high dimensional rates recently obtained for the ML Flip-flop algorithm for Kronecker product covariance estimation. We show that a class of block Toeplitz covariance matrices is approximatable by low separation rank and give bounds on the minimal separation rank $r$ that ensures a given level of bias. Simulations are presented to validate the theoretical bounds. As a real world application, we illustrate the utility of the proposed Kronecker covariance estimator for spatio-temporal linear least squares prediction of multivariate wind speed measurements.