The Traveling Thief Problem (TTP) combines the Traveling Salesman Problem and the 0/1 Knapsack Problem through a load-dependent speed function that tightly couples routing and packing decisions. Exact models fail beyond a few dozen cities, and the strongest heuristics rely on problem-specific operators. This paper focuses on the formulation and models the TTP for a general-purpose solver, Hexaly, by representing the objective as a directed acyclic graph (DAG) over a sequence variable. The main contribution is M-LEG (Lean Expression-Graph model), which replaces the conditional (iif) arcs of a direct encoding with array lookups (at) and a linear cumulative weight chain. This reduces the graph from cubic to near-quadratic size and speeds up the search. The paper also clarifies which interval bound drives the reported gap. The conditional encoding inflates the upper travel-time bound by a factor that grows quadratically with the number of cities, but the gap depends on the lower bound, so the sound gap of M-LEG is only slightly smaller. Finally, the experiments provide a reproducible baseline. On toy instances, M-LEG reaches its best solution four times faster than the direct formulation and matches or improves the Gurobi incumbent, which is proven optimal on only two instances. On the 40 smallest of 60 categorized instances, which favors M-LEG, it ranks fifth of ten up to 130 cities when all methods are scored on their mean over ten runs, and its best run beats the best CoCo+ run on five of them. Beyond that size it ranks last.
Rapidly evolving market conditions call for real-time risk monitoring, but estimating it online remains challenging. In this paper, we study the online estimation of one of the most widely used risk measures, Value at Risk (VaR). Under the offline simulation online application (OSOA) framework, we formulate online VaR estimation as a conditional quantile learning problem with two key requirements: estimation accuracy and coverage reliability. To address the former, we employ quantile regression forests (QRF) to capture nonlinear and high-dimensional relationships between risk factors and conditional VaR. To address the latter, we adapt conformal prediction, conventionally used for prediction intervals, into a one-sided calibration procedure for the VaR estimator. We establish pointwise and L2 consistency for the QRF estimator, finite-sample model-free marginal coverage validity for the conformalized estimator, and asymptotic conditional coverage validity. Numerical experiments on a multi-asset option portfolio show that our proposed method becomes particularly competitive as the risk-factor dimension increases, while conformal calibration substantially corrects undercoverage. The proposed estimators also enable fast online prediction, supporting real-time risk monitoring.
The Capacitated Arc Routing Problem (CARP), introduced by Golden and Wong in 1981, is an important arc routing problem in Operations Research, which generalizes the famous Capacitated Vehicle Routing Problem (CVRP). When every customer has a unit demand, the best known approximation ratio for CARP, given by Jansen in 1993, remains 5/2-1.5/k, where k denotes the vehicle capacity. Based on recent progress in approximating CVRP, we improve this result by proposing a (5/2-Θ(1/√(k)))-approximation algorithm, which to the best of our knowledge constitutes the first improvement over Jansen's bound.
