In this paper, we study twelve stochastic input models for online problems and reveal the differences between their competitive ratios for the models. The competitive ratio is defined as the worst ratio between the expected optimal value and the expected profit of the solution obtained by the online algorithm, where the input distribution is restricted according to the input model. Our purpose is to clarify the difference among the input models from the view of the competitive ratio. To handle a broad class of online problems, we use a framework called request-answer games, which were introduced by Ben-David et al. The stochastic input models fit into two categories: known distribution and unknown distribution. We consider six natural classes of distribution for each category: dependent distributions, deterministic input, independent distributions, identical independent distribution, random order of a deterministic input, and random order of independent distributions. Using each input model, we consider two fundamental online problems, which are variants of the secretary problem and the prophet inequality problem. We use these problems to observe the difference in the competitive ratios of each input model.

In the component order connectivity problem, we are given a finite undirected graph G = (V,E) and non-negative integers k, ℓ. The goal of this problem is to determine whether there exists a subset S ⊆ V such that |S| ≤ k and the size of every connected component of the subgraph of G induced by V \ S is at most ℓ. In this paper, we consider the generalization of the component order connectivity problem where the condition on the sizes of connected components is generalized by non-decreasing subadditive functions defined on the subsets of V. We prove that the kernelization techniques for the component order connectivity problem proposed by Xiao can be generalized to our setting.

The Wardrop equilibrium model is widely used for traffic pattern prediction in transportation networks. One of the reasons is that this model can be converted into an equivalent optimization problem, which makes the numerical evaluation of the equilibrium straightforward and efficient. This conversion approach has been extended to networks with link capacities. It is known that the solution of the resulting constrained optimization problem is a flow pattern satisfying an extension of the Wardrop equilibrium. However, the converse does not hold true. We resolve this gap by equilibrium refinement.

This paper proposes an analytical model for analyzing the effect of road pricing in a grid network. The model is based on a continuous approximation where origins and destinations are uniformly distributed and the travel demand is expressed as the exponential function of the travel cost. Analytical expressions for the traffic volume in the toll area and the toll revenue are obtained for a rectangular city with a grid network. The analytical expressions demonstrate how the size and shape of the toll area and the toll level affect the traffic volume and the toll revenue. The model explicitly considers through, inward, outward, and city traffic, thus allowing us to assess the effect of road pricing on each traffic separately. Comparisons between cordon and area pricing and between grid and radial-arc networks show that area pricing is superior to cordon pricing and that reducing the traffic volume is easier in the grid network than in the radial-arc network.