arXiv subjects

Liwei Zeng

Publications and source records attributed to Liwei Zeng.

5 recordsLinked to original sources

arXiv2020

A Bounded Formulation for The School Bus Scheduling Problem

This paper proposes a new formulation for the school bus scheduling problem (SBSP) which optimizes school start times and bus operation times to minimize transportation cost. Our goal is to minimize the number of buses to serve all bus routes such that each route arrives in a time window before school starts. We present a new time-indexed integer linear programming (ILP) formulation for this problem. Based on a strengthened version of the linear relaxation of the ILP, we develop a dependent randomized rounding algorithm that yields near-optimal solutions for large-scale problem instances. We also generalize our methodologies to solve a robust version of the SBSP.

math.OC↗

arXiv2019

Generalizing the Covering Path Problem on a Grid

We study the covering path problem on a grid of R^{2}. We generalize earlier results on a rectangular grid and prove that the covering path cost can be bounded by the area and perimeter of the grid. We provide (2+ε) and (1+ε)-approximations for the problem on a general grid and on a convex grid, respectively.

cs.DS↗

arXiv2019

The Covering Path Problem on a Grid

This paper introduces the covering path problem on a grid (CPPG) which finds the cost-minimizing path connecting a subset of points in a grid such that each point that needs to be covered is within a predetermined distance of a point from the chosen subset. We leverage the geometric properties of the grid graph which captures the road network structure in many transportation problems, including our motivating setting of school bus routing. As defined in this paper, the CPPG is a bi-objective optimization problem comprised of one cost term related to path length and one cost term related to stop count. We develop a trade-off constraint which quantifies the trade-off between path length and stop count and provides a lower bound for the bi-objective optimization problem. We introduce simple construction techniques to provide feasible paths that match the lower bound within a constant factor. Importantly, this solution approach uses transformations of the general CPPG to either a discrete CPPG or continuous CPPG based on the value of the coverage radius. For both the discrete and continuous versions, we provide fast constant-factor approximations, thus solving the general CPPG.

math.OC↗

arXiv2018

On the Inequalities of Projected Volumes and the Constructible Region

We study the following geometry problem: given a $2^n-1$ dimensional vector $π=\{π_S\}_{S\subseteq [n], S\ne \emptyset}$, is there an object $T\subseteq\mathbb{R}^n$ such that $\log(\mathsf{vol}(T_S))= π_S$, for all $S\subseteq [n]$, where $T_S$ is the projection of $T$ to the subspace spanned by the axes in $S$? If $π$ does correspond to an object in $\mathbb{R}^n$, we say that $π$ is {\em constructible}. We use $Ψ_n$ to denote the constructible region, i.e., the set of all constructible vectors in $\mathbb{R}^{2^n-1}$. In 1995, Bollobás and Thomason showed that $Ψ_n$ is contained in a polyhedral cone, defined a class of so called uniform cover inequalities. We propose a new set of natural inequalities, called nonuniform-cover inequalities, which generalize the BT inequalities. We show that any linear inequality that all points in $Ψ_n$ satisfy must be a nonuniform-cover inequality. Based on this result and an example by Bollobás and Thomason, we show that constructible region $Ψ_n$ is not even convex, and thus cannot be fully characterized by linear inequalities. We further show that some subclasses of the nonuniform-cover inequalities are not correct by various combinatorial constructions, which refutes a previous conjecture about $Ψ_n$. Finally, we conclude with an interesting conjecture regarding the convex hull of $Ψ_n$.

cs.DM↗

arXiv2014

Construction of Directed Strongly Regular Graphs as Generalized Cayley Graphs

Directed strongly regular graphs were introduced by Duval in 1998 as one of the possible generalization of classical strongly regular graphs to the directed case. Duval also provided several construction methods for directed strongly regular graphs. In this paper, an infinite family of directed strongly regular graphs is constructed, as generalized Cayley graphs of cyclic groups.

math.CO↗