Quickest Paths: Faster Algorithms and Dynamization

Given a network $N=(V,E,{c},{l})$, where $G=(V,E)$, $|V|=n$ and $|E|=m$, is a directed graph, ${c}(e) \u3e 0$ is the capacity and ${l}(e) \ge 0$ is the lead time (or delay) for each edge $e\in E$, the quickest path problem is to find a path for a given source--destination pair such that the total lead time plus the inverse of the minimum edge capacity of the path is minimal. The problem has applications to fast data transmissions in communication networks. The best previous algorithm for the single--pair quickest path problem runs in time $O(r m+r n \log n)$, where $r$ is the number of distinct capacities of $N$ \cite{ROS}. In this paper, we present algorithms for general, sparse and planar networks that have significantly lower running times. For general networks, we show that the time complexity can be reduced to $O(r^{\ast} m+r^{\ast} n \log n)$, where $r^{\ast}$ is at most the number of capacities greater than the capacity of the shortest (with respect to lead time) path in $N$. For sparse networks, we present an algorithm with time complexity $O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma})$, where $\tilde{\gamma}$ is a topological measure of $N$. Since for sparse networks $\tilde{\gamma}$ ranges from $1$ up to $\Theta(n)$, this constitutes an improvement over the previously known bound of $O(r n \log n)$ in all cases that $\tilde{\gamma}=o(n)$. For planar networks, the complexity becomes $O(n \log n + n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma})$. Similar improvements are obtained for the all--pairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem

Quickest paths: faster algorithms and dynamization

Given a network $N=(V,E,{c},{l})$, where $G=(V,E)$, $|V|=n$ and $|E|=m$, is a directed graph, ${c}(e) > 0$ is the capacity and ${l}(e) \ge 0$ is the lead time (or delay) for each edge $e\in E$, the quickest path problem is to find a path for a given source--destination pair such that the total lead time plus the inverse of the minimum edge capacity of the path is minimal. The problem has applications to fast data transmissions in communication networks. The best previous algorithm for the single pair quickest path problem runs in time $O(r m+r n \log n)$, where $r$ is the number of distinct capacities of $N$. In this paper, we present algorithms for general, sparse and planar networks that have significantly lower running times. For general networks, we show that the time complexity can be reduced to $O(r^{\ast} m+r^{\ast} n \log n)$, where $r^{\ast}$ is at most the number of capacities greater than the capacity of the shortest (with respect to lead time) path in $N$. For sparse networks, we present an algorithm with time complexity $O(n \log n + r^{\ast} n + r^{\ast} \tilde{\gamma} \log \tilde{\gamma})$, where $\tilde{\gamma}$ is a topological measure of $N$. Since for sparse networks $\tilde{\gamma}$ ranges from $1$ up to $\Theta(n)$, this constitutes an improvement over the previously known bound of $O(r n \log n)$ in all cases that $\tilde{\gamma}=o(n)$. For planar networks, the complexity becomes $O(n \log n + n\log^3 \tilde{\gamma}+ r^{\ast} \tilde{\gamma})$. Similar improvements are obtained for the all--pairs quickest path problem. We also give the first algorithm for solving the dynamic quickest path problem

Incremental Network Design with Minimum Spanning Trees

Given an edge-weighted graph $G=(V,E)$ and a set $E_0\subset E$, the incremental network design problem with minimum spanning trees asks for a sequence of edges $e'_1,\ldots,e'_T\in E\setminus E_0$ minimizing $\sum_{t=1}^Tw(X_t)$ where $w(X_t)$ is the weight of a minimum spanning tree $X_t$ for the subgraph $(V,E_0\cup\{e'_1,\ldots,e'_t\})$ and $T=\lvert E\setminus E_0\rvert$. We prove that this problem can be solved by a greedy algorithm.Comment: 9 pages, minor revision based on reviewer comment

Selfish Routing on Dynamic Flows

Selfish routing on dynamic flows over time is used to model scenarios that vary with time in which individual agents act in their best interest. In this paper we provide a survey of a particular dynamic model, the deterministic queuing model, and discuss how the model can be adjusted and applied to different real-life scenarios. We then examine how these adjustments affect the computability, optimality, and existence of selfish routings.Comment: Oberlin College Computer Science Honors Thesis. Supervisor: Alexa Sharp, Oberlin Colleg

Second best toll and capacity optimisation in network: solution algorithm and policy implications

This paper looks at the first and second-best jointly optimal toll and road capacity investment problems from both policy and technical oriented perspectives. On the technical side, the paper investigates the applicability of the constraint cutting algorithm for solving the second-best problem under elastic demand which is formulated as a bilevel programming problem. The approach is shown to perform well despite several problems encountered by our previous work in Shepherd and Sumalee (2004). The paper then applies the algorithm to a small sized network to investigate the policy implications of the first and second-best cases. This policy analysis demonstrates that the joint first best structure is to invest in the most direct routes while reducing capacities elsewhere. Whilst unrealistic this acts as a useful benchmark. The results also show that certain second best policies can achieve a high proportion of the first best benefits while in general generating a revenue surplus. We also show that unless costs of capacity are known to be low then second best tolls will be affected and so should be analysed in conjunction with investments in the network