An adaptive discretization method for the shortest path problem with time windows

The Shortest Path Problem with Time Windows (SPPTW) is an important generalization of the classical shortest path problem. SPPTW has been extensively studied in practical problems, such as transportation optimization, scheduling, and routing problems. It also appears as a sub-problem in the column-generation process of the vehicle routing problem with time windows. In SPPTW, we consider a time-constrained graph, where each node is assigned with a time window, each edge is assigned with a cost and a travel time. The objective is to find the shortest path from a source node to a destination node while respecting the time window constraints. When the graph contains negative cycles, the problem becomes Elementary Shortest Path Problem with Time Windows (ESPPTW). In this thesis, we adopt the time-expanded network approach, extend it by incorporating the adaptive expansion idea and propose a new approach: Adaptive Time Window Discretization(ATWD) method. We demonstrate that the ATWD method can be easily combined with label setting algorithms and label correcting algorithms for solving SPPTW. We further extend the ATWD embedded label correcting algorithm by adding k-cycle elimination to solve ESPPTW on graphs with negative cycles. We also propose an ATWD based integer programming solution for solving ESPPTW. The objective of our study is to show that optimal solutions in a time-constrained network can be found without first constructing the entire time-expanded network

Modelling and solving complex combinatorial optimization problems : quorumcast routing, elementary shortest path, elementary longest path and agricultural land allocation

The feasible solution set of a Combinatorial Optimization Problem (COP) is discrete and finite. Solving a COP is to find optimal solutions in the set of feasible solutions such that the value of a given cost function is minimized or maximized. In the literature, there exist both complete and incomplete methods for solving COPs. The complete (or exact) methods return the optimal solutions with the proof of the optimality, for example the branch-and-cut search. The incomplete methods try to find hight-quality solutions which are as close to the optimal solutions as possible, for example local search. In this thesis we focus on solving four distinct COPs: the Quorumcast Routing Problem (QRP), the Elementary Shortest Path Problem on graphs with negative-cost cycles (ESPP), the Elementary Longest Path Problem on graphs with positive-cost cycles (ELPP), and the Agricultural Land Allocation Problem (ALAP). In order to solve these problems with the complete methods, we use the Branch-and-Infer search, the Branch-and-Cut search, and the Branch-and-Price search. We also solve ALAP by the incomplete methods, such as Local Search, Tabu Search, Constraints-Based Local Search that combine with metaheuristics. The experimental evaluations on well-known benchmarks show that all proposed algorithms for all the first three COPs (QRP, ESPP and ELPP) are better than the-state-the art algorithms. Specially, we describe ALAP, formulate it as a combination of three COPs, and propose several complete and incomplete algorithms for these COPs.(FSA - Sciences de l'ingénieur) -- UCL, 201

Finding kk Simple Shortest Paths and Cycles

The problem of finding multiple simple shortest paths in a weighted directed graph $G=(V,E)$ has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed in the paths. Even for a single source-sink pair, it is known that two simple shortest paths cannot be found in time polynomially smaller than $n^3$ (where $n=|V|$) unless the All-Pairs Shortest Paths problem can be solved in a similar time bound. The latter is a well-known open problem in algorithm design. We consider the all-pairs version of the problem, and we give a new algorithm to find $k$ simple shortest paths for all pairs of vertices. For $k=2$, our algorithm runs in $O(mn + n^2 \log n)$ time (where $m=|E|$), which is almost the same bound as for the single pair case, and for $k=3$ we improve earlier bounds. Our approach is based on forming suitable path extensions to find simple shortest paths; this method is different from the `detour finding' technique used in most of the prior work on simple shortest paths, replacement paths, and distance sensitivity oracles. Enumerating simple cycles is a well-studied classical problem. We present new algorithms for generating simple cycles and simple paths in $G$ in non-decreasing order of their weights; the algorithm for generating simple paths is much faster, and uses another variant of path extensions. We also give hardness results for sparse graphs, relative to the complexity of computing a minimum weight cycle in a graph, for several variants of problems related to finding $k$ simple paths and cycles.Comment: The current version includes new results for undirected graphs. In Section 4, the notion of an (m,n) reduction is generalized to an f(m,n) reductio

Faster Parametric Shortest Path and Minimum Balance Algorithms

The parametric shortest path problem is to find the shortest paths in graph where the edge costs are of the form w_ij+lambda where each w_ij is constant and lambda is a parameter that varies. The problem is to find shortest path trees for every possible value of lambda. The minimum-balance problem is to find a ``weighting'' of the vertices so that adjusting the edge costs by the vertex weights yields a graph in which, for every cut, the minimum weight of any edge crossing the cut in one direction equals the minimum weight of any edge crossing the cut in the other direction. The paper presents fast algorithms for both problems. The algorithms run in O(nm+n^2 log n) time. The paper also describes empirical studies of the algorithms on random graphs, suggesting that the expected time for finding a minimum-mean cycle (an important special case of both problems) is O(n log(n) + m)