A simple efficient approximation scheme for the restricted shortest path problem

Lorenz, Dean; Raz, Danny

doi:10.1016/s0167-6377(01)00069-4

Cited by 229 publications

(209 citation statements)

References 6 publications

Supporting

Mentioning

205

Contrasting

Unclassified

Order By: Relevance

“…shortest path [42] (see also [20,27]), and matching [6]. The approach in [35] easily generalizes to the case of matroid basis.…”

Section: Introductionmentioning

confidence: 99%

“…One basic approach is combining dynamic programming (which solves the problem for polynomial weights and lengths) with rounding and scaling techniques (to reduce the problem to the case of polynomial quantities). This leads for example to the FPTAS for 1-budgeted shortest path [20,27,42]. Another fundamental technique is the Lagrangian relaxation method.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

New approaches to multi-objective optimization

et al. 2013

View full text Add to dashboard Cite

A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Many classical optimization problems, such as maximum spanning tree and forest, shortest path, maximum weight (perfect) matching, maximum weight independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. Not much is known however about the case of two or more budgets: filling this gap, at least partially, is the main goal of this paper. In more detail, we obtain the following main results:• Using iterative rounding for the first time in multi-objective optimization, we obtain multi-criteria PTASs (which slightly violate the budget constraints) for spanning tree, matroid basis, and bipartite matching with k = O(1) budget constraints.• We present a simple mechanism to transform multi-criteria approximation schemes into pure approximation schemes for problems whose feasible solutions define an independence system. This gives improved algorithms for several problems. In particular, this mechanism can be applied to the above bipartite matching algorithm, hence obtaining a pure PTAS.• We show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set. • We present a deterministic approximation scheme for k-budgeted matching (in general graphs), where k = O(1). Interestingly, to show that our procedure works, we rely on a non-constructive result by Stromquist and Woodall, which is based on the Ham Sandwich Theorem.

show abstract

“…shortest path [42] (see also [20,27]), and matching [6]. The approach in [35] easily generalizes to the case of matroid basis.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

New approaches to multi-objective optimization

et al. 2013

View full text Add to dashboard Cite

show abstract

“…Ergun et al [2] gave another (1 + ε; 1)-FPTAS with improved running time of O(mn/ε). For general graphs Lorenz and Raz [9] proposed an (1 + ε; 1)-FPTAS with time complexity of O(mn(log log n + 1/ε)). Goel et al [5] gave an (1; 1 + ε)-FPTAS with running time O((m + n log n)n/ε).…”

Section: Introductionmentioning

confidence: 99%

“…, 1 + ε)-FPTAS. Our multi-criteria FPTAS is a dynamic programming algorithm, similar to the SPPP algorithm of Lorenz and Raz [9], with a combination of the rounding technique of Song and Sahni [11]. The time complexity of our multi-criteria approximation scheme is O(m(n/ε) k ), which for k = 1 matches that of Ergun et al [2], who provided an (1 + ε; 1)-FPTAS for the RCSPP restricted to acyclic graphs.…”

Section: Introductionmentioning

confidence: 99%

Multi-criteria approximation schemes for the resource constrained shortest path problem

Horváth

Kis

2017

Optim Lett

View full text Add to dashboard Cite

In the resource constrained shortest path problem we are given a directed graph along with a source node and a destination node, and each arc has a cost and a vector of weights specifying its requirements from a set of resources with finite budget limits. A minimum cost source-destination path is sought such that the total consumption of the arcs from each resource does not exceed its budget limit. In the case of constant number of weight functions we give a fully polynomial time multi-criteria approximation scheme for the problem which returns a source-destination path of cost at most the optimum, however, the path may slightly violate the budget limits. On the negative side, we show that there does not exist polynomial time multi-criteria approximation scheme for the problem if the number of weight functions is not a constant. The latter result applies to a broad class of problem as well, including the multi-dimensional knapsack, the multi-budgeted spanning tree, the multi-budgeted matroid basis and the multi-budgeted bipartite perfect matching problems.

show abstract

“…However, to compare results of the RSP-based search in the G D graph to the simple shortest-path search in the G C graph we developed a much simpler dynamic programming solution, provided below. The computational complexity of this algorithm is pseudo-polynomial, but it can be turned into FPAS (fully polynomial approximation scheme) by using the approximation from [5]. In the following algorithm, C[v, t] denotes a vector associated with each node v, which stores the minimum uncertainty on any path from v s to v, that has a total cost of t. T max is the maximum cost of a path from v s to v g in the graph, obtained by search w.r.t the cost information only.…”

Section: Planning the Positioning Actionsmentioning

confidence: 99%

Planning Positioning Actions of a Mobile Robot Cooperating with Distributed Sensors

Skrzypczyński

Advances in Soft Computing

View full text Add to dashboard Cite

show abstract

A simple efficient approximation scheme for the restricted shortest path problem

Cited by 229 publications

References 6 publications

New approaches to multi-objective optimization

New approaches to multi-objective optimization

Multi-criteria approximation schemes for the resource constrained shortest path problem

Planning Positioning Actions of a Mobile Robot Cooperating with Distributed Sensors

Contact Info

Product

Resources

About