The Complexity of Optimization Problems

Ausiello, Giorgio; Marchetti-Spaccamela, Alberto; Crescenzi, Pierluigi; Gambosi, Giorgio; Protasi, Marco; Kann, Viggo

doi:10.1007/978-3-642-58412-1_1

Cited by 60 publications

(66 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…An APTAS (Ausiello et al, 1999) is an asymptotic polynomial time approximation scheme, which is an algorithm whose performance ratio approaches 1 as the number of objects increases. A problem is said to be MAX-SNP-hard (Papadimitriou and Yannakakis 1991), abbreviated MAX-SNP in the table, if there is a constant c > 1 such that it is impossible to approximate the problem with performance ratio smaller than c unless P = N P .…”

Section: Our Resultsmentioning

confidence: 99%

“…Since for every scenario k we can augment this tree to include t k at cost no more than 2r * (k), our approximation ratio follows. 2 …”

Section: Theoremmentioning

confidence: 99%

“…As the size of the models being solved increases in scale, this solution approach gains in importance. For a detailed exposition of this topic, the reader is referred to Vazirani (2001) and Ausiello et al (1999).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems

Ravi

Sinha

2005

Math. Program.

View full text Add to dashboard Cite

We initiate the design of approximation algorithms for stochastic combinatorial optimization problems; we formulate the problems in the framework of two-stage stochastic optimization, and provide nearly tight approximation algorithms. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios.The approximation ratio of the stochastic variant of a typical problem is of the same order of magnitude as its deterministic counterpart. Furthermore, common techniques for designing approximation algorithms such as LP rounding, the primal-dual method, and the greedy algorithm, can be carefully adapted to obtain these results.

show abstract

Section: Our Resultsmentioning

confidence: 99%

“…Since for every scenario k we can augment this tree to include t k at cost no more than 2r * (k), our approximation ratio follows. 2 …”

Section: Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems

Ravi

Sinha

2005

Math. Program.

View full text Add to dashboard Cite

show abstract

“…Since the vertex-coloring problem is NP -hard which has no polynomial-time algorithm with a constant approximation performance ratio [2], we use a simple heuristic based on the sequential coloring approach proposed in [11]. Algorithm C Wavelength Assignment Algorithm …”

Section: Wavelength Assignment For Multi-treesmentioning

confidence: 99%

“…An important objective of multicast routing is to minimize the network cost of the routing tree, which is defined as the sum of costs of all the links in the tree. The problem of finding a tree in a general topology network, such that it connects a set of nodes and has the minimal cost, is known as the Steiner tree problem [2].…”

Section: Introductionmentioning

confidence: 99%

Multicast routing and wavelength assignment in WDM networks with limited drop-offs

Hu¹,

Shuai²,

Jia

et al.

Ieee Infocom 2004

View full text Add to dashboard Cite

Abstract-In WDM networks with limited drop-offs, the route of a multicast connection consists of a set of light-trees. Each of the light-tree is rooted at the source node and contains no more than a limited number, say k, destination nodes due to the power loss of dropping optical signals off at destination nodes. We call such a light-tree k-drop light-tree. In this paper we study the multicast routing problem of constructing a set of k-drop lighttrees that have the minimal network cost. The network cost of a set of light-trees is defined as the summation of the link cost of all the light-trees. We first prove that this problem is polynomialtime solvable for k = 2 and NP -hard for k ≥ 3. We then propose a 4-approximation algorithm for the problem for k ≥ 3. A wavelength assignment algorithm is also proposed to assign wavelengths to the light-trees of a multicast connection. In the end we give simulation results showing that k-drop multi-tree routing can significantly save not only the network cost but also wavelengths used. Moreover, when k ≥ 5 its performance is very close to the case where k is infinite (i.e., the case of using a single tree for a multicast connection).

show abstract

Pricing bridges to cross a river

Bouhtou

Grigoriev

Hoesel

et al. 2007

Naval Research Logistics

View full text Add to dashboard Cite

Abstract:We consider a pricing problem in directed, uncapacitated networks. Tariffs must be defined by an operator, the leader, for a subset of m arcs, the tariff arcs. Costs of all other arcs in the network are assumed to be given. There are n clients, the followers, and after the tariffs have been determined, the clients route their demands independent of each other on paths with minimal total cost. The problem is to find tariffs that maximize the operator's revenue. Motivated by applications in telecommunication networks, we consider a restricted version of this problem, assuming that each client utilizes at most one of the operator's tariff arcs. The problem is equivalent to pricing bridges that clients can use in order to cross a river. We prove that this problem is APX-hard. Moreover, we analyze the effect of uniform pricing, proving that it yields both an m-approximation and a (1 + ln D)-approximation. Here, D is upper bounded by the total demand of all clients. In addition, we consider the problem under the additional restriction that the operator must not reject any of the clients. We prove that this problem does not admit approximation algorithms with any reasonable performance guarantee, unless P = NP, and we prove the existence of an n-approximation algorithm.

show abstract

The Complexity of Optimization Problems

Cited by 60 publications

References 0 publications

Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems

Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems

Multicast routing and wavelength assignment in WDM networks with limited drop-offs

Pricing bridges to cross a river

Contact Info

Product

Resources

About