omid amini

Traffic grooming is a central problem in optical networks. It refers to pack low rate signals into higher speed streams, in order to improve bandwidth utilization and reduce network cost. In WDM networks, the most accepted criterion is to minimize the number of electronic terminations, namely the number of SONET Add-Drop Multiplexers (ADMs). In this article we focus on ring and path topologies. On the one hand, we provide the first inapproximability result for Traffic Grooming for fixed values of the grooming factor g, answering affirmatively the conjecture of Chow and Lin (Networks, 44:194-202, 2004). More precisely, we prove that Ring Traffic Grooming for fixed g ≥ 1 and Path Traffic Grooming for fixed g ≥ 2 are APX-complete. That is, they do not accept a PTAS unless P= NP. Both results rely on the fact that finding the maximum number of edge-disjoint triangles in a graph (and more generally cycles of length 2g + 1 in a graph of girth 2g + 1) is APX-complete. On the other hand, we provide a polynomial-time approximation algorithm for Ring and Path Traffic Grooming, based on a greedy cover algorithm, with an approximation ratio independent of g. Namely, the approximation guarantee is $\mathcal{O}(n^{1/3} \log^2 n)$ for any g ≥ 1, n being the size of the network. This is useful in practical applications, since in backbone networks the grooming factor is usually greater than the network size. As far as we know, this is the first approximation algorithm with this property. Finally, we improve this approximation ratio under some extra assumptions about the request graph.

... Bermond 1 and F. Giroire 2 and F. Huc 1 and S. Pérennes 1 1Projet Mascotte, CNRS/INRIA/UNSA, INRIA Sophia-Antipolis, 2004 route des Lucioles BP 93, 06902 Sophia-Antipolis Cedex, France 2Projet Algorithmes, INRIA Rocquencourt, F-78153 Le Chesnay, France ...

Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well-known problems like Set Cover, Vertex

A general instance of a Degree-Constrained Subgraph problem consists of an edge-weighted or vertex-weighted graph G and the objective is to find an optimal weighted subgraph, subject to certain degree constraints on the vertices of the subgraph. This paper considers two natural Degree-Constrained Subgraph problems and studies their behavior in terms of approximation algorithms. These problems take as input an undirected graph G = (V,E), with |V| = n and |E| = m. Our results, together with the definition of the two problems, are listed below. The Maximum Degree-Bounded Connected Subgraph problem (MDBCS d ) takes as input a weight function $\omega : E \rightarrow \mathbb R^+$ and an integer d ≥ 2, and asks for a subset E′ ⊆ E such that the subgraph G′ = (V,E′) is connected, has maximum degree at most d, and ∑ e ∈ E′ω(e) is maximized. This problem is one of the classical NP-hard problems listed by Garey and Johnson in [Computers and Intractability, W.H. Freeman, 1979], but there were no results in the literature except for d = 2. We prove that MDBCS d is not in Apx for any d ≥ 2 (this was known only for d = 2) and we provide a $(\min \{m/ \log n,\ nd/(2 \log n)\})$ -approximation algorithm for unweighted graphs, and a $(\min\{n/2,\ m/d\})$ -approximation algorithm for weighted graphs. We also prove that when G has a low-degree spanning tree, in terms of d, MDBCS d can be approximated within a small constant factor in unweighted graphs. The Minimum Subgraph of Minimum Degree  ≥ d (MSMD d ) problem requires finding a smallest subgraph of G (in terms of number of vertices) with minimum degree at least d. We prove that MSMD d is not in Apx for any d ≥ 3 and we provide an $\mathcal{O}(n/\log n)$ -approximation algorithm for the class of graphs excluding a fixed graph as a minor, using dynamic programming techniques and a known structural result on graph minors.

Traffic grooming is a central problem in optical networks. It refers to pack low rate signals into higher speed streams, in order to improve bandwidth utilization and reduce network cost. In WDM networks, the most accepted criterion is to minimize the number of electronic terminations, namely the number of SONET Add-Drop Multiplexers (ADMs). In this article we focus on ring and path topologies. On the one hand, we provide the first inapproximability result for Traffic Grooming for fixed values of the grooming factor g, answering affirmatively the conjecture of Chow and Lin (Networks, 44:194-202, 2004). More precisely, we prove that Ring Traffic Grooming for fixed g ≥ 1 and Path Traffic Grooming for fixed g ≥ 2 are APX-complete. That is, they do not accept a PTAS unless P= NP. Both results rely on the fact that finding the maximum number of edge-disjoint triangles in a graph (and more generally cycles of length 2g + 1 in a graph of girth 2g + 1) is APX-complete. On the other hand, we provide a polynomial-time approximation algorithm for Ring and Path Traffic Grooming, based on a greedy cover algorithm, with an approximation ratio independent of g. Namely, the approximation guarantee is $\mathcal{O}(n^{1/3} \log^2 n)$ for any g ≥ 1, n being the size of the network. This is useful in practical applications, since in backbone networks the grooming factor is usually greater than the network size. As far as we know, this is the first approximation algorithm with this property. Finally, we improve this approximation ratio under some extra assumptions about the request graph.

... Bermond 1 and F. Giroire 2 and F. Huc 1 and S. Pérennes 1 1Projet Mascotte, CNRS/INRIA/UNSA, INRIA Sophia-Antipolis, 2004 route des Lucioles BP 93, 06902 Sophia-Antipolis Cedex, France 2Projet Algorithmes, INRIA Rocquencourt, F-78153 Le Chesnay, France ...

Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well-known problems like Set Cover, Vertex

A general instance of a Degree-Constrained Subgraph problem consists of an edge-weighted or vertex-weighted graph G and the objective is to find an optimal weighted subgraph, subject to certain degree constraints on the vertices of the subgraph. This paper considers two natural Degree-Constrained Subgraph problems and studies their behavior in terms of approximation algorithms. These problems take as input an undirected graph G = (V,E), with |V| = n and |E| = m. Our results, together with the definition of the two problems, are listed below. The Maximum Degree-Bounded Connected Subgraph problem (MDBCS d ) takes as input a weight function $\omega : E \rightarrow \mathbb R^+$ and an integer d ≥ 2, and asks for a subset E′ ⊆ E such that the subgraph G′ = (V,E′) is connected, has maximum degree at most d, and ∑ e ∈ E′ω(e) is maximized. This problem is one of the classical NP-hard problems listed by Garey and Johnson in [Computers and Intractability, W.H. Freeman, 1979], but there were no results in the literature except for d = 2. We prove that MDBCS d is not in Apx for any d ≥ 2 (this was known only for d = 2) and we provide a $(\min \{m/ \log n,\ nd/(2 \log n)\})$ -approximation algorithm for unweighted graphs, and a $(\min\{n/2,\ m/d\})$ -approximation algorithm for weighted graphs. We also prove that when G has a low-degree spanning tree, in terms of d, MDBCS d can be approximated within a small constant factor in unweighted graphs. The Minimum Subgraph of Minimum Degree  ≥ d (MSMD d ) problem requires finding a smallest subgraph of G (in terms of number of vertices) with minimum degree at least d. We prove that MSMD d is not in Apx for any d ≥ 3 and we provide an $\mathcal{O}(n/\log n)$ -approximation algorithm for the class of graphs excluding a fixed graph as a minor, using dynamic programming techniques and a known structural result on graph minors.