Results 1 
9 of
9
Maximizing a Monotone Submodular Function subject to a Matroid Constraint
, 2008
"... Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)app ..."
Abstract

Cited by 25 (0 self)
 Add to MetaCart
Let f: 2 X → R+ be a monotone submodular set function, and let (X, I) be a matroid. We consider the problem maxS∈I f(S). It is known that the greedy algorithm yields a 1/2 approximation [14] for this problem. For certain special cases, e.g. max S≤k f(S), the greedy algorithm yields a (1 − 1/e)approximation. It is known that this is optimal both in the value oracle model (where the only access to f is through a black box returning f(S) for a given set S) [28], and also for explicitly posed instances assuming P � = NP [10]. In this paper, we provide a randomized (1 − 1/e)approximation for any monotone submodular function and an arbitrary matroid. The algorithm works in the value oracle model. Our main tools are a variant of the pipage rounding technique of Ageev and Sviridenko [1], and a continuous greedy process that might be of independent interest. As a special case, our algorithm implies an optimal approximation for the Submodular Welfare Problem in the value oracle model [32]. As a second application, we show that the Generalized Assignment Problem (GAP) is also a special case; although the reduction requires X  to be exponential in the original problem size, we are able to achieve a (1 − 1/e − o(1))approximation for GAP, simplifying previously known algorithms. Additionally, the reduction enables us to obtain approximation algorithms for variants of GAP with more general constraints.
Two New Approximation Algorithms for the Maximum Planar Subgraph Problem
, 2006
"... The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NPhard and it has applications in graph drawing and resource location optimization. Călinescu et al. [J. Alg. 27, 269302 (1998)] presented the first approximation a ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
The maximum planar subgraph problem (MPS) is defined as follows: given a graph G, find a largest planar subgraph of G. The problem is NPhard and it has applications in graph drawing and resource location optimization. Călinescu et al. [J. Alg. 27, 269302 (1998)] presented the first approximation algorithms for MPS with nontrivial performance ratios. Two algorithms were given, a simple algorithm which runs in linear time for boundeddegree graphs with a ratio 7/18 and a more complicated algorithm with a ratio 4/9. Both algorithms produce outerplanar subgraphs. In this article we present two new versions of the simpler algorithm. The first new algorithm still runs in the same time, produces outerplanar subgraphs, has at least the same performance ratio as the original algorithm, but in practice it finds larger planar subgraphs than the original algorithm. The second new algorithm has similar properties to the first algorithm, but it produces only planar subgraphs. We conjecture that the performance ratios of our algorithms are at least 4/9 for MPS. We experimentally compare the new algorithms against the original simple algorithm. We also apply the new algorithms for approximating the thickness and outerthickness of a graph. Experiments show that the new algorithms produce clearly better approximations than the original simple algorithm by Călinescu et al.
Research Project: Packing Problems in Combinatorial Optimization
"... submitted to the Probral DAAD–CAPES Program ..."
(Show Context)
Otimização Combinatória: Teoria, Projeto de
"... O objetivo desse projeto é o estudo de problemas de otimização combinatória. Temos particular interesse em projeto de algoritmos de aproximação e em resultados de inaproximabilidade para tais problemas. Consideraremos em nossas pesquisas problemas clássicos da teoria dos grafos, problemas de empacot ..."
Abstract
 Add to MetaCart
(Show Context)
O objetivo desse projeto é o estudo de problemas de otimização combinatória. Temos particular interesse em projeto de algoritmos de aproximação e em resultados de inaproximabilidade para tais problemas. Consideraremos em nossas pesquisas problemas clássicos da teoria dos grafos, problemas de empacotamento e problemas em biologia computacional. A equipe do projeto consiste de 8 pesquisadores que já interagem há algum tempo, com experiência na área em que o projeto se enquadra, envolve cerca de 10 alunos de pósgraduação, e 4 pesquisadores colaboradores. Duas das instituições envolvidas (USP e UFRJ) são bem estabelecidas na área e uma é emergente (UFMS). Dentro do projeto, pretendemos fortalecer a colaboração já existente, através de realização de oficinas semestrais, e expor os alunos envolvidos a um ambiente de pesquisa prolífero. 2 Projeto de pesquisa e metodologia Problemas de otimização têm o objetivo de encontrar um ponto ótimo (mínimo ou máximo) de uma função definida sobre um certo domínio. Os problemas de
A Polynomial Time Randomized Parallel Approximation Algorithm for Finding Heavy Planar Subgraphs
, 2006
"... We provide an approximation algorithm for the Maximum Weight Planar Subgraph problem, the NPhard problem of finding a heaviest planar subgraph in an edgeweighted graph G. In the general case our algorithm has performance ratio at least 1/3 + 1/72 matching the best algorithm known so far, though in ..."
Abstract
 Add to MetaCart
We provide an approximation algorithm for the Maximum Weight Planar Subgraph problem, the NPhard problem of finding a heaviest planar subgraph in an edgeweighted graph G. In the general case our algorithm has performance ratio at least 1/3 + 1/72 matching the best algorithm known so far, though in several special cases we prove stronger results. In particular, we obtain performance ratio 2/3 (instead of 7/12) for the NPhard Maximum Weight Outerplanar Subgraph problem meeting the performance ratio of the best algorithm for the unweighted case. When the maximum weight planar subgraph is one of several special types of Hamiltonian graphs, we show performance ratios at least 2/5 and 4/9 (instead of 1/3 + 1/72), and 1/2 (instead of 4/9) for the unweighted case.
Noname manuscript No. (will be inserted by the editor) Maximum SeriesParallel Subgraph
, 2009
"... Abstract Consider the NPhard problem of, given a simple graph G, to find a seriesparallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a 1 2approximation. Indeed, if n is the number of vertices in G, any spanning tr ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract Consider the NPhard problem of, given a simple graph G, to find a seriesparallel subgraph of G with the maximum number of edges. The algorithm that, given a connected graph G, outputs a spanning tree of G, is a 1 2approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has n−1 edges and any seriesparallel graph on n vertices has at most 2n−3 edges. We present a 7 12approximation for this problem and results showing the limits of our approach. 1
On the krestricted structure ratio in planar and outerplanar graphs
"... A planar krestricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar krestricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar krestricted ratio is the infimum, over simple plan ..."
Abstract
 Add to MetaCart
A planar krestricted structure is a simple graph whose blocks are planar and each has at most k vertices. Planar krestricted structures are used by approximation algorithms for Maximum Weight Planar Subgraph, which motivates this work. The planar krestricted ratio is the infimum, over simple planar graphs H, of the ratio of the number of edges in a maximum krestricted structure subgraph of H to the number edges of H. We prove that, as k tends to infinity, the planar krestricted ratio tends to 1/2. The same result holds for the weighted version. Our results are based on analyzing the analogous ratios for outerplanar and weighted outerplanar graphs. Here both ratios tend to 1 as k goes to infinity, and we provide good estimates of the rates of convergence, showing that they differ in the weighted from the unweighted case.