On Approximate Solutions for Combinatorial Optimization Problems

Simon, Hans-Ulrich

doi:10.1137/0403025

Cited by 85 publications

(36 citation statements)

References 12 publications

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“…The approximation-preserving reduction to and from set cover, together with the result of Simon [32], gives us the following interesting corollary: Corollary 2. Given an algorithm that, in every iteration of the greedy algorithm, selects the factor F i ∈ D that approximates the best choice by a factor of O(h(n)), we can approximate the smallest binary sub-decomposition by a factor of O(h(n) ln n).…”

Section: Approximating Smallest Sub-decompositionsmentioning

confidence: 99%

Generalized Matrix Factorizations as a Unifying Framework for Pattern Set Mining: Complexity Beyond Blocks

Miettinen

2015

Machine Learning and Knowledge Discovery in Databases

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Abstract. Matrix factorizations are a popular tool to mine regularities from data. There are many ways to interpret the factorizations, but one particularly suited for data mining utilizes the fact that a matrix product can be interpreted as a sum of rank-1 matrices. Then the factorization of a matrix becomes the task of finding a small number of rank-1 matrices, sum of which is a good representation of the original matrix. Seen this way, it becomes obvious that many problems in data mining can be expressed as matrix factorizations with correct definitions of what a rank-1 matrix and a sum of rank-1 matrices mean. This paper develops a unified theory, based on generalized outer product operators, that encompasses many pattern set mining tasks. The focus is on the computational aspects of the theory and studying the computational complexity and approximability of many problems related to generalized matrix factorizations. The results immediately apply to a large number of data mining problems, and hopefully allow generalizing future results and algorithms, as well.

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Section: Approximating Smallest Sub-decompositionsmentioning

confidence: 99%

Generalized Matrix Factorizations as a Unifying Framework for Pattern Set Mining: Complexity Beyond Blocks

Miettinen

2015

Machine Learning and Knowledge Discovery in Databases

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“…It consists of solving a minimization problem (the master one) by iteratively solving a maximization one (the slave problem) (for more details on this technique, cf., [1,12,19]). This kind of technique has a very natural application in the case of minimum coloring where the slave problem is the maximum independent set.…”

Section: Lemmamentioning

confidence: 99%

Probabilistic Coloring of Bipartite and Split Graphs

Croce

Escoffier

Murat

et al. 2005

Computational Science and Its Applications – ICCSA 2005

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“…Proof. Consider the following reduction from Independent Set to Bipartite Subgraph given in [34]. Let G(V, E) be an instance of Independent Set of order n. Construct a graph G (V , E ) for Bipartite Subgraph by taking two distinct copies of G (denote them by G 1 and G 2 , respectively) and adding the following edges: a vertex v i1 of copy G 1 is linked with a vertex v j2 of G 2 , if and only if either i = j or (v i , v j ) ∈ E. The graph G has 2n vertices.…”

Section: Ethmentioning

confidence: 99%

On Subexponential and FPT-Time Inapproximability

Bonnet

Escoffier

Kim

et al. 2013

Parameterized and Exact Computation

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Abstract. Fixed-parameter algorithms, approximation algorithms and moderately exponential algorithms are three major approaches to algorithm design. While each of them being very active in its own, there is an increasing attention to the connection between these different frameworks. In particular, whether Independent Set would be better approximable once endowed with subexponential-time or FPT-time is a central question. In this article, we provide new insights to this question using two complementary approaches; the former makes a strong link between the linear PCP conjecture and inapproximability; the latter builds a class of equivalent problems under approximation in subexponential time.

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On Approximate Solutions for Combinatorial Optimization Problems

Cited by 85 publications

References 12 publications

Generalized Matrix Factorizations as a Unifying Framework for Pattern Set Mining: Complexity Beyond Blocks

Generalized Matrix Factorizations as a Unifying Framework for Pattern Set Mining: Complexity Beyond Blocks

Probabilistic Coloring of Bipartite and Split Graphs

On Subexponential and FPT-Time Inapproximability

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