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A Generic Algorithm for Approximately Solving Stochastic Graph Optimization Problems

  • Ei Ando
  • Hirotaka Ono
  • Masafumi Yamashita
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5792)

Abstract

Given a (directed or undirected) graph G = (V,E), a mutually independent random variable X e obeying a normal distribution for each edge e ∈ E that represents its edge weight, and a property \(\cal P\) on graph, a stochastic graph maximization problem asks the distribution function F MAX(x) of random variable \(X_{\rm MAX} = \max_{P \in {\cal P}} \sum_{e \in A} X_e\), where property \({\cal P}\) is identified with the set of subgraphs P = (U,A) of G having \(\cal P\). This paper proposes a generic algorithm for computing an elementary function \(\tilde F(x)\) that approximates F MAX(x). It is applicable to any \(\cal P\) and runs in time \(O(T_{A_{\rm MAX}} ({\cal P})+ T_{A_{\rm CNT}} ({\cal P}))\), provided the existence of an algorithm A MAX that solves the (deterministic) graph maximization problem for \(\cal P\) in time \(T_{A_{\rm MAX}} ({\cal P})\) and an algorithm A CNT that outputs an upper bound on \(|{\cal P}|\) in time \(T_{A_{\rm CNT}} ({\cal P})\). We analyze the approximation ratio and apply it to three graph maximization problems. In case no efficient algorithms are known for solving the graph maximization problem for \(\cal P\), an approximation algorithm A APR can be used instead of A MAX to reduce the time complexity, at the expense of increase of approximation ratio. Our algorithm can be modified to handle minimization problems.

Keywords

Independent Random Variable Approximation Ratio Edge Weight Stochastic Optimization Problem Span Tree Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ei Ando
    • 1
  • Hirotaka Ono
    • 1
    • 2
  • Masafumi Yamashita
    • 1
    • 2
  1. 1.Dept. Computer Sci. and Communication Eng.Kyushu UniversityFukuokaJapan
  2. 2.Institute of Systems, Information Technologies and Nanotechnologies, FukuokaFukuokaJapan

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