Finding Good Decompositions for Dynamic Programming on Dense Graphs

  • Eivind Magnus Hvidevold
  • Sadia Sharmin
  • Jan Arne Telle
  • Martin Vatshelle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7112)


It is well-known that for graphs with high edge density the tree-width is always high while the clique-width can be low. Boolean-width is a new parameter that is never higher than tree-width or clique-width and can in fact be as small as logarithmic in clique-width. Boolean-width is defined using a decomposition tree by evaluating the number of neighborhoods across the resulting cuts of the graph. Several NP-hard problems can be solved efficiently by dynamic programming when given a decomposition of boolean-width k, e.g. Max Weight Independent Set in time O(n 2 k22k ) and Min Weight Dominating Set in time O(n 2 + nk23k ). Finding decompositions of low boolean-width is therefore of practical interest. There is evidence that computing boolean-width is hard, while the existence of a useful approximation algorithm is still open. In this paper we introduce and study a heuristic algorithm that finds a reasonably good decomposition to be used for dynamic programming based on boolean-width. On a set of graphs of practical relevance, specifically graphs in TreewidthLIB, the best known upper bound on their tree-width is compared to the upper bound on their boolean-width given by our heuristic. For the large majority of the graphs on which we made the tests, the tree-width bound is at least twice as big as the boolean-width bound, and boolean-width compares better the higher the edge density. This means that, for problems like Dominating Set, using boolean-width should outperform dynamic programming by tree-width, at least for graphs of edge density above a certain bound. In view of the amount of previous work on heuristics for tree-width these results indicate that boolean-width could in the future outperform tree-width in practice for a large class of graphs and problems.


Local Search Dynamic Programming Edge Density Decomposition Tree Dense Graph 
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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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eivind Magnus Hvidevold
    • 1
  • Sadia Sharmin
    • 1
  • Jan Arne Telle
    • 1
  • Martin Vatshelle
    • 1
  1. 1.Department of InformaticsUniversity of BergenNorway

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