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Exact Algorithms on Graphs of Bounded Average Degree

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  • First Online:
Encyclopedia of Algorithms

  • 73 Accesses

Years and Authors of Summarized Original Work

  • 2010; Björklund, Husfeldt, Kaski, Koivisto

  • 2012; Björklund, Husfeldt, Kaski, Koivisto

  • 2013; Cygan, Pilipczuk

  • 2014; Golovnev, Kulikov, Mihajlin

Problem Definition

We focus on the following question: how an assumption on the sparsity of an input graph, such as bounded (average) degree, can help in designing exact (exponential-time) algorithms for NP-hard problems. The following classic problems are studied:

Traveling Salesman Problem :

Find a minimum-length Hamiltonian cycle in an input graph with edge weights.

Chromatic Number :

Find a minimum number k for which the vertices of an input graph can be colored with k colors such that no two adjacent vertices receive the same color.

Counting Perfect Matchings :

Find the number of perfect matchings in an input graph.

Key Results

The classic algorithms of Bellman [1] and Held and Karp [10] for traveling salesman problem run in \(2^{n}n^{\mathcal{O}(1)}\) time for n-vertex graphs. Using the...

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Recommended Reading

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  2. Björklund A (2012) Counting perfect matchings as fast as ryser. In: Rabani Y (ed) SODA, Kyoto. SIAM, pp 914–921

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Author information

Authors and Affiliations

  1. Institute of Informatics, University of Bergen, Bergen, Norway

    Marcin Pilipczuk

  2. Institute of Informatics, University of Warsaw, Warsaw, Poland

    Marcin Pilipczuk

Authors
  1. Marcin Pilipczuk
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Corresponding author

Correspondence to Marcin Pilipczuk .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL, USA

    Ming-Yang Kao

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Pilipczuk, M. (2016). Exact Algorithms on Graphs of Bounded Average Degree. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2864-4_516

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