On the algorithmic effectiveness of digraph decompositions and complexity measures

  • Authors:

  • Michael Lampis;Georgia Kaouri;Valia Mitsou

  • Affiliations:

  • Department of Computer Science, Graduate Center, City University of New York, 365 5th Avenue, New York, NY 10016, USA;Computation and Reasoning lab, School of Electrical and Computer Engineering, National Technical University of Athens, Greece;Department of Computer Science, Graduate Center, City University of New York, 365 5th Avenue, New York, NY 10016, USA

  • Venue:
  • Discrete Optimization
  • Year:
  • 2011

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Abstract

We place our focus on the gap between treewidth's success in producing fixed-parameter polynomial algorithms for hard graph problems, and specifically Hamiltonian Circuit and Max Cut, and the failure of its directed variants (directed treewidth (Johnson et al., 2001 [13]), DAG-width (Obdrzalek, 2006 [14]) and Kelly-width (Hunter and Kreutzer, 2007 [15]) to replicate it in the realm of digraphs. We answer the question of why this gap exists by giving two hardness results: we show that Directed Hamiltonian Circuit is W[2]-hard when the parameter is the width of the input graph, for any of these widths, and that Max Di Cut remains NP-hard even when restricted to DAGs, which have the minimum possible width under all these definitions. Along the way, we extend our reduction for Directed Hamiltonian Circuit to show that the related Minimum Leaf Outbranching problem is also W[2]-hard when naturally parameterized by the number of leaves of the solution, even if the input graph has constant width. All our results also apply to directed pathwidth and cycle rank.