Two dichotomies for model-checking in multi-layer structures.

Multi-layer graphs can capture qualitatively different types of connection between entities, and networks of this kind are prevalent in biological and social systems: for example, a social contact network typically involves both virtual and face-to-face interactions between individuals. Since each layer is likely to exhibit stronger and/or more easily identifiable structurally properties than the overall system, it is natural to ask whether we can exploit the structural properties of individual layers to solve NP-hard problems on the overall network. In this paper we provide a complete characterisation of the structural properties required in each layer to guarantee the existence of an FPT algorithm to solve problems definable in either first-order or monadic second-order logic on the overall system, subject to the assumption that the structural properties are preserved under deletion of vertices and/or edges.

[1]  Christian Komusiewicz,et al.  Assessing the computational complexity of multilayer subgraph detection , 2016, Network Science.

[2]  S. Shen-Orr,et al.  Network motifs: simple building blocks of complex networks. , 2002, Science.

[3]  Mason A. Porter,et al.  Multilayer networks , 2013, J. Complex Networks.

[4]  Charalampos E. Tsourakakis.  Fast Counting of Triangles in Large Real Networks without Counting: Algorithms and Laws , 2008, 2008 Eighth IEEE International Conference on Data Mining.

[5]  Jessica Enright,et al.  Deleting Edges to Restrict the Size of an Epidemic: A New Application for Treewidth , 2015, Algorithmica.

[6]  Detlef Seese,et al.  Easy Problems for Tree-Decomposable Graphs , 1991, J. Algorithms.

[7]  Michael R. Fellows,et al.  Fundamentals of Parameterized Complexity , 2013 .

[8]  Stephan Kreutzer,et al.  Algorithmic Meta-theorems , 2008, IWPEC.

[9]  Nicole Schweikardt,et al.  First-Order Query Evaluation with Cardinality Conditions , 2017, PODS.

[10]  Jörg Flum,et al.  The Parameterized Complexity of Counting Problems , 2004, SIAM J. Comput..

[11]  Michael R. Fellows,et al.  Fixed-Parameter Tractability and Completeness II: On Completeness for W[1] , 1995, Theor. Comput. Sci..

[12]  Mercedes Pascual,et al.  The multilayer nature of ecological networks , 2015, Nature Ecology &Evolution.

[13]  Stefan Kratsch,et al.  Multi-budgeted directed cuts , 2018, IPEC.

[14]  Mark Jerrum,et al.  The parameterised complexity of counting connected subgraphs and graph motifs , 2013, J. Comput. Syst. Sci..

[15]  Robin Thomas,et al.  Deciding First-Order Properties for Sparse Graphs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[16]  Bruno Courcelle,et al.  On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic , 2001, Discret. Appl. Math..