The Hardness of Embedding Grids and Walls

The dichotomy conjecture for the parameterized embedding problem states that the problem of deciding whether a given graph G from some class \(\mathcal K\) of “pattern graphs” can be embedded into a given graph H (that is, is isomorphic to a subgraph of H) is fixed-parameter tractable if \(\mathcal K\) is a class of graphs of bounded tree width and \(W [1]\)-complete otherwise.

[1]  Bingkai Lin,et al.  The Parameterized Complexity of k-Biclique , 2014, SODA.

[2]  B. Monien.  How to Find Long Paths Efficiently , 1985 .

[3]  Manuel Bodirsky,et al.  Non-dichotomies in Constraint Satisfaction Complexity , 2008, ICALP.

[4]  Julian R. Ullmann,et al.  An Algorithm for Subgraph Isomorphism , 1976, J. ACM.

[5]  M. Lucertini,et al.  Analysis and design of algorithms for combinatorial problems , 1985 .

[6]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.

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

[8]  D. Matula.  Subtree Isomorphism in O(n5/2) , 1978 .

[9]  Yijia Chen,et al.  Understanding the Complexity of Induced Subgraph Isomorphisms , 2008, ICALP.

[10]  Phokion G. Kolaitis,et al.  Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics , 2002, CP.

[11]  Bernd Voigt,et al.  Finding Minimally Weighted Subgraphs , 1991, WG.

[12]  Dániel Marx,et al.  Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels , 2015, SODA.

[13]  Mam Riess Jones.  Color Coding , 1962, Human factors.

[14]  Martin Grohe.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2007, JACM.

[15]  Thomas Schwentick,et al.  When is the evaluation of conjunctive queries tractable? , 2001, STOC '01.

[16]  David Eppstein,et al.  The Polyhedral Approach to the Maximum Planar Subgraph Problem: New Chances for Related Problems , 1994, GD.

[17]  Michal Pilipczuk,et al.  Parameterized Algorithms , 2015, Springer International Publishing.

[18]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[19]  Michal Pilipczuk,et al.  Everything you always wanted to know about the parameterized complexity of Subgraph Isomorphism (but were afraid to ask) , 2013, STACS.

[20]  Hubie Chen,et al.  One Hierarchy Spawns Another , 2013, ACM Trans. Comput. Log..

[21]  Paul D. Seymour,et al.  Graph minors. V. Excluding a planar graph , 1986, J. Comb. Theory B.

[22]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .