The parameterised complexity of counting even and odd induced subgraphs

We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory.

[1]  Venkatesh Raman,et al.  Approximation Algorithms for Some Parameterized Counting Problems , 2002, ISAAC.

[2]  H. Niederreiter,et al.  Finite Fields: Encyclopedia of Mathematics and Its Applications. , 1997 .

[3]  K. Conrad,et al.  Finite Fields , 2018, Series and Products in the Development of Mathematics.

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

[5]  Yijia Chen,et al.  On Parameterized Path and Chordless Path Problems , 2007, Twenty-Second Annual IEEE Conference on Computational Complexity (CCC'07).

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

[7]  Mark Jerrum,et al.  Some Hard Families of Parameterized Counting Problems , 2013, ACM Trans. Comput. Theory.

[8]  Pentti Haukkanen,et al.  On meet matrices on posets , 1996 .

[9]  Michael R. Fellows,et al.  On the parameterized complexity of multiple-interval graph problems , 2009, Theor. Comput. Sci..

[10]  Dániel Marx,et al.  Complexity of Counting Subgraphs: Only the Boundedness of the Vertex-Cover Number Counts , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[11]  Radu Curticapean,et al.  Counting Matchings of Size k Is W[1]-Hard , 2013, ICALP.

[12]  B. V. Rajarama Bhat,et al.  On greatest common divisor matrices and their applications , 1991 .

[13]  Kitty Meeks,et al.  The challenges of unbounded treewidth in parameterised subgraph counting problems , 2014, Discret. Appl. Math..

[14]  Leslie Ann Goldberg,et al.  A Complexity Dichotomy for Partition Functions with Mixed Signs , 2008, SIAM J. Comput..

[15]  G. Szekeres,et al.  A combinatorial problem in geometry , 2009 .

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

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

[18]  Rudolf Lide,et al.  Finite fields , 1983 .

[19]  A. Ehrenfeucht,et al.  The Computational Complexity of ({\it XOR, AND\/})-Counting Problems , 1990 .