116 research outputs found
Counting Homomorphisms to -minor-free Graphs, modulo 2
We study the problem of computing the parity of the number of homomorphisms
from an input graph to a fixed graph . Faben and Jerrum [ToC'15]
introduced an explicit criterion on the graph and conjectured that, if
satisfied, the problem is solvable in polynomial time and, otherwise, the
problem is complete for the complexity class of parity
problems. We verify their conjecture for all graphs that exclude the
complete graph on vertices as a minor. Further, we rule out the existence
of a subexponential-time algorithm for the -complete cases,
assuming the randomised Exponential Time Hypothesis. Our proofs introduce a
novel method of deriving hardness from globally defined substructures of the
fixed graph . Using this, we subsume all prior progress towards resolving
the conjecture (Faben and Jerrum [ToC'15]; G\"obel, Goldberg and Richerby
[ToCT'14,'16]). As special cases, our machinery also yields a proof of the
conjecture for graphs with maximum degree at most , as well as a full
classification for the problem of counting list homomorphisms, modulo
On Counting (Quantum-)Graph Homomorphisms in Finite Fields of Prime Order
We study the problem of counting the number of homomorphisms from an input
graph to a fixed (quantum) graph in any finite field of prime
order . The subproblem with graph was introduced by Faben and
Jerrum~[ToC'15] and its complexity is still uncharacterised despite active
research, e.g. the very recent work of Focke, Goldberg, Roth, and
Zivn\'y~[SODA'21]. Our contribution is threefold. First, we introduce the study
of quantum graphs to the study of modular counting homomorphisms. We show that
the complexity for a quantum graph collapses to the complexity
criteria found at dimension 1: graphs. Second, in order to prove cases of
intractability we establish a further reduction to the study of bipartite
graphs. Lastly, we establish a dichotomy for all bipartite
-free graphs by a thorough structural
study incorporating both local and global arguments. This result subsumes all
results on bipartite graphs known for all prime moduli and extends them
significantly. Even for the subproblem with this establishes new results.Comment: 84 pages, revised title and mainly the Introduction and the section
on partially surjective homomorphism
Parameterized (Modular) Counting and Cayley Graph Expanders
We study the problem #EdgeSub(?) of counting k-edge subgraphs satisfying a given graph property ? in a large host graph G. Building upon the breakthrough result of Curticapean, Dell and Marx (STOC 17), we express the number of such subgraphs as a finite linear combination of graph homomorphism counts and derive the complexity of computing this number by studying its coefficients.
Our approach relies on novel constructions of low-degree Cayley graph expanders of p-groups, which might be of independent interest. The properties of those expanders allow us to analyse the coefficients in the aforementioned linear combinations over the field ?_p which gives us significantly more control over the cancellation behaviour of the coefficients. Our main result is an exhaustive and fine-grained complexity classification of #EdgeSub(?) for minor-closed properties ?, closing the missing gap in previous work by Roth, Schmitt and Wellnitz (ICALP 21).
Additionally, we observe that our methods also apply to modular counting. Among others, we obtain novel intractability results for the problems of counting k-forests and matroid bases modulo a prime p. Furthermore, from an algorithmic point of view, we construct algorithms for the problems of counting k-paths and k-cycles modulo 2 that outperform the best known algorithms for their non-modular counterparts.
In the course of our investigations we also provide an exhaustive parameterized complexity classification for the problem of counting graph homomorphisms modulo a prime p
Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths
We systematically investigate the complexity of counting subgraph patterns
modulo fixed integers. For example, it is known that the parity of the number
of -matchings can be determined in polynomial time by a simple reduction to
the determinant. We generalize this to an -time algorithm to
compute modulo the number of subgraph occurrences of patterns that are
vertices away from being matchings. This shows that the known
polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry
over into the setting of counting modulo .
Complementing our algorithm, we also give a simple and self-contained proof
that counting -matchings modulo odd integers is Mod_q-W[1]-complete and
prove that counting -paths modulo is Parity-W[1]-complete, answering an
open question by Bj\"orklund, Dell, and Husfeldt (ICALP 2015).Comment: 23 pages, to appear at ESA 202
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