Affiliation:
1. CISPA Helmholtz Center for Information Security, Germany
2. Warsaw University of Technology, Faculty of Mathematics and Information Science, Poland and Warsaw, Institute of Informatics, Poland
Abstract
The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs
G
,
H
, and lists
L
(
v
)⊆
V
(
H
) for every
v
∈
V
(
G
), a
list homomorphism
is a function
f
:
V
(
G
) →
V
(
H
) that preserves the edges (i.e.,
uv
∈
E
(
G
) implies
f
(
u
)
f
(
v
) ∈
E
(
H
)) and respects the lists (i.e.,
f
(
v
) ∈
L
(
v
)). Standard techniques show that if
G
is given with a tree decomposition of width
t
, then the number of list homomorphisms can be counted in time
\(|V(H)|^t\cdot n^{\mathcal {O}(1)} \)
. Our main result is determining, for every fixed graph
H
, how much the base |
V
(
H
)| in the running time can be improved. For a connected graph
H
we define
\(\operatorname{irr}(H) \)
in the following way: if
H
has a loop or is nonbipartite, then
\(\operatorname{irr}(H) \)
is the maximum size of a set
S
⊆
V
(
H
) where any two vertices have different neighborhoods; if
H
is bipartite, then
\(\operatorname{irr}(H) \)
is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected
H
, we define
\(\operatorname{irr}(H) \)
as the maximum of
\(\operatorname{irr}(C) \)
over every connected component
C
of
H
. It follows from earlier results that if
\(\operatorname{irr}(H)=1 \)
, then the problem of counting list homomorphisms to
H
is polynomial-time solvable, and otherwise it is #P-hard. We show that, for every fixed graph
H
, the number of list homomorphisms from (
G
,
L
) to
H
•
can be counted in time
\(\operatorname{irr}(H)^t\cdot n^{\mathcal {O}(1)} \)
if a tree decomposition of
G
having width at most
t
is given in the input, and
•
given that
\(\operatorname{irr}(H)\ge 2 \)
, cannot be counted in time
\((\operatorname{irr}(H)-\epsilon)^t\cdot n^{\mathcal {O}(1)} \)
for any ϵ > 0, even if a tree decomposition of
G
having width at most
t
is given in the input, unless the Counting Strong Exponential-Time Hypothesis (#SETH) fails.
Thereby we give a precise and complete complexity classification featuring matching upper and lower bounds for all target graphs with or without loops.
Publisher
Association for Computing Machinery (ACM)
Subject
Mathematics (miscellaneous)