4,979 research outputs found
The complexity of the list homomorphism problem for graphs
We completely classify the computational complexity of the list H-colouring
problem for graphs (with possible loops) in combinatorial and algebraic terms:
for every graph H the problem is either NP-complete, NL-complete, L-complete or
is first-order definable; descriptive complexity equivalents are given as well
via Datalog and its fragments. Our algebraic characterisations match important
conjectures in the study of constraint satisfaction problems.Comment: 12 pages, STACS 201
List homomorphism problems for signed graphs
We consider homomorphisms of signed graphs from a computational perspective.
In particular, we study the list homomorphism problem seeking a homomorphism of
an input signed graph , equipped with lists , of allowed images, to a fixed target signed graph . The
complexity of the similar homomorphism problem without lists (corresponding to
all lists being ) has been previously classified by Brewster and
Siggers, but the list version remains open and appears difficult. We illustrate
this difficulty by classifying the complexity of the problem when is a tree
(with possible loops). The tools we develop will be useful for classifications
of other classes of signed graphs, and we illustrate this by classifying the
complexity of irreflexive signed graphs in which the unicoloured edges form
some simple structures, namely paths or cycles. The structure of the signed
graphs in the polynomial cases is interesting, suggesting they may constitute a
nice class of signed graphs analogous to the so-called bi-arc graphs (which
characterize the polynomial cases of list homomorphisms to unsigned graphs).Comment: various changes + rewritten section on path- and cycle-separable
graphs based on a new conference submission (split possible in future
Min orderings and list homomorphism dichotomies for signed and unsigned graphs
The CSP dichotomy conjecture has been recently established, but a number of
other dichotomy questions remain open, including the dichotomy classification
of list homomorphism problems for signed graphs. Signed graphs arise naturally
in many contexts, including for instance nowhere-zero flows for graphs embedded
in non-orientable surfaces. For a fixed signed graph , the list
homomorphism problem asks whether an input signed graph with
lists admits a
homomorphism to with all .
Usually, a dichotomy classification is easier to obtain for list
homomorphisms than for homomorphisms, but in the context of signed graphs a
structural classification of the complexity of list homomorphism problems has
not even been conjectured, even though the classification of the complexity of
homomorphism problems is known.
Kim and Siggers have conjectured a structural classification in the special
case of "weakly balanced" signed graphs. We confirm their conjecture for
reflexive and irreflexive signed graphs; this generalizes previous results on
weakly balanced signed trees, and weakly balanced separable signed graphs. In
the reflexive case, the result was first presented in a paper of Kim and
Siggers, where the proof relies on a result in this paper. The irreflexive
result is new, and its proof depends on first deriving a theorem on extensions
of min orderings of (unsigned) bipartite graphs, which is interesting on its
own
List Homomorphism Problems for Signed Graphs
We consider homomorphisms of signed graphs from a computational perspective. In particular, we study the list homomorphism problem seeking a homomorphism of an input signed graph (G,?), equipped with lists L(v) ? V(H), v ? V(G), of allowed images, to a fixed target signed graph (H,?). The complexity of the similar homomorphism problem without lists (corresponding to all lists being L(v) = V(H)) has been previously classified by Brewster and Siggers, but the list version remains open and appears difficult. Both versions (with lists or without lists) can be formulated as constraint satisfaction problems, and hence enjoy the algebraic dichotomy classification recently verified by Bulatov and Zhuk. By contrast, we seek a combinatorial classification for the list version, akin to the combinatorial classification for the version without lists completed by Brewster and Siggers. We illustrate the possible complications by classifying the complexity of the list homomorphism problem when H is a (reflexive or irreflexive) signed tree. It turns out that the problems are polynomial-time solvable for certain caterpillar-like trees, and are NP-complete otherwise. The tools we develop will be useful for classifications of other classes of signed graphs, and we mention some follow-up research of this kind; those classifications are surprisingly complex
Testing List H-Homomorphisms
Let be an undirected graph. In the List -Homomorphism Problem, given
an undirected graph with a list constraint for each
variable , the objective is to find a list -homomorphism , that is, for every and whenever .
We consider the following problem: given a map as an oracle
access, the objective is to decide with high probability whether is a list
-homomorphism or \textit{far} from any list -homomorphisms. The
efficiency of an algorithm is measured by the number of accesses to .
In this paper, we classify graphs with respect to the query complexity
for testing list -homomorphisms and show the following trichotomy holds: (i)
List -homomorphisms are testable with a constant number of queries if and
only if is a reflexive complete graph or an irreflexive complete bipartite
graph. (ii) List -homomorphisms are testable with a sublinear number of
queries if and only if is a bi-arc graph. (iii) Testing list
-homomorphisms requires a linear number of queries if is not a bi-arc
graph
Full Complexity Classification of the List Homomorphism Problem for Bounded-Treewidth Graphs
A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is assigned with a list L(v) of vertices of H. We ask whether there exists a homomorphism h from G to H, which respects lists L, i.e., for every v ? V(G) it holds that h(v) ? L(v).
The complexity dichotomy for LHom(H) was proven by Feder, Hell, and Huang [JGT 2003]. The authors showed that the problem is polynomial-time solvable if H belongs to the class called bi-arc graphs, and for all other graphs H it is NP-complete.
We are interested in the complexity of the LHom(H) problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rz??ewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs H, i.e., if every vertex has a loop.
In this paper we extend and generalize their results for all relevant graphs H, i.e., those, for which the LHom(H) problem is NP-hard. For every such H we find a constant k = k(H), such that the LHom(H) problem on instances G with n vertices and treewidth t
- can be solved in time k^t ? n^?(1), provided that G is given along with a tree decomposition of width t,
- cannot be solved in time (k-?)^t ? n^?(1), for any ? > 0, unless the SETH fails. For some graphs H the value of k(H) is much smaller than the trivial upper bound, i.e., |V(H)|.
Obtaining matching upper and lower bounds shows that the set of algorithmic tools that we have discovered cannot be extended in order to obtain faster algorithms for LHom(H) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of the LHom(H) problem, e.g. with different parameterizations
List Locally Surjective Homomorphisms in Hereditary Graph Classes
A locally surjective homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H) that is surjective in the neighborhood of each vertex in G. In the list locally surjective homomorphism problem, denoted by LLSHom(H), the graph H is fixed and the instance consists of a graph G whose every vertex is equipped with a subset of V(H), called list. We ask for the existence of a locally surjective homomorphism from G to H, where every vertex of G is mapped to a vertex from its list. In this paper, we study the complexity of the LLSHom(H) problem in F-free graphs, i.e., graphs that exclude a fixed graph F as an induced subgraph. We aim to understand for which pairs (H,F) the problem can be solved in subexponential time.
We show that for all graphs H, for which the problem is NP-hard in general graphs, it cannot be solved in subexponential time in F-free graphs for F being a bounded-degree forest, unless the ETH fails. The initial study reveals that a natural subfamily of bounded-degree forests F, that might lead to some tractability results, is the family ? consisting of forests whose every component has at most three leaves. In this case, we exhibit the following dichotomy theorem: besides the cases that are polynomial-time solvable in general graphs, the graphs H ? {P?,C?} are the only connected ones that allow for a subexponential-time algorithm in F-free graphs for every F ? ? (unless the ETH fails)
Fine-Grained Complexity of the List Homomorphism Problem: Feedback Vertex Set and Cutwidth
For graphs G,H, a homomorphism from G to H is an edge-preserving mapping from V(G) to V(H). In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is equipped with a list L(v) ? V(H), and we need to determine whether there exists a homomorphism from G to H which additionally respects the lists L. List homomorphisms are a natural generalization of (list) colorings.
Very recently Okrasa, Piecyk, and Rz??ewski [ESA 2020] studied the fine-grained complexity of the problem, parameterized by the treewidth of the instance graph G. They defined a new invariant i^*(H), and proved that for every relevant graph H, i.e., such that LHom(H) is NP-hard, this invariant is the correct base of the exponent in the running time of any algorithm solving the LHom(H) problem.
In this paper we continue this direction and study the complexity of the problem under different parameterizations. As the first result, we show that i^*(H) is also the right complexity base if the parameter is the size of a minimum feedback vertex set of G, denoted by fvs(G). In particular, for every relevant graph H, the LHom(H) problem
- can be solved in time i^*(H)^fvs(G) ? |V(G)|^?(1), if a minimum feedback vertex set of G is given,
- cannot be solved in time (i^*(H) - ?)^fvs(G) ? |V(G)|^?(1), for any ? > 0, unless the SETH fails. Then we turn our attention to a parameterization by the cutwidth ctw(G) of G. Jansen and Nederlof [TCS 2019] showed that List k-Coloring (i.e., LHom(K_k)) can be solved in time c^ctw(G) ? |V(G)|^?(1) for an absolute constant c, i.e., the base of the exponential function does not depend on the number of colors. Jansen asked whether this behavior extends to graph homomorphisms. As the main result of the paper, we answer the question in the negative. We define a new graph invariant mim^*(H), closely related to the size of a maximum induced matching in H, and prove that for all relevant graphs H, the LHom(H) problem cannot be solved in time (mim^*(H)-?)^{ctw(G)}? |V(G)|^?(1) for any ? > 0, unless the SETH fails. In particular, this implies that, assuming the SETH, there is no constant c, such that for every odd cycle the non-list version of the problem can be solved in time c^ctw(G) ? |V(G)|^?(1)
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