23 research outputs found
Complexity of the List Homomorphism Problem in Hereditary Graph Classes
A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). For a fixed graph 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). We ask if there exists a homomorphism f from G to H, in which f(v) ? L(v) for every v ? V(G). Feder, Hell, and Huang [JGT 2003] proved that LHom(H) is polynomial time-solvable if H is a so-called bi-arc-graph, and NP-complete otherwise.
We are interested in the complexity of the LHom(H) problem in F-free graphs, i.e., graphs excluding a copy of some fixed graph F as an induced subgraph. It is known that if F is connected and is not a path nor a subdivided claw, then for every non-bi-arc graph the LHom(H) problem is NP-complete and cannot be solved in subexponential time, unless the ETH fails. We consider the remaining cases for connected graphs F.
If F is a path, we exhibit a full dichotomy. We define a class called predacious graphs and show that if H is not predacious, then for every fixed t the LHom(H) problem can be solved in quasi-polynomial time in P_t-free graphs. On the other hand, if H is predacious, then there exists t, such that the existence of a subexponential-time algorithm for LHom(H) in P_t-free graphs would violate the ETH.
If F is a subdivided claw, we show a full dichotomy in two important cases: for H being irreflexive (i.e., with no loops), and for H being reflexive (i.e., where every vertex has a loop). Unless the ETH fails, for irreflexive H the LHom(H) problem can be solved in subexponential time in graphs excluding a fixed subdivided claw if and only if H is non-predacious and triangle-free. On the other hand, if H is reflexive, then LHom(H) cannot be solved in subexponential time whenever H 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
On digraphs without onion star immersions
The -onion star is the digraph obtained from a star with leaves by
replacing every edge by a triple of arcs, where in triples we orient two
arcs away from the center, and in the remaining triples we orient two arcs
towards the center. Note that the -onion star contains, as an immersion,
every digraph on vertices where each vertex has outdegree at most and
indegree at most , or vice versa. We investigate the structure in digraphs
that exclude a fixed onion star as an immersion. The main discovery is that in
such digraphs, for some duality statements true in the undirected setting we
can prove their directed analogues. More specifically, we show the next two
statements.
There is a function satisfying the
following: If a digraph contains a set of vertices such that for
any there are arc-disjoint paths from to , then
contains the -onion star as an immersion.
There is a function
satisfying the following: If and is a pair of vertices in a digraph
such that there are at least arc-disjoint paths from to and
there are at least arc-disjoint paths from to , then either
contains the -onion star as an immersion, or there is a family of
pairwise arc-disjoint paths with paths from to and paths from
to .Comment: 14 pages, 5 figure
Vertex Deletion into Bipartite Permutation Graphs
A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines ?? and ??, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis [John M. Lewis and Mihalis Yannakakis, 1980].
We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time f(k)n^O(1), and also give a polynomial-time 9-approximation algorithm
Vertex deletion into bipartite permutation graphs
A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines and , one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis [20]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time , and also give a polynomial-time 9-approximation algorithm
Vertex deletion into bipartite permutation graphs
A permutation graph can be defined as an intersection graph of segments whose endpoints lie on two parallel lines \u1d4c1₁ and \u1d4c1₂, one on each. A bipartite permutation graph is a permutation graph which is bipartite. In this paper we study the parameterized complexity of the bipartite permutation vertex deletion problem, which asks, for a given n-vertex graph, whether we can remove at most k vertices to obtain a bipartite permutation graph. This problem is NP-complete by the classical result of Lewis and Yannakakis [John M. Lewis and Mihalis Yannakakis, 1980]. We analyze the structure of the so-called almost bipartite permutation graphs which may contain holes (large induced cycles) in contrast to bipartite permutation graphs. We exploit the structural properties of the shortest hole in a such graph. We use it to obtain an algorithm for the bipartite permutation vertex deletion problem with running time f(k)n^O(1), and also give a polynomial-time 9-approximation algorithm
Vertex deletion into bipartite permutation graphs
A permutation graph can be defined as an intersection graph of segments whose
endpoints lie on two parallel lines and , one on each. A bipartite
permutation graph is a permutation graph which is bipartite. In this paper we
study the parameterized complexity of the bipartite permutation vertex deletion
problem, which asks, for a given n-vertex graph, whether we can remove at most
k vertices to obtain a bipartite permutation graph. This problem is NP-complete
by the classical result of Lewis and Yannakakis. We analyze the structure of
the so-called almost bipartite permutation graphs which may contain holes
(large induced cycles) in contrast to bipartite permutation graphs. We exploit
the structural properties of the shortest hole in a such graph. We use it to
obtain an algorithm for the bipartite permutation vertex deletion problem with
running time , and also give a polynomial-time 9-approximation
algorithm.Comment: Extended abstract accepted to International Symposium on
Parameterized and Exact Computation (IPEC'20