8 research outputs found
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)
The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid
We show that computing the Tutte polynomial of a linear matroid of dimension
on points over a field of elements requires
time unless the \#ETH---a counting extension of the Exponential
Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell {\em et al.}
[ACM TALG 2014]---is false. This holds also for linear matroids that admit a
representation where every point is associated to a vector with at most two
nonzero coordinates. We also show that the same is true for computing the Tutte
polynomial of a binary matroid of dimension on points with at
most three nonzero coordinates in each point's vector. This is in sharp
contrast to computing the Tutte polynomial of a -vertex graph (that is, the
Tutte polynomial of a {\em graphic} matroid of dimension ---which is
representable in dimension over the binary field so that every vector has
two nonzero coordinates), which is known to be computable in
time [Bj\"orklund {\em et al.}, FOCS 2008]. Our lower-bound proofs proceed via
(i) a connection due to Crapo and Rota [1970] between the number of tuples of
codewords of full support and the Tutte polynomial of the matroid associated
with the code; (ii) an earlier-established \#ETH-hardness of counting the
solutions to a bipartite -CSP on vertices in time; and
(iii) new embeddings of such CSP instances as questions about codewords of full
support in a linear code. We complement these lower bounds with two algorithm
designs. The first design computes the Tutte polynomial of a linear matroid of
dimension~ on points in operations. The second design
generalizes the Bj\"orklund~{\em et al.} algorithm and runs in
time for linear matroids of dimension defined over the
-element field by points with at most two nonzero coordinates
each.Comment: This version adds Theorem
Hedonic Seat Arrangement Problems
In this paper, we study a variant of hedonic games, called \textsc{Seat
Arrangement}. The model is defined by a bijection from agents with preferences
to vertices in a graph. The utility of an agent depends on the neighbors
assigned in the graph. More precisely, it is the sum over all neighbors of the
preferences that the agent has towards the agent assigned to the neighbor. We
first consider the price of stability and fairness for different classes of
preferences. In particular, we show that there is an instance such that the
price of fairness ({\sf PoF}) is unbounded in general. Moreover, we show an
upper bound and an almost tight lower bound
of {\sf PoF}, where is the average degree of an input graph.
Then we investigate the computational complexity of problems to find certain
``good'' seat arrangements, say \textsc{Maximum Welfare Arrangement},
\textsc{Maximin Utility Arrangement}, \textsc{Stable Arrangement}, and
\textsc{Envy-free Arrangement}. We give dichotomies of computational complexity
of four \textsc{Seat Arrangement} problems from the perspective of the maximum
order of connected components in an input graph. For the parameterized
complexity, \textsc{Maximum Welfare Arrangement} can be solved in time
, while it cannot be solved in time
under ETH, where is the vertex cover number of an input graph.
Moreover, we show that \textsc{Maximin Utility Arrangement} and
\textsc{Envy-free Arrangement} are weakly NP-hard even on graphs of bounded
vertex cover number. Finally, we prove that determining whether a stable
arrangement can be obtained from a given arrangement by swaps is W[1]-hard
when parameterized by , whereas it can be solved in time
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
Computation of Hadwiger Number and Related Contraction Problems: Tight Lower Bounds
We prove that the Hadwiger number of an n-vertex graph G (the maximum size of a clique minor in G) cannot be computed in time n^o(n), unless the Exponential Time Hypothesis (ETH) fails. This resolves a well-known open question in the area of exact exponential algorithms. The technique developed for resolving the Hadwiger number problem has a wider applicability. We use it to rule out the existence of n^o(n)-time algorithms (up to ETH) for a large class of computational problems concerning edge contractions in graphs