296 research outputs found
Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth
We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small edge separators.
Let p,q ? ? such that p is a prime and q ? 3. We show:
- If p divides q-1, there is a (q-1)^{ctw}n^{O(1)} time algorithm for counting list q-colorings modulo p of n-vertex graphs of cutwidth ctw. Furthermore, there is no ? > 0 for which there is a (q-1-?)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming the Strong Exponential Time Hypothesis (SETH).
- If p does not divide q-1, there is no ? > 0 for which there exists a (q-?)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming SETH. The lower bounds are in stark contrast with the existing 2^{ctw}n^{O(1)} time algorithm to compute the chromatic number of a graph by Jansen and Nederlof [Theor. Comput. Sci.\u2718].
Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no ? > 0 for which there is an algorithm that, given a graph G and a cutwidth ordering of cutwidth ctw, counts the number of spanning connected edge sets of G modulo p in time (p - ?)^{ctw} n^{O(1)}, assuming SETH. We also give an algorithm with matching running time for this problem.
Before our work, even for the treewidth parameterization, the best conditional lower bound by Dell et al. [ACM Trans. Algorithms\u2714] only excluded 2^{o(tw)}n^{O(1)} time algorithms for this problem.
Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain "compatibility matrix" in a non-trivial way
Weighted counting of solutions to sparse systems of equations
Given complex numbers , we define the weight of a
set of 0-1 vectors as the sum of over all
vectors in . We present an algorithm, which for a set
defined by a system of homogeneous linear equations with at most
variables per equation and at most equations per variable, computes
within relative error in time
provided for an absolute constant and all . A similar algorithm is constructed for computing
the weight of a linear code over . Applications include counting
weighted perfect matchings in hypergraphs, counting weighted graph
homomorphisms, computing weight enumerators of linear codes with sparse code
generating matrices, and computing the partition functions of the ferromagnetic
Potts model at low temperatures and of the hard-core model at high fugacity on
biregular bipartite graphs.Comment: The exposition is improved, a couple of inaccuracies correcte
Data Reduction for Graph Coloring Problems
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.Comment: Author-accepted manuscript of the article that will appear in the FCT
2011 special issue of Information & Computatio
Using contracted solution graphs for solving reconfiguration problems.
We introduce a dynamic programming method for solving reconfiguration problems, based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. As an example, we consider a well-studied problem: given two k-colorings alpha and beta of a graph G, can alpha be modified into beta by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k-2)-connected chordal graphs
Using Contracted Solution Graphs for Solving Reconfiguration Problems
We introduce a dynamic programming method for solving reconfiguration problems, based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. As an example, we consider a well-studied problem: given two k-colorings alpha and beta of a graph G, can alpha be modified into beta by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k-2)-connected chordal graphs
Using contracted solution graphs for solving reconfiguration problems
We introduce a dynamic programming method for solving reconfiguration problems, based on contracted solution graphs, which are obtained from solution graphs by performing an appropriate series of edge contractions that decrease the graph size without losing any critical information needed to solve the reconfiguration problem under consideration. As an example, we consider a well-studied problem: given two k-colorings alpha and beta of a graph G, can alpha be modified into beta by recoloring one vertex of G at a time, while maintaining a k-coloring throughout? By applying our method in combination with a thorough exploitation of the graph structure we obtain a polynomial-time algorithm for (k-2)-connected chordal graphs
Reconfiguration of Dominating Sets
We explore a reconfiguration version of the dominating set problem, where a
dominating set in a graph is a set of vertices such that each vertex is
either in or has a neighbour in . In a reconfiguration problem, the goal
is to determine whether there exists a sequence of feasible solutions
connecting given feasible solutions and such that each pair of
consecutive solutions is adjacent according to a specified adjacency relation.
Two dominating sets are adjacent if one can be formed from the other by the
addition or deletion of a single vertex.
For various values of , we consider properties of , the graph
consisting of a vertex for each dominating set of size at most and edges
specified by the adjacency relation. Addressing an open question posed by Haas
and Seyffarth, we demonstrate that is not necessarily
connected, for the maximum cardinality of a minimal dominating set
in . The result holds even when graphs are constrained to be planar, of
bounded tree-width, or -partite for . Moreover, we construct an
infinite family of graphs such that has exponential
diameter, for the minimum size of a dominating set. On the positive
side, we show that is connected and of linear diameter for any
graph on vertices having at least independent edges.Comment: 12 pages, 4 figure
Tight Bounds for Counting Colorings and Connected Edge Sets Parameterized by Cutwidth
We study the fine-grained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. While decompositions of small treewidth decompose the graph with small vertex separators, decompositions with small cutwidth decompose the graph with small edge separators. Let p,q ∈ ℕ such that p is a prime and q ≥ 3. We show: - If p divides q-1, there is a (q-1)^{ctw}n^{O(1)} time algorithm for counting list q-colorings modulo p of n-vertex graphs of cutwidth ctw. Furthermore, there is no ε > 0 for which there is a (q-1-ε)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming the Strong Exponential Time Hypothesis (SETH). - If p does not divide q-1, there is no ε > 0 for which there exists a (q-ε)^{ctw} n^{O(1)} time algorithm that counts the number of list q-colorings modulo p of n-vertex graphs of cutwidth ctw, assuming SETH. The lower bounds are in stark contrast with the existing 2^{ctw}n^{O(1)} time algorithm to compute the chromatic number of a graph by Jansen and Nederlof [Theor. Comput. Sci.'18]. Furthermore, by building upon the above lower bounds, we obtain the following lower bound for counting connected spanning edge sets: there is no ε > 0 for which there is an algorithm that, given a graph G and a cutwidth ordering of cutwidth ctw, counts the number of spanning connected edge sets of G modulo p in time (p - ε)^{ctw} n^{O(1)}, assuming SETH. We also give an algorithm with matching running time for this problem. Before our work, even for the treewidth parameterization, the best conditional lower bound by Dell et al. [ACM Trans. Algorithms'14] only excluded 2^{o(tw)}n^{O(1)} time algorithms for this problem. Both our algorithms and lower bounds employ use of the matrix rank method, by relating the complexity of the problem to the rank of a certain "compatibility matrix" in a non-trivial way
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