
Close relatives of Feedback Vertex Set without singleexponential algorithms parameterized by treewidth
The Cut Count technique and the rankbased approach have lead to sin...
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Detecting and Counting Small Patterns in Planar Graphs in Subexponential Parameterized Time
We present an algorithm that takes as input an nvertex planar graph G a...
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A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even k, the matchings connectivity matrix M_k encodes which pairs of...
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On exploiting degeneracy to count subgraphs
Motivated by practical applications, we study the complexity of counting...
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Fast Witness Counting
We study the witnesscounting problem: given a set of vectors V in the d...
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The FineGrained Complexity of Computing the Tutte Polynomial of a Linear Matroid
We show that computing the Tutte polynomial of a linear matroid of dimen...
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Counting and localizing defective nodes by Boolean network tomography
Identifying defective items in larger sets is a main problem with many a...
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Tight bounds for counting colorings and connected edge sets parameterized by cutwidth
We study the finegrained complexity of counting the number of colorings and connected spanning edge sets parameterized by the cutwidth and treewidth of the graph. Let p,q ∈ℕ such that p is a prime and q ≥ 3. We show:  If p divides q1, there is a (q1)^ctwn^O(1) time algorithm for counting list qcolorings modulo p of nvertex graphs of cutwidth ctw. Furthermore, no algorithm can count the number of distinct qcolorings modulo p in time (q1ε)^ctw n^O(1) for some ε>0, assuming the Strong Exponential Time Hypothesis (SETH).  If p does not divide q1, no algorithm can count the number of distinct qcolorings modulo p in time (qε)^ctw n^O(1) for some ε>0, assuming SETH. The lower bounds are in stark contrast with the existing 2^ctwn^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.
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