Partitions and Coverings of Trees by Bounded-Degree Subtrees

This paper addresses the following questions for a given tree $T$ and integer $d\geq2$: (1) What is the minimum number of degree-$d$ subtrees that partition $E(T)$? (2) What is the minimum number of degree-$d$ subtrees that cover $E(T)$? We answer the first question by providing an explicit formula for the minimum number of subtrees, and we describe a linear time algorithm that finds the corresponding partition. For the second question, we present a polynomial time algorithm that computes a minimum covering. We then establish a tight bound on the number of subtrees in coverings of trees with given maximum degree and pathwidth. Our results show that pathwidth is the right parameter to consider when studying coverings of trees by degree-3 subtrees. We briefly consider coverings of general graphs by connected subgraphs of bounded degree

TREEWIDTH and PATHWIDTH parameterized by vertex cover

After the number of vertices, Vertex Cover is the largest of the classical graph parameters and has more and more frequently been used as a separate parameter in parameterized problems, including problems that are not directly related to the Vertex Cover. Here we consider the TREEWIDTH and PATHWIDTH problems parameterized by k, the size of a minimum vertex cover of the input graph. We show that the PATHWIDTH and TREEWIDTH can be computed in O*(3^k) time. This complements recent polynomial kernel results for TREEWIDTH and PATHWIDTH parameterized by the Vertex Cover

On space efficiency of algorithms working on structural decompositions of graphs

Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition's width. Following the work of Allender et al. [Theory of Computing, '14], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely related to a conjecture that the Longest Common Subsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we complete the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth. We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition's depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non-deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new version is augmented with a space-efficient algorithm for Dominating Set using the Chinese remainder theore

An FPT algorithm and a polynomial kernel for Linear Rankwidth-1 Vertex Deletion

Linear rankwidth is a linearized variant of rankwidth, introduced by Oum and Seymour [Approximating clique-width and branch-width. J. Combin. Theory Ser. B, 96(4):514--528, 2006]. Motivated from recent development on graph modification problems regarding classes of graphs of bounded treewidth or pathwidth, we study the Linear Rankwidth-1 Vertex Deletion problem (shortly, LRW1-Vertex Deletion). In the LRW1-Vertex Deletion problem, given an $n$-vertex graph $G$ and a positive integer $k$, we want to decide whether there is a set of at most $k$ vertices whose removal turns $G$ into a graph of linear rankwidth at most $1$ and find such a vertex set if one exists. While the meta-theorem of Courcelle, Makowsky, and Rotics implies that LRW1-Vertex Deletion can be solved in time $f(k)\cdot n^3$ for some function $f$, it is not clear whether this problem allows a running time with a modest exponential function. We first establish that LRW1-Vertex Deletion can be solved in time $8^k\cdot n^{\mathcal{O}(1)}$. The major obstacle to this end is how to handle a long induced cycle as an obstruction. To fix this issue, we define necklace graphs and investigate their structural properties. Later, we reduce the polynomial factor by refining the trivial branching step based on a cliquewidth expression of a graph, and obtain an algorithm that runs in time $2^{\mathcal{O}(k)}\cdot n^4$. We also prove that the running time cannot be improved to $2^{o(k)}\cdot n^{\mathcal{O}(1)}$ under the Exponential Time Hypothesis assumption. Lastly, we show that the LRW1-Vertex Deletion problem admits a polynomial kernel.Comment: 29 pages, 9 figures, An extended abstract appeared in IPEC201

Computing cutwidth and pathwidth of semi-complete digraphs via degree orderings

The notions of cutwidth and pathwidth of digraphs play a central role in the containment theory for tournaments, or more generally semi-complete digraphs, developed in a recent series of papers by Chudnovsky, Fradkin, Kim, Scott, and Seymour (Maria Chudnovsky, Alexandra Fradkin, and Paul Seymour, 2012; Maria Chudnovsky, Alex Scott, and Paul Seymour, 2011; Maria Chudnovsky and Paul D. Seymour, 2011; Alexandra Fradkin and Paul Seymour, 2010; Alexandra Fradkin and Paul Seymour, 2011; Ilhee Kim and Paul Seymour, 2012). In this work we introduce a new approach to computing these width measures on semi-complete digraphs, via degree orderings. Using the new technique we are able to reprove the main results of (Maria Chudnovsky, Alexandra Fradkin, and Paul Seymour, 2012; Alexandra Fradkin and Paul Seymour, 2011) in a unified and significantly simplified way, as well as obtain new results. First, we present polynomial-time approximation algorithms for both cutwidth and pathwidth, faster and simpler than the previously known ones; the most significant improvement is in case of pathwidth, where instead of previously known O(OPT)-approximation in fixed-parameter tractable time (Fedor V. Fomin and Michal Pilipczuk, 2013) we obtain a constant-factor approximation in polynomial time. Secondly, by exploiting the new set of obstacles for cutwidth and pathwidth, we show that topological containment and immersion in semi-complete digraphs can be tested in single-exponential fixed-parameter tractable time. Finally, we present how the new approach can be used to obtain exact fixed-parameter tractable algorithms for cutwidth and pathwidth, with single-exponential running time dependency on the optimal width.publishedVersio

A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank

For even $k$, the matchings connectivity matrix $\mathbf{M}_k$ encodes which pairs of perfect matchings on $k$ vertices form a single cycle. Cygan et al. (STOC 2013) showed that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_2$ is $\Theta(\sqrt 2^k)$ and used this to give an $O^*((2+\sqrt{2})^{\mathsf{pw}})$ time algorithm for counting Hamiltonian cycles modulo $2$ on graphs of pathwidth $\mathsf{pw}$. The same authors complemented their algorithm by an essentially tight lower bound under the Strong Exponential Time Hypothesis (SETH). This bound crucially relied on a large permutation submatrix within $\mathbf{M}_k$, which enabled a "pattern propagation" commonly used in previous related lower bounds, as initiated by Lokshtanov et al. (SODA 2011). We present a new technique for a similar pattern propagation when only a black-box lower bound on the asymptotic rank of $\mathbf{M}_k$ is given; no stronger structural insights such as the existence of large permutation submatrices in $\mathbf{M}_k$ are needed. Given appropriate rank bounds, our technique yields lower bounds for counting Hamiltonian cycles (also modulo fixed primes $p$) parameterized by pathwidth. To apply this technique, we prove that the rank of $\mathbf{M}_k$ over the rationals is $4^k / \mathrm{poly}(k)$. We also show that the rank of $\mathbf{M}_k$ over $\mathbb{Z}_p$ is $\Omega(1.97^k)$ for any prime $p\neq 2$ and even $\Omega(2.15^k)$ for some primes. As a consequence, we obtain that Hamiltonian cycles cannot be counted in time $O^*((6-\epsilon)^{\mathsf{pw}})$ for any $\epsilon>0$ unless SETH fails. This bound is tight due to a $O^*(6^{\mathsf{pw}})$ time algorithm by Bodlaender et al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be counted modulo primes $p\neq 2$ in time $O^*(3.97^\mathsf{pw})$, indicating that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in SODA 201