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Register a new userAn optimal XP algorithm for Hamiltonian cycle on graphs of bounded clique-width
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O-Joung Kwon
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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In this paper, we prove that, given a clique-width $k$-expression of an $n$-vertex graph, textsc{Hamiltonian Cycle} can be solved in time $n^{mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{mathcal{O}(k^2)}$ by Espelage et al. (WG 2001), and it also matches with the lower bound result by Fomin et al. that, unless the Exponential Time Hypothesis fails, there is no algorithm running in time $n^{o(k)}$ (SIAM. J. Computing 2014). We present a technique of representative sets using two-edge colored multigraphs on $k$ vertices. The essential idea is that, for a two-edge colored multigraph, the existence of an Eulerian trail that uses edges with different colors alternately can be determined by two information: the number of colored edges incident with each vertex, and the connectedness of the multigraph. With this idea, we avoid the bottleneck of the naive algorithm, which stores all the possible multigraphs on $k$ vertices with at most $n$ edges.
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The maximum/minimum bisection problems are, given an edge-weighted graph, to find a bipartition of the vertex set into two sets whose sizes differ by at most one, such that the total weight of edges between the two sets is maximized/minimized. Although these two problems are known to be NP-hard, there is an efficient algorithm for bounded-treewidth graphs. In particular, Jansen et al. (SIAM J. Comput. 2005) gave an $O(2^tn^3)$-time algorithm when given a tree decomposition of width $t$ of the input graph, where $n$ is the number of vertices of the input graph. Eiben et al. (ESA 2019) improved the dependency of $n$ in the running time by giving an $O(8^tt^5n^2log n)$-time algorithm. Moreover, they showed that there is no $O(n^{2-varepsilon})$-time algorithm for trees under some reasonable complexity assumption. In this paper, we show an $O(2^t(tn)^2)$-time algorithm for both problems, which is asymptotically tight to their conditional lower bound. We also show that the exponential dependency of the treewidth is asymptotically optimal under the Strong Exponential Time Hypothesis. Finally, we discuss the (in)tractability of both problems with respect to special graph classes.
The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time $2^{O(rw^3)}cdot n^{4}$, $2^{O(q^2log(q))}cdot n^{4}$, and $n^{O(k^2)}$ where $rw$ is the rank-width, $q$ the $mathbb{Q}$-rank-width, and $k$ the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica, 2019) concerning an XP algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, $k$-Polygon, Dilworth-$k$ and Co-$k$-Degenerate graphs for fixed $k$; and also on Leaf Power graphs if a leaf root is given as input, on $H$-Graphs for fixed $H$ if an $H$-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.
By a well known result the treewidth of k-outerplanar graphs is at most 3k-1. This paper gives, besides a rigorous proof of this fact, an algorithmic implementation of the proof, i.e. it is shown that, given a k-outerplanar graph G, a tree decomposition of G of width at most 3k-1 can be found in O(kn) time and space. Similarly, a branch decomposition of a k-outerplanar graph of width at most 2k+1 can be also obtained in O(kn) time, the algorithm for which is also analyzed.
For a clustered graph, i.e, a graph whose vertex set is recursively partitioned into clusters, the C-Planarity Testing problem asks whether it is possible to find a planar embedding of the graph and a representation of each cluster as a region homeomorphic to a closed disk such that 1. the subgraph induced by each cluster is drawn in the interior of the corresponding disk, 2. each edge intersects any disk at most once, and 3. the nesting between clusters is reflected by the representation, i.e., child clusters are properly contained in their parent cluster. The computational complexity of this problem, whose study has been central to the theory of graph visualization since its introduction in 1995 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. ESA95], has only been recently settled [Radoslav Fulek and Csaba D. Toth. Atomic Embeddability, Clustered Planarity, and Thickenability. To appear at SODA20]. Before such a breakthrough, the complexity question was still unsolved even when the graph has a prescribed planar embedding, i.e, for embedded clustered graphs. We show that the C-Planarity Testing problem admits a single-exponential single-parameter FPT algorithm for embedded clustered graphs, when parameterized by the carving-width of the dual graph of the input. This is the first FPT algorithm for this long-standing open problem with respect to a single notable graph-width parameter. Moreover, in the general case, the polynomial dependency of our FPT algorithm is smaller than the one of the algorithm by Fulek and Toth. To further strengthen the relevance of this result, we show that the C-Planarity Testing problem retains its computational complexity when parameterized by several other graph-width parameters, which may potentially lead to faster algorithms.
An $(m, n)$-colored mixed graph is a graph having arcs of $m$ different colors and edges of $n$ different colors. A graph homomorphism of an $(m, n$)-colored mixed graph $G$ to an $(m, n)$-colored mixed graph $H$ is a vertex mapping such that if $uv$ is an arc (edge) of color $c$ in $G$, then $f(u)f(v)$ is also an arc (edge) of color $c$. The ($m, n)$-colored mixed chromatic number of an $(m, n)$-colored mixed graph $G$, introduced by Nev{s}etv{r}il and Raspaud [J. Combin. Theory Ser. B 2000] is the order (number of vertices) of the smallest homomorphic image of $G$. Later Bensmail, Duffy and Sen [Graphs Combin. 2017] introduced another parameter related to the $(m, n)$-colored mixed chromatic number, namely, the $(m, n)$-relative clique number as the maximum cardinality of a vertex subset which, pairwise, must have distinct images with respect to any colored homomorphism. In this article, we study the $(m, n$)-relative clique number for the family of subcubic graphs, graphs with maximum degree $Delta$, planar graphs and triangle-free planar graphs and provide new improved bounds in each of the cases. In particular, for subcubic graphs we provide exact value of the parameter.
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