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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 Subgraph Isomorphism problem is of considerable importance in computer science. We examine the problem when the pattern graph H is of bounded treewidth, as occurs in a variety of applications. This problem has a well-known algorithm via color-cod
We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems are essential to data analysis and used for example in road networks and data visualization. A coreset
We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable in graphs of bounded treewidth. These schemes apply in all fractionally treewidth-
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
Given a graph $G=(V,E)$ and a positive integer $tgeq2$, the task in the vertex cover $P_t$ ($VCP_t$) problem is to find a minimum subset of vertices $Fsubseteq V$ such that every path of order $t$ in $G$ contains at least one vertex from $F$. The $VC