Unit Interval Editing is Fixed-Parameter Tractable

Given a graph~$G$ and integers $k_1$, $k_2$, and~$k_3$, the unit interval editing problem asks whether $G$ can be transformed into a unit interval graph by at most $k_1$ vertex deletions, $k_2$ edge deletions, and $k_3$ edge additions. We give an algorithm solving this problem in time $2^{O(k\log k)}\cdot (n+m)$, where $k := k_1 + k_2 + k_3$, and $n, m$ denote respectively the numbers of vertices and edges of $G$. Therefore, it is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm implies the fixed-parameter tractability of the unit interval edge deletion problem, for which we also present a more efficient algorithm running in time $O(4^k \cdot (n + m))$. Another result is an $O(6^k \cdot (n + m))$-time algorithm for the unit interval vertex deletion problem, significantly improving the algorithm of van 't Hof and Villanger, which runs in time $O(6^k \cdot n^6)$.Comment: An extended abstract of this paper has appeared in the proceedings of ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an appendix is provided for a brief overview of related graph classe

An approximation result for a periodic allocation problem

AbstractIn this paper we study a periodic allocation problem which is a generalization of the dynamic storage allocation problem to the case in which the arrival and departure time of each item is periodically repeated. These problems are equivalent to the interval coloring problem on weighted graphs in which each feasible solution corresponds to an acyclic orientation, and the solution value is equal to the length of the longest weighted path of the oriented graph. Optimal solutions correspond to acyclic orientations having the length of longest weighted path as small as possible. We prove that for the interval coloring problem on a class of circular arc graphs, and hence for a periodic allocation problem, there exists an approximation algorithm that finds a feasible solution whose value is at most two times the optimal