4 research outputs found
Improved FPT Algorithms for Deletion to Forest-Like Structures
The Feedback Vertex Set problem is undoubtedly one of the most well-studied problems in Parameterized Complexity. In this problem, given an undirected graph G and a non-negative integer k, the objective is to test whether there exists a subset S ? V(G) of size at most k such that G-S is a forest. After a long line of improvement, recently, Li and Nederlof [SODA, 2020] designed a randomized algorithm for the problem running in time ?^?(2.7^k). In the Parameterized Complexity literature, several problems around Feedback Vertex Set have been studied. Some of these include Independent Feedback Vertex Set (where the set S should be an independent set in G), Almost Forest Deletion and Pseudoforest Deletion. In Pseudoforest Deletion, each connected component in G-S has at most one cycle in it. However, in Almost Forest Deletion, the input is a graph G and non-negative integers k,? ? ?, and the objective is to test whether there exists a vertex subset S of size at most k, such that G-S is ? edges away from a forest. In this paper, using the methodology of Li and Nederlof [SODA, 2020], we obtain the current fastest algorithms for all these problems. In particular we obtain following randomized algorithms.
1) Independent Feedback Vertex Set can be solved in time ?^?(2.7^k).
2) Pseudo Forest Deletion can be solved in time ?^?(2.85^k).
3) Almost Forest Deletion can be solved in ?^?(min{2.85^k ? 8.54^?, 2.7^k ? 36.61^?, 3^k ? 1.78^?})
Tight Approximation Algorithms for Two Dimensional Guillotine Strip Packing
In the Strip Packing problem (SP), we are given a vertical half-strip
and a set of axis-aligned rectangles of width at
most . The goal is to find a non-overlapping packing of all rectangles into
the strip such that the height of the packing is minimized. A well-studied and
frequently used practical constraint is to allow only those packings that are
guillotine separable, i.e., every rectangle in the packing can be obtained by
recursively applying a sequence of edge-to-edge axis-parallel cuts (guillotine
cuts) that do not intersect any item of the solution. In this paper, we study
approximation algorithms for the Guillotine Strip Packing problem (GSP), i.e.,
the Strip Packing problem where we require additionally that the packing needs
to be guillotine separable. This problem generalizes the classical Bin Packing
problem and also makespan minimization on identical machines, and thus it is
already strongly NP-hard. Moreover, due to a reduction from the Partition
problem, it is NP-hard to obtain a polynomial-time
-approximation algorithm for GSP for any
(exactly as Strip Packing). We provide a matching polynomial time
-approximation algorithm for GSP. Furthermore, we present a
pseudo-polynomial time -approximation algorithm for GSP. This
is surprising as it is NP-hard to obtain a -approximation
algorithm for (general) Strip Packing in pseudo-polynomial time. Thus, our
results essentially settle the approximability of GSP for both the polynomial
and the pseudo-polynomial settings.Comment: 32 pages, 9 figure