In the Strip Packing (SP) problem, we are given a vertical half-strip \([0,W]\times[0,\infty)\) and a set of \( n \) axis-aligned rectangles of width at most \( W \) . 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 article, we study approximation algorithms for the Guillotine Strip Packing (GSP) problem, i.e., the SP 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 \(\mathsf{NP}\) -hard. Moreover, due to a reduction from the Partition problem, it is \(\mathsf{NP}\) -hard to obtain a polynomial-time \((3/2-\varepsilon)\) -approximation algorithm for GSP for any \(\varepsilon > 0\) (exactly as SP ). We provide a matching polynomial time \((3/2+\varepsilon)\) -approximation algorithm for GSP. Furthermore, we present a pseudo-polynomial time \((1+\varepsilon)\) -approximation algorithm for GSP. This is surprising as it is \(\mathsf{NP}\) -hard to obtain a \((5/4-\varepsilon)\) -approximation algorithm for (general) SP in pseudo-polynomial time. Thus, our results essentially settle the approximability of GSP for both the polynomial and the pseudo-polynomial settings.

Set cover and hitting set are fundamental problems in combinatorial optimization which are well-studied in the offline, online, and dynamic settings. We study the geometric versions of these problems and present new online and dynamic algorithms for them. In the online version of set cover (resp. hitting set), $m$ sets (resp.~$n$ points) are give $n$ points (resp.~$m$ sets) arrive online, one-by-one. In the dynamic versions, points (resp. sets) can arrive as well as depart. Our goal is to maintain a set cover (resp. hitting set), minimizing the size of the computed solution. For online set cover for (axis-parallel) squares of arbitrary sizes, we present a tight $O(\log n)$-competitive algorithm. In the same setting for hitting set, we provide a tight $O(\log N)$-competitive algorithm, assuming that all points have integral coordinates in $[0,N)^{2}$. No online algorithm had been known for either of these settings, not even for unit squares (apart from the known online algorithms for arbitrary set systems). For both dynamic set cover and hitting set with $d$-dimensional hyperrectangles, we obtain $(\log m)^{O(d)}$-approximation algorithms with $(\log m)^{O(d)}$ worst-case update time. This partially answers an open question posed by Chan et al. [SODA'22]. Previously, no dynamic algorithms with polylogarithmic update time were known even in the setting of squares (for either of these problems). Our main technical contributions are an \emph{extended quad-tree }approach and a \emph{frequency reduction} technique that reduces geometric set cover instances to instances of general set cover with bounded frequency.

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 $S\subseteq V(G)$ 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 $\mathcal{O}^{\star}(2.7^k)$. 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 $k,\ell \in \mathbb{N}$, and the objective is to test whether there exists a vertex subset $S$ of size at most $k$, such that $G-S$ is $\ell$ 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 $\mathcal{O}^{\star}(2.7^k)$. 2) Pseudo Forest Deletion can be solved in time $\mathcal{O}^{\star}(2.85^k)$. 3) Almost Forest Deletion can be solved in $\mathcal{O}^{\star}(\min\{2.85^k \cdot 8.54^\ell,2.7^k \cdot 36.61^\ell,3^k \cdot 1.78^\ell\})$.