Makespan Scheduling of Unit Jobs with Precedence Constraints in O(1.995n)O(1.995^n) time

In a classical scheduling problem, we are given a set of $n$ jobs of unitlength along with precedence constraints and the goal is to find a schedule ofthese jobs on $m$ identical machines that minimizes the makespan. This problemis well-known to be NP-hard for an unbounded number of machines. Using standard3-field notation, it is known as $P|\text{prec}, p_j=1|C_{\max}$. We present an algorithm for this problem that runs in $O(1.995^n)$ time.Before our work, even for $m=3$ machines the best known algorithms ran in$O^\ast(2^n)$ time. In contrast, our algorithm works when the number ofmachines $m$ is unbounded. A crucial ingredient of our approach is an algorithmwith a runtime that is only single-exponential in the vertex cover of thecomparability graph of the precedence constraint graph. This heavily relies oninsights from a classical result by Dolev and Warmuth (Journal of Algorithms1984) for precedence graphs without long chains.<br

Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments

In the Maximum Weight Independent Set of Rectangles problem (MWISR) we aregiven a weighted set of $n$ axis-parallel rectangles in the plane. The task isto find a subset of pairwise non-overlapping rectangles with the maximumpossible total weight. This problem is NP-hard and the best-knownpolynomial-time approximation algorithm, due to by Chalermsook and Walczak(SODA 2021), achieves approximation factor $O(\log\log n )$. While in theunweighted setting, constant factor approximation algorithms are known, due toMitchell (FOCS 2021) and to G\'alvez et al. (SODA 2022), it remains open toextend these techniques to the weighted setting. In this paper, we consider MWISR through the lens of parameterizedapproximation. Grandoni et al. (ESA 2019) gave a $(1-\epsilon)$-approximationalgorithm with running time $k^{O(k/\epsilon^8)} n^{O(1/\epsilon^8)}$ time,where $k$ is the number of rectangles in an optimum solution. Unfortunately,their algorithm works only in the unweighted setting and they left it as anopen problem to give a parameterized approximation scheme in the weightedsetting. Our contribution is a partial answer to the open question of Grandoni et al.(ESA 2019). We give a parameterized approximation algorithm for MWISR thatgiven a parameter $k$, finds a set of non-overlapping rectangles of weight atleast $(1-\epsilon) \text{opt}_k$ in $2^{O(k \log(k/\epsilon))}n^{O(1/\epsilon)}$ time, where $\text{opt}_k$ is the maximum weight of asolution of cardinality at most $k$. Note that thus, our algorithm may return asolution consisting of more than $k$ rectangles. To complement this apparentweakness, we also propose a parameterized approximation scheme with runningtime $2^{O(k^2 \log(k/\epsilon))} n^{O(1)}$ that finds a solution withcardinality at most $k$ and total weight at least $(1-\epsilon)\text{opt}_k$for the special case of axis-parallel segments.<br

Bounding Generalized Coloring Numbers of Planar Graphs Using Coin Models

We study Koebe orderings of planar graphs: vertex orderings obtained bymodelling the graph as the intersection graph of pairwise internally-disjointdiscs in the plane, and ordering the vertices by non-increasing radii of theassociated discs. We prove that for every $d\in \mathbb{N}$, any such orderinghas $d$-admissibility bounded by $O(d/\ln d)$ and weak $d$-coloring numberbounded by $O(d^4 \ln d)$. This in particular shows that the $d$-admissibilityof planar graphs is bounded by $O(d/\ln d)$, which asymptotically matches aknown lower bound due to Dvo\v{r}\'ak and Siebertz.<br

A Gap-{ETH}-Tight Approximation Scheme for Euclidean {TSP}

We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running time $(1/\varepsilon)^{\mathcal{O}(1/\varepsilon^{d-1})}n \log n$ of the algorithm by Rao and Smith (STOC 1998). We also show that a $2^{o(1/\varepsilon^{d-1})}\text{poly}(n)$ algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree

Independence Number of Intersection Graphs of Axis-Parallel Segments

We prove that for any triangle-free intersection graph of $n$ axis-parallelsegments in the plane, the independence number $\alpha$ of this graph is atleast $\alpha \ge n/4 + \Omega(\sqrt{n})$. We complement this with aconstruction of a graph in this class satisfying $\alpha \le n/4 + c \sqrt{n}$for an absolute constant $c$, which demonstrates the optimality of our result.<br

Dynamic Data Structures for Parameterized String Problems

We revisit classic string problems considered in the area of parameterized complexity, and study them through the lens of dynamic data structures. That is, instead of asking for a static algorithm that solves the given instance efficiently, our goal is to design a data structure that efficiently maintains a solution, or reports a lack thereof, upon updates in the instance. We first consider the Closest String problem, for which we design randomized dynamic data structures with amortized update times $d^{\mathcal{O}(d)}$ and $|\Sigma|^{\mathcal{O}(d)}$, respectively, where $\Sigma$ is the alphabet and $d$ is the assumed bound on the maximum distance. These are obtained by combining known static approaches to Closest String with color-coding. Next, we note that from a result of Frandsen et al.~[J. ACM'97] one can easily infer a meta-theorem that provides dynamic data structures for parameterized string problems with worst-case update time of the form $\mathcal{O}(\log \log n)$, where $k$ is the parameter in question and $n$ is the length of the string. We showcase the utility of this meta-theorem by giving such data structures for problems Disjoint Factors and Edit Distance. We also give explicit data structures for these problems, with worst-case update times $\mathcal{O}(k2^{k}\log \log n)$ and $\mathcal{O}(k^2\log \log n)$, respectively. Finally, we discuss how a lower bound methodology introduced by Amarilli et al.~[ICALP'21] can be used to show that obtaining update time $\mathcal{O}(f(k))$ for Disjoint Factors and Edit Distance is unlikely already for a constant value of the parameter $k$.Comment: 28 page

Approximating APSP without Scaling: Equivalence of Approximate Min-Plus and Exact Min-Max

Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time $\widetilde{O}(\frac{n^\omega}{\varepsilon} \log{W})$, where $\omega \le 2.373$ is the exponent of matrix multiplication and $W$ denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor $\log W$ in the running time. In this paper, we study whether APSP and related problems admit approximation schemes avoiding the scaling technique. That is, the number of arithmetic operations should be independent of $W$; this is called strongly polynomial. Our main results are as follows. - We design approximation schemes in strongly polynomial time $O(\frac{n^\omega}{\varepsilon} \text{polylog}(\frac{n}{\varepsilon}))$ for APSP on undirected graphs as well as for the graph characteristics diameter, radius, median, minimum-weight triangle, and minimum-weight cycle on directed or undirected graphs. - For APSP on directed graphs we design an approximation scheme in strongly polynomial time $O(n^{\frac{\omega + 3}{2}} \varepsilon^{-1} \text{polylog}(\frac{n}{\varepsilon}))$. This is significantly faster than the best exact algorithm. - We explain why our approximation scheme for APSP on directed graphs has a worse exponent than $\omega$: Any improvement over our exponent $\frac{\omega + 3}{2}$ would improve the best known algorithm for Min-Max Product In fact, we prove that approximating directed APSP and exactly computing the Min-Max Product are equivalent