An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs

In this paper, we consider the task of computing an independent set of maximum weight in a given $d$-claw free graph $G=(V,E)$ equipped with a positive weight function $w:V\rightarrow\mathbb{R}^+$. In doing so, $d\geq 2$ is considered a constant. The previously best known approximation algorithm for this problem is the local improvement algorithm SquareImp proposed by Berman. It achieves a performance ratio of $\frac{d}{2}+\epsilon$ in time $\mathcal{O}(|V(G)|^{d+1}\cdot(|V(G)|+|E(G)|)\cdot (d-1)^2\cdot \left(\frac{d}{2\epsilon}+1\right)^2)$ for any $\epsilon>0$, which has remained unimproved for the last twenty years. By considering a broader class of local improvements, we obtain an approximation ratio of $\frac{d}{2}-\frac{1}{63,700,992}+\epsilon$ for any $\epsilon>0$ at the cost of an additional factor of $\mathcal{O}(|V(G)|^{(d-1)^2})$ in the running time. In particular, our result implies a polynomial time $\frac{d}{2}$-approximation algorithm. Furthermore, the well-known reduction from the weighted $k$-Set Packing Problem to the Maximum Weight Independent Set Problem in $k+1$-claw free graphs provides a $\frac{k+1}{2}-\frac{1}{63,700,992}+\epsilon$-approximation algorithm for the weighted $k$-Set Packing Problem for any $\epsilon>0$. This improves on the previously best known approximation guarantee of $\frac{k+1}{2}+\epsilon$ originating from the result of Berman.Comment: full version of the paper "An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs" published in the proceedings of STACS 2021, 30 pages, 4 figure

The Pareto Cover Problem

We introduce the problem of finding a set B of k points in [0,1]? such that the expected cost of the cheapest point in B that dominates a random point from [0,1]? is minimized. We study the case where the coordinates of the random points are independently distributed and the cost function is linear. This problem arises naturally in various application areas where customers\u27 requests are satisfied based on predefined products, each corresponding to a subset of features. We show that the problem is NP-hard already for k = 2 when each coordinate is drawn from {0,1}, and obtain an FPTAS for general fixed k under mild assumptions on the distributions

Improved guarantees for the a priori TSP

We revisit the a priori TSP (with independent activation) and prove stronger approximation guarantees than were previously known. In the a priori TSP, we are given a metric space $(V,c)$ and an activation probability $p(v)$ for each customer $v\in V$. We ask for a TSP tour $T$ for $V$ that minimizes the expected length after cutting $T$ short by skipping the inactive customers. All known approximation algorithms select a nonempty subset $S$ of the customers and construct a master route solution, consisting of a TSP tour for $S$ and two edges connecting every customer $v\in V\setminus S$ to a nearest customer in $S$. We address the following questions. If we randomly sample the subset $S$, what should be the sampling probabilities? How much worse than the optimum can the best master route solution be? The answers to these questions (we provide almost matching lower and upper bounds) lead to improved approximation guarantees: less than 3.1 with randomized sampling, and less than 5.9 with a deterministic polynomial-time algorithm.Comment: 39 pages, 6 figures, extended abstract to appear in the proceedings of ISAAC 202

The 22-33-Set Packing problem and a 43\frac{4}{3}-approximation for the Maximum Leaf Spanning Arborescence problem in rooted dags

The weighted $3$-Set Packing problem is defined as follows: As input, we are given a collection $\mathcal{S}$ of sets, each of cardinality at most $3$ and equipped with a positive weight. The task is to find a disjoint sub-collection of maximum total weight. Already the special case of unit weights is known to be NP-hard, and the state-of-the-art are $\frac{4}{3}+\epsilon$-approximations by Cygan and F\"urer and Yu. In this paper, we study the $2$-$3$-Set Packing problem, a generalization of the unweighted $3$-Set Packing problem, where our set collection may contain sets of cardinality $3$ and weight $2$, as well as sets of cardinality $2$ and weight $1$. Building upon the state-of-the-art works in the unit weight setting, we manage to provide a $\frac{4}{3}+\epsilon$-approximation also for the more general $2$-$3$-Set Packing problem. We believe that this result can be a good starting point to identify classes of weight functions to which the techniques used for unit weights can be generalized. Using a reduction by Fernandes and Lintzmayer, our result further implies a $\frac{4}{3}+\epsilon$-approximation for the Maximum Leaf Spanning Arborescence problem (MLSA) in rooted directed acyclic graphs, improving on the previously known $\frac{7}{5}$-approximation by Fernandes and Lintzmayer. By exploiting additional structural properties of the instance constructed in their reduction, we can further get the approximation guarantee for the MLSA down to $\frac{4}{3}$. The MLSA has applications in broadcasting where a message needs to be transferred from a source node to all other nodes along the arcs of an arborescence in a given network.Comment: 49 pages, 10 figure