A Polynomial-time Algorithm for Outerplanar Diameter Improvement

The Outerplanar Diameter Improvement problem asks, given a graph $G$ and an integer $D$, whether it is possible to add edges to $G$ in a way that the resulting graph is outerplanar and has diameter at most $D$. We provide a dynamic programming algorithm that solves this problem in polynomial time. Outerplanar Diameter Improvement demonstrates several structural analogues to the celebrated and challenging Planar Diameter Improvement problem, where the resulting graph should, instead, be planar. The complexity status of this latter problem is open.Comment: 24 page

Improved Approximation Algorithms for Steiner Connectivity Augmentation Problems

The Weighted Connectivity Augmentation Problem is the problem of augmenting the edge-connectivity of a given graph by adding links of minimum total cost. This work focuses on connectivity augmentation problems in the Steiner setting, where we are not interested in the connectivity between all nodes of the graph, but only the connectivity between a specified subset of terminals. We consider two related settings. In the Steiner Augmentation of a Graph problem ($k$-SAG), we are given a $k$-edge-connected subgraph $H$ of a graph $G$. The goal is to augment $H$ by including links and nodes from $G$ of minimum cost so that the edge-connectivity between nodes of $H$ increases by 1. In the Steiner Connectivity Augmentation Problem ($k$-SCAP), we are given a Steiner $k$-edge-connected graph connecting terminals $R$, and we seek to add links of minimum cost to create a Steiner $(k+1)$-edge-connected graph for $R$. Note that $k$-SAG is a special case of $k$-SCAP. All of the above problems can be approximated to within a factor of 2 using e.g. Jain's iterative rounding algorithm for Survivable Network Design. In this work, we leverage the framework of Traub and Zenklusen to give a $(1 + \ln{2} +\varepsilon)$-approximation for the Steiner Ring Augmentation Problem (SRAP): given a cycle $H = (V(H),E)$ embedded in a larger graph $G = (V, E \cup L)$ and a subset of terminals $R \subseteq V(H)$, choose a subset of links $S \subseteq L$ of minimum cost so that $(V, E \cup S)$ has 3 pairwise edge-disjoint paths between every pair of terminals. We show this yields a polynomial time algorithm with approximation ratio $(1 + \ln{2} + \varepsilon)$ for $2$-SCAP. We obtain an improved approximation guarantee of $(1.5+\varepsilon)$ for SRAP in the case that $R = V(H)$, which yields a $(1.5+\varepsilon)$-approximation for $k$-SAG for any $k$

Combinatorial Optimization

Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

Exact Lagrangian immersions with one double point revisited

We study exact Lagrangian immersions with one double point of a closed orientable manifold K into n-complex-dimensional Euclidean space. Our main result is that if the Maslov grading of the double point does not equal 1 then K is homotopy equivalent to the sphere, and if, in addition, the Lagrangian Gauss map of the immersion is stably homotopic to that of the Whitney immersion, then K bounds a parallelizable (n+1)-manifold. The hypothesis on the Gauss map always holds when n=2k or when n=8k-1. The argument studies a filling of K obtained from solutions to perturbed Cauchy-Riemann equations with boundary on the image f(K) of the immersion. This leads to a new and simplified proof of some of the main results of arXiv:1111.5932, which treated Lagrangian immersions in the case n=2k by applying similar techniques to a Lagrange surgery of the immersion, as well as to an extension of these results to the odd-dimensional case.Comment: 39 pages, 2 figures. Version 2: A lengthy appendix now contains a detailed and largely self-contained proof of the existence of a C1-smooth structure on a certain compactified moduli space of Floer disks. The rest of the text is accordingly somewhat re-organised. Version 3: minor further change

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Combinatorial Optimization

Combinatorial Optimization is a very active field that benefits from bringing together ideas from different areas, e.g., graph theory and combinatorics, matroids and submodularity, connectivity and network flows, approximation algorithms and mathematical programming, discrete and computational geometry, discrete and continuous problems, algebraic and geometric methods, and applications. We continued the long tradition of triannual Oberwolfach workshops, bringing together the best researchers from the above areas, discovering new connections, and establishing new and deepening existing international collaborations