arXiv subjects

Felix Hommelsheim

Publications and source records attributed to Felix Hommelsheim.

At least 19 recordsLinked to original sources

arXiv2026

A Fast and Simple $(1+ε)$-Approximation for Minimum Spanning Trees in Doubling Metrics

The minimum spanning tree (MST) problem is one of the most basic optimization problems on metric spaces and graphs. We study the problem of computing a $(1+ε)$-approximation to the MST of an $n$-point metric space $(X, \mathbf{d})$ of doubling dimension $\mathrm{ddim}$. In doubling metrics, previous deterministic algorithms incur a running time with dependence $ε^{-O(\mathrm{ddim})}$. We give a deterministic algorithm that computes a $(1+ε)$-approximation to MST in time $2^{O(\mathrm{ddim})} n \bigl(\log n + ε^{-1} \log^4(1/ε)\bigr)$. For bounded doubling dimension, this improves the previous dependence on $ε$ from $ε^{-O(\mathrm{ddim})}$ to essentially linear in $ε^{-1}$. Moreover, as a special case, our result improves the previous best deterministic running time for bounded-dimensional Euclidean metrics due to Arya and Mount~[SODA'16] by almost a factor of $ε^{-1}$. We also show that, unlike in bounded-dimensional Euclidean spaces, MSTs in bounded doubling metrics can have arbitrarily large maximum degree, while every doubling metric nevertheless admits a $(1+ε)$-approximate MST of maximum degree $2^{O(\mathrm{ddim})}\log(1/ε)$.

cs.DS↗

arXiv2026

Approximation Algorithms for Discounted Graph Search with Norm Objectives

We introduce a unified framework for classical search and routing problems, including pathwise search, expanding search, the minimum spanning tree problem, and the traveling salesperson problem. The framework is based on two parameters. The first is a discount factor $α\in [0,1]$: the first traversal of an edge incurs its full cost, whereas each subsequent traversal incurs only an $α$-fraction of this cost. For a path starting at a designated root vertex, the $α$-latency of a vertex is the discounted cost accumulated until the vertex is first visited. The second parameter is a norm parameter $p\geq 1$. The objective is to find a root-starting path that visits all vertices and minimizes the $p$-norm of the resulting vector of $α$-latencies. The model interpolates between several well-studied objectives. For $p=1$ and $α=1$, it recovers pathwise search; for $p=1$ and $α=0$, it recovers expanding search. As $p$ tends to infinity, the objective converges to a makespan-type criterion. At the endpoints $α=1$ and $α=0$, this limiting objective corresponds to TSP-type and MST-type behavior, respectively. For $p=1$, we give polynomial-time constant-factor approximation algorithms for all $α\in[0,1]$, matching the best known guarantees for expanding search at $α=0$ and pathwise search at $α=1$. For general $p\geq 1$, we obtain a randomized constant-factor approximation algorithm and a derandomized pseudo-polynomial-time algorithm with the same guarantee.

cs.DS↗

arXiv2026

Recoverable Robust Optimization with Commitment

We propose a model for recoverable robust optimization with commitment. Given a combinatorial optimization problem and uncertainty about elements that may fail, we ask for a robust solution that, after the failing elements are revealed, can be augmented in a limited way. However, we commit to preserve the non-failing elements of the initial solution. We settle the computational complexity of such a robust counterpart of various classical polynomial-time solvable combinatorial optimization problems. We show, for the weighted matroid independent set problem, that an optimal solution to the nominal problem is also optimal for its robust counterpart. Indeed, matroids are provably the only structures with this strong property. Robust counterparts of other problems are \NP-hard such as the matching problem and the stable set problem, even in bipartite graphs. However, we establish polynomial-time algorithms for the robust counterparts of the unweighted stable set problem in bipartite graphs and the weighted stable set problem in interval graphs, also known as the interval scheduling problem.

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arXiv2026

A Complexity Dichotomy for Generalized Rainbow Matchings Based on Color Classes

Given an edge-colored graph, the Maximum Rainbow Matching problem asks for a maximum-cardinality matching of the graph that contains at most one edge from each color. We provide the following complexity dichotomy for this problem based on the structure of the color classes: Maximum Rainbow Matching admits a polynomial-time algorithm if almost every color class is a complete multipartite graph and it is NP-hard otherwise. To prove the NP-hardness-part of the dichotomy, we first show that the problem remains NP-hard even if every color class is a subgraph on four vertices that is either a matching of size two, a path on four vertices or a paw. We then leverage this result to all color classes that are not complete multipartite graphs. For this purpose, we introduce color-closed graph classes, which seem to be an appropriate notion for obtaining complexity classifications for rainbow problems and may be of independent interest. To prove the positive part of the dichotomy, we show that the problem essentially reduces to computing a maximum $(l, u)$-matching, where we heavily exploit that almost all color classes are complete multipartite graphs. In the case where all color classes are complete multipartite, we provide a polynomial-time algorithm that computes a maximum matching containing at most $m_i$ edges from each color class $i$.

cs.DM↗

arXiv2025

Two-Edge Connectivity via Pac-Man Gluing

We study the 2-edge-connected spanning subgraph (2-ECSS) problem: Given a graph $G$, compute a connected subgraph $H$ of $G$ with the minimum number of edges such that $H$ is spanning, i.e., $V(H) = V(G)$, and $H$ is 2-edge-connected, i.e., $H$ remains connected upon the deletion of any single edge, if such an $H$ exists. The $2$-ECSS problem is known to be NP-hard. In this work, we provide a polynomial-time $(\frac 5 4 + \varepsilon)$-approximation for the problem for an arbitrarily small $\varepsilon>0$, improving the previous best approximation ratio of $\frac{13}{10}+\varepsilon$. Our improvement is based on two main innovations: First, we reduce solving the problem on general graphs to solving it on structured graphs with high vertex connectivity. This high vertex connectivity ensures the existence of a 4-matching across any bipartition of the vertex set with at least 10 vertices in each part. Second, we exploit this property in a later gluing step, where isolated 2-edge-connected components need to be merged without adding too many edges. Using the 4-matching property, we can repeatedly glue a huge component (containing at least 10 vertices) to other components. This step is reminiscent of the Pac-Man game, where a Pac-Man (a huge component) consumes all the dots (other components) as it moves through a maze. These two innovations lead to a significantly simpler algorithm and analysis for the gluing step compared to the previous best approximation algorithm, which required a long and tedious case analysis.

cs.DS↗

arXiv2025

A Better-Than-$5/4$-Approximation for Two-Edge Connectivity

The 2-Edge-Connected Spanning Subgraph Problem (2ECSS) is a fundamental problem in survivable network design. Given an undirected $2$-edge-connected graph, the goal is to find a $2$-edge-connected spanning subgraph with the minimum number of edges; a graph is 2-edge-connected if it is connected after the removal of any single edge. 2ECSS is APX-hard and has been extensively studied in the context of approximation algorithms. Very recently, Bosch-Calvo, Garg, Grandoni, Hommelsheim, Jabal Ameli, and Lindermayr showed the currently best-known approximation ratio of $\frac{5}{4}$ [STOC 2025]. This factor is tight for many of their techniques and arguments, and it was not clear whether $\frac{5}{4}$ can be improved. We break this natural barrier and present a $(\frac{5}{4} - η)$-approximation algorithm, for some constant $η\geq 10^{-6}$. On a high level, we follow the approach of previous works: take a triangle-free $2$-edge cover and transform it into a 2-edge-connected spanning subgraph by adding only a few additional edges. For $\geq \frac{5}{4}$-approximations, one can heavily exploit that a $4$-cycle in the 2-edge cover can ``buy'' one additional edge. This enables simple and nice techniques, but immediately fails for our improved approximation ratio. To overcome this, we design two complementary algorithms that perform well for different scenarios: one for few $4$-cycles and one for many $4$-cycles. Besides this, there appear more obstructions when breaching $\frac54$, which we surpass via new techniques such as colorful bridge covering, rich vertices, and branching gluing paths.

cs.DS↗

arXiv2025

Improved Approximation Algorithms for Path and Forest Augmentation via a Novel Relaxation

The Forest Augmentation Problem (FAP) asks for a minimum set of additional edges (links) that make a given forest 2-edge-connected while spanning all vertices. A key special case is the Path Augmentation Problem (PAP), where the input forest consists of vertex-disjoint paths. Grandoni, Jabal Ameli, and Traub [STOC'22] recently broke the long-standing 2-approximation barrier for FAP, achieving a 1.9973-approximation. A crucial component of this result was their 1.9913-approximation for PAP; the first better-than-2 approximation for PAP. In this work, we improve these results and provide a 1.9412-approximation for PAP, which implies a 1.9955-approximation for FAP. One of our key innovations is a $(\frac{7}{4} + \varepsilon)$-approximation preserving reduction to so-called structured instances, which simplifies the problem and enables our improved approximation. Additionally, we introduce a new relaxation inspired by 2-edge covers and analyze it via a corresponding packing problem, where the relationship between the two problems is similar to the relationship between 2-edge covers and 2-matchings. Using a factor-revealing LP, we bound the cost of our solution to the packing problem w.r.t. the relaxation and derive a strong initial solution. We then transform this solution into a feasible PAP solution, combining techniques from FAP and related connectivity augmentation problems, along with new insights. A key aspect of our approach is leveraging the properties of structured PAP instances to achieve our final approximation guarantee. Our reduction framework and relaxation may be of independent interest in future work on connectivity augmentation problems.

cs.DS↗

arXiv2025

Improved Approximation Algorithms for the Expanding Search Problem

A searcher is tasked with exploring a graph with edge lengths and vertex weights, starting from a designated vertex. Initially, only the starting vertex is considered explored. At each step, the searcher adds an edge to the solution, connecting an unexplored vertex to an explored one. The time required to add an edge equals its length. The objective is to minimize the weighted sum of exploration times for all vertices. We demonstrate that this problem is hard to approximate and present algorithms with improved approximation guarantees. Specifically, we provide a $(2\mathrm{e} + \varepsilon)$-approximation for any $\varepsilon > 0$ for the general case. On instances where the vertex weights are binary, we achieve a $2\mathrm{e}$-approximation. Finally, we develop a polynomial-time approximation scheme (PTAS) for Euclidean graphs. Previously, only an $8$-approximation was known for all these cases.

cs.DS↗

arXiv2025

A $5/4$-Approximation for Two-Edge Connectivity

The 2-Edge-Connected Spanning Subgraph problem (2ECSS) is among the most basic survivable network design problems: given an undirected and unweighted graph, the task is to find a spanning subgraph with the minimum number of edges that is 2-edge-connected (i.e., it remains connected after the removal of any single edge). 2ECSS is an NP-hard problem that has been extensively studied in the context of approximation algorithms. The best known approximation ratio for 2ECSS prior to this work was $1.3+\varepsilon$, for any constant $\varepsilon>0$ [Garg, Grandoni, Jabal-Ameli'23; Kobayashi, Noguchi'23]. In this paper, we present a 5/4-approximation algorithm. Our algorithm is also faster for small values of $\varepsilon$: its running time is $n^{O(1)}$ instead of $n^{O(1/\varepsilon)}$.

cs.DS↗

arXiv2025

Protecting the Connectivity of a Graph Under Non-Uniform Edge Failures

We study the problem of guaranteeing the connectivity of a given graph by protecting or strengthening edges. Herein, a protected edge is assumed to be robust and will not fail, which features a non-uniform failure model. We introduce the $(p,q)$-Steiner-Connectivity Preservation problem where we protect a minimum-cost set of edges such that the underlying graph maintains $p$-edge-connectivity between given terminal pairs against edge failures, assuming at most $q$ unprotected edges can fail. We design polynomial-time exact algorithms for the cases where $p$ and $q$ are small and approximation algorithms for general values of $p$ and $q$. Additionally, we show that when both $p$ and $q$ are part of the input, even deciding whether a given solution is feasible is NP-complete. This hardness also carries over to Flexible Network Design, a research direction that has gained significant attention. In particular, previous work focuses on problem settings where either $p$ or $q$ is constant, for which our new hardness result now provides justification.

cs.DS↗

arXiv2024

Accelerating Matroid Optimization through Fast Imprecise Oracles

Querying complex models for precise information (e.g. traffic models, database systems, large ML models) often entails intense computations and results in long response times. Thus, weaker models which give imprecise results quickly can be advantageous, provided inaccuracies can be resolved using few queries to a stronger model. In the fundamental problem of computing a maximum-weight basis of a matroid, a well-known generalization of many combinatorial optimization problems, algorithms have access to a clean oracle to query matroid information. We additionally equip algorithms with a fast but dirty oracle modelling an unknown, potentially different matroid. We design and analyze practical algorithms which only use few clean queries w.r.t. the quality of the dirty oracle, while maintaining robustness against arbitrarily poor dirty matroids, approaching the performance of classic algorithms for the given problem. Notably, we prove that our algorithms are, in many respects, best-possible. Further, we outline extensions to other matroid oracle types, non-free dirty oracles and other matroid problems.

cs.DS↗

arXiv2024

A Tight ($3/2 + \varepsilon$)-Approximation Algorithm for Demand Strip Packing

We consider the Demand Strip Packing problem (DSP), in which we are given a set of jobs, each specified by a processing time and a demand. The task is to schedule all jobs such that they are finished before some deadline $D$ while minimizing the peak demand, i.e., the maximum total demand of tasks executed at any point in time. DSP is closely related to the Strip Packing problem (SP), in which we are given a set of axis-aligned rectangles that must be packed into a strip of fixed width while minimizing the maximum height. DSP and SP are known to be NP-hard to approximate to within a factor below $\frac{3}{2}$. To achieve the essentially best possible approximation guarantee, we prove a structural result. Any instance admits a solution with peak demand at most $\big(\frac32+\varepsilon\big)OPT$ satisfying one of two properties. Either (i) the solution leaves a gap for a job with demand $OPT$ and processing time $\mathcal O(\varepsilon D)$ or (ii) all jobs with demand greater than $\frac{OPT}2$ appear sorted by demand in immediate succession. We then provide two efficient algorithms that find a solution with maximum demand at most $\big(\frac32+\varepsilon\big)OPT$ in the respective case. A central observation, which sets our approach apart from previous ones for DSP, is that the properties (i) and (ii) need not be efficiently decidable: We can simply run both algorithms and use whichever solution is the better one.

cs.DS↗

arXiv2023

Matching Augmentation via Simultaneous Contractions

We consider the matching augmentation problem (MAP), where a matching of a graph needs to be extended into a $2$-edge-connected spanning subgraph by adding the minimum number of edges to it. We present a polynomial-time algorithm with an approximation ratio of $13/8 = 1.625$ improving upon an earlier $5/3$-approximation. The improvement builds on a new $α$-approximation preserving reduction for any $α\geq 3/2$ from arbitrary MAP instances to well-structured instances that do not contain certain forbidden structures like parallel edges, small separators, and contractible subgraphs. We further introduce, as key ingredients, the technique of repeated simultaneous contractions and provide improved lower bounds for instances that cannot be contracted.

cs.DS↗

arXiv2022

On the Complexity of the Bilevel Minimum Spanning Tree Problem

We consider the bilevel minimum spanning tree (BMST) problem where the leader and the follower choose a spanning tree together, according to different objective functions. By showing that this problem is NP-hard in general, we answer an open question stated in by Shi et al. We prove that BMST remains hard even in the special case where the follower only controls a matching. Moreover, by a polynomial reduction from the vertex-disjoint Steiner trees problem, we give some evidence that BMST might even remain hard in case the follower controls only few edges. On the positive side, we present a polynomial-time $(|V|-1)$-approximation algorithm for BMST, where $|V|$ is the number of vertices in the input graph. Moreover, considering the number of edges controlled by the follower as parameter, we show that 2-approximating BMST is fixed-parameter tractable and that, in case of uniform costs on leader's edges, even solving BMST exactly is fixed-parameter tractable. We finally consider bottleneck variants of BMST and settle the complexity landscape of all combinations of sum or bottleneck objective functions for the leader and follower, for the optimistic as well as the pessimistic setting.

cs.DS↗

arXiv2021

On the Complexity of Robust Bilevel Optimization With Uncertain Follower's Objective

We investigate the complexity of bilevel combinatorial optimization with uncertainty in the follower's objective, in a robust optimization approach. We show that the robust counterpart of the bilevel problem under interval uncertainty can be $Σ^{\text P}_2$-hard, even when the certain bilevel problem is NP-equivalent and the follower's problem is tractable. On the contrary, in the discrete uncertainty case, the robust bilevel problem is at most one level harder than the follower's problem.

math.OC↗

arXiv2020

Fixed-Parameter Algorithms for Graph Constraint Logic

Non-deterministic constraint logic (NCL) is a simple model of computation based on orientations of a constraint graph with edge weights and vertex demands. NCL captures \PSPACE\xspace and has been a useful tool for proving algorithmic hardness of many puzzles, games, and reconfiguration problems. In particular, its usefulness stems from the fact that it remains \PSPACE-complete even under severe restrictions of the weights (e.g., only edge-weights one and two are needed) and the structure of the constraint graph (e.g., planar \textsc{and/or}\xspace graphs of bounded bandwidth). While such restrictions on the structure of constraint graphs do not seem to limit the expressiveness of NCL, the building blocks of the constraint graphs cannot be limited without losing expressiveness: We consider as parameters the number of weight-one edges and the number of weight-two edges of a constraint graph, as well as the number of \textsc{and}\xspace or \textsc{or}\xspace vertices of an \textsc{and/or}\xspace constraint graph. We show that NCL is fixed-parameter tractable (FPT) for any of these parameters. In particular, for NCL parameterized by the number of weight-one edges or the number of \textsc{and}\xspace vertices, we obtain a linear kernel. It follows that, in a sense, NCL as introduced by Hearn and Demaine is defined in the most economical way for the purpose of capturing \PSPACE.

cs.DS↗

arXiv2020

Fault-Tolerant Edge-Disjoint Paths -- Beyond Uniform Faults

The overwhelming majority of survivable (fault-tolerant) network design models assume a uniform fault model. Such a model assumes that every subset of the network resources (edges or vertices) of a given cardinality $k$ may fail. While this approach yields problems with clean combinatorial structure and good algorithms, it often fails to capture the true nature of the scenario set coming from applications. One natural refinement of the uniform model is obtained by partitioning the set of resources into vulnerable and safe resources. The scenario set contains every subset of at most $k$ faulty resources. This work studies the Fault-Tolerant Path (FTP) problem, the counterpart of the Shortest Path problem in this fault model and the Fault-Tolerant Flow problem (FTF), the counterpart of the $\ell$-disjoint Shortest $s$-$t$ Path problem. We present complexity results alongside exact and approximation algorithms for both models. We emphasize the vast increase in the complexity of the problem with respect to the uniform analogue, the Edge-Disjoint Paths problem.

cs.DS↗

arXiv2020

Flexible Graph Connectivity: Approximating Network Design Problems Between 1- and 2-connectivity

Graph connectivity and network design problems are among the most fundamental problems in combinatorial optimization. The minimum spanning tree problem, the two edge-connected spanning subgraph problem (2-ECSS) and the tree augmentation problem (TAP) are all examples of fundamental well-studied network design tasks that postulate different initial states of the network and different assumptions on the reliability of network components. In this paper we motivate and study \emph{Flexible Graph Connectivity} (FGC), a problem that mixes together both the modeling power and the complexities of all aforementioned problems and more. In a nutshell, FGC asks to design a connected network, while allowing to specify different reliability levels for individual edges. While this non-uniform nature of the problem makes it appealing from the modeling perspective, it also renders most existing algorithmic tools for dealing with network design problems unfit for approximating FGC. In this paper we develop a general algorithmic approach for approximating FGC that yields approximation algorithms with ratios that are very close to the best known bounds for many special cases, such as 2-ECSS and TAP. Our algorithm and analysis combine various techniques including a weight-scaling algorithm, a charging argument that uses a variant of exchange bijections between spanning trees and a factor revealing min-max-min optimization problem.

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