arXiv subjects

Christoph Grüne

Publications and source records attributed to Christoph Grüne.

10 recordsLinked to original sources

arXiv2026

Continuous Defensive Domination Problems

The problem Defensive $δ$-Covering, for some covering range $δ> 0$, is a continuous facility location problem on undirected graphs where all edges have unit length. It is a generalization of Defensive Dominating Set and $δ$-Covering. An attack and defense are sets of points, which are on vertices or on the interior of an edge. A defense counters an attack, if there is a matching of the points in the defense to the points in the attack, such that any matched points have distance at most $δ$, and every point in the attack is matched. The task is, given a graph $G$ and numbers $\ell, k \in \mathbb N$, to find a defense of size at most $\ell$ that counters every possible attack of size at most $k$. We study the complexity of this problem in various different settings. We show that if the attack is restricted to vertices, the problem is $Σ^P_2$-complete for large $δ$, but if the attack may consist of any points on the graph, it is NP-complete. Additionally, we analyze how the complexity changes if the attacks or defenses may be a multiset. If the defense is allowed to be a multiset, the complexity does not change in any case we consider, while if the attack is allowed to be a multiset, the problem often becomes easier. To show containment in the various complexity classes, we introduce a number of discretization arguments, which show that solutions with a regular structure must always exist.

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arXiv2026

Completeness in the Polynomial Hierarchy and PSPACE for many natural problems derived from NP

Many natural optimization problems derived from $\sf NP$ admit bilevel and multilevel extensions in which decisions are made sequentially by multiple players with conflicting objectives, as in interdiction, adversarial selection, and adjustable robust optimization. Such problems are naturally modeled by alternating quantifiers and, therefore, lie beyond $\sf NP$, typically in the polynomial hierarchy or $\sf PSPACE$. Despite extensive study of these problem classes, relatively few natural completeness results are known at these higher levels. We introduce a general framework for proving completeness in the polynomial hierarchy and $\sf PSPACE$ for problems derived from $\sf NP$. Our approach is based on a refinement of $\sf NP$, which we call $\sf NP$ with solutions ($\sf NP$-$\sf S$), in which solutions are explicit combinatorial objects, together with a restricted class of reductions -- solution-embedding reductions -- that preserve solution structure. We define $\sf NP$-$\sf S$-completeness and show that a large collection of classical $\sf NP$-complete problems, including Clique, Vertex Cover, Knapsack, and Traveling Salesman, are $\sf NP$-$\sf S$-complete. Using this framework, we establish general meta-theorems showing that if a problem is $\sf NP$-$\sf S$-complete, then its natural two-level extensions are $Σ_2^p$-complete, its three-level extensions are $Σ_3^p$-complete, and its $k$-level extensions are $Σ_k^p$-complete. When the number of levels is unbounded, the resulting problems are $\sf PSPACE$-complete. Our results subsume nearly all previously known completeness results for multilevel optimization problems derived from $\sf NP$ and yield many new ones simultaneously, demonstrating that high computational complexity is a generic feature of multilevel extensions of $\sf NP$-complete problems.

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arXiv2025

The Complexity Classes of Hamming Distance Recoverable Robust Problems

In the well-known complexity class NP are combinatorial problems, whose optimization counterparts are important for many practical settings. These problems typically consider full knowledge about the input. In practical settings, however, uncertainty in the input data is a usual phenomenon, whereby this is normally not covered in optimization versions of NP problems. One concept to model the uncertainty in the input data, is recoverable robustness. The instance of the recoverable robust version of a combinatorial problem P is split into a base scenario $σ_0$ and an uncertainty scenario set $\textsf{S}$. The base scenario and all members of the uncertainty scenario set are instances of the original combinatorial problem P. The task is to calculate a solution $s_0$ for the base scenario $σ_0$ and solutions $s$ for all uncertainty scenarios $σ\in \textsf{S}$ such that $s_0$ and $s$ are not too far away from each other according to a distance measure, so $s_0$ can be easily adapted to $s$. This paper introduces Hamming Distance Recoverable Robustness, in which solutions $s_0$ and $s$ have to be calculated, such that $s_0$ and $s$ may only differ in at most $κ$ elements. We survey the complexity of Hamming distance recoverable robust versions of optimization problems, typically found in NP for different scenario encodings. The complexity is primarily situated in the lower levels of the polynomial hierarchy. The main contribution of the paper is a gadget reduction framework that shows that the recoverable robust versions of problems in a large class of combinatorial problems is $Σ^P_{3}$-complete. This class includes problems such as Vertex Cover, Coloring or Subset Sum. Additionally, we expand the results to $Σ^P_{2m+1}$-completeness for multi-stage recoverable robust problems with $m \in \mathbb{N}$ stages.

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arXiv2025

Completeness in the Polynomial Hierarchy for many natural Problems in Bilevel and Robust Optimization

In bilevel and robust optimization we are concerned with combinatorial min-max problems, for example from the areas of min-max regret robust optimization, network interdiction, most vital vertex problems, blocker problems, and two-stage adjustable robust optimization. Even though these areas are well-researched for over two decades and one would naturally expect many (if not most) of the problems occurring in these areas to be complete for the classes $Σ^p_2$ or $Σ^p_3$ from the polynomial hierarchy, almost no hardness results in this regime are currently known. However, such complexity insights are important, since they imply that no polynomial-sized integer program for these min-max problems exist, and hence conventional IP-based approaches fail. We address this lack of knowledge by introducing over 70 new $Σ^p_2$-complete and $Σ^p_3$-complete problems. The majority of all earlier publications on $Σ^p_2$- and $Σ^p_3$-completeness in said areas are special cases of our meta-theorem. Precisely, we introduce a large list of problems for which the meta-theorem is applicable (including clique, vertex cover, knapsack, TSP, facility location and many more). We show that for every single of these problems, the corresponding min-max (i.e. interdiction/regret) variant is $Σ^p_2$- and the min-max-min (i.e. two-stage) variant is $Σ^p_3$-complete.

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arXiv2025

The Complexity of Stackelberg Pricing Games

We consider Stackelberg pricing games, which are also known as bilevel pricing problems, or combinatorial price-setting problems. This family of problems consists of games between two players: the leader and the follower. There is a market that is partitioned into two parts: the part of the leader and the part of the leader's competitors. The leader controls one part of the market and can freely set the prices for products. By contrast, the prices of the competitors' products are fixed and known in advance. The follower, then, needs to solve a combinatorial optimization problem in order to satisfy their own demands, while comparing the leader's offers to the offers of the competitors. Therefore, the leader has to hit the intricate balance of making an attractive offer to the follower, while at the same time ensuring that their own profit is maximized. Pferschy, Nicosia, Pacifici, and Schauer considered the Stackelberg pricing game where the follower solves a knapsack problem. They raised the question whether this problem is complete for the second level of the polynomial hierarchy, i.e., $Σ^p_2$-complete. The same conjecture was also made by Böhnlein, Schaudt, and Schauer. In this paper, we positively settle this conjecture. Moreover, we show that this result holds actually in a much broader context: The Stackelberg pricing game is $Σ^p_2$-complete for over 50 NP-complete problems, including most classics such as TSP, vertex cover, clique, subset sum, etc. This result falls in line of recent meta-theorems about higher complexity in the polynomial hierarchy by Grüne and Wulf.

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arXiv2025

A Compendium of Reductions: reductions.network

The website reductions.network serves as a comprehensive database for exploring problems and reductions between them. It presents several complexity classes in the form of an interconnected graph where problems are represented as vertices, while edges represent reductions between them. This graphical perspective allows for identifying problem clusters and simplifying finding problem candidates to reduce from. Moreover, users can easily search for existing problems via a dedicated search bar, and various filters allow them to focus on specific subgraphs of interest. The design of the website enables users to contribute by adding new problems and reductions to the database. Furthermore, the software architecture allows for the integration of additional graphs corresponding to new complexity classes. In the current state, the following networks with their respective complexity classes are included: - classical complexity including the classes NP, #P, and SSP-NP - parameterized complexity including the classes W[1], W[2] - gap-preserving reductions under the PCP-Theorem and the Unique Games Conjecture.

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arXiv2025

The Complexity of Blocking All Solutions

We consider the general problem of blocking all solutions of some given combinatorial problem with only few elements. For example, the problem of destroying all maximum cliques of a given graph by forbidding only few vertices. Problems of this kind are so fundamental that they have been studied under many different names in many different disjoint research communities already since the 90s. Depending on the context, they have been called the interdiction, most vital vertex, most vital edge, blocker, or vertex deletion problem. Despite their apparent popularity, surprisingly little is known about the computational complexity of interdiction problems in the case where the original problem is already NP-complete. In this paper, we fill that gap of knowledge by showing that a large amount of interdiction problems are even harder than NP-hard. Namely, they are complete for the second stage of Stockmeyer's polynomial hierarchy, the complexity class $Σ^p_2$. Such complexity insights are important because they imply that all these problems can not be modelled by a compact integer program (unless the unlikely conjecture NP $= Σ_2^p$ holds). Concretely, we prove $Σ^p_2$-completeness of the following interdiction problems: satisfiability, 3satisfiability, dominating set, set cover, hitting set, feedback vertex set, feedback arc set, uncapacitated facility location, $p$-center, $p$-median, independent set, clique, subset sum, knapsack, Hamiltonian path/cycle (directed/undirected), TSP, $k$ directed vertex disjoint path ($k \geq 2$), Steiner tree. We show that all of these problems share an abstract property which implies that their interdiction counterpart is $Σ_2^p$-complete. Thus, all of these problems are $Σ_2^p$-complete \enquote{for the same reason}. Our result extends a recent framework by Grüne and Wulf.

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arXiv2024

The Complexity of Graph Exploration Games

Graph Exploration problems ask a searcher to explore an unknown environment. The environment is modeled as a graph, where the searcher needs to visit each vertex beginning at some vertex. Treasure Hunt problems are a variation of Graph Exploration, in which the searcher needs to find a hidden treasure, which is located at a designated vertex. Usually these problems are modeled as online problems, and any online algorithm performs poorly because it has too little knowledge about the instance to react adequately to the requests of the adversary. Thus, the impact of a priori knowledge is of interest. One form of a priori knowledge is an unlabeled map, which is an isomorphic copy of the graph. We analyze Graph Exploration and Treasure Hunt problems with an unlabeled map that is provided to the searcher. For this, we formulate decision variants of both problems by interpreting the online problems as a game between the online algorithm (the searcher) and the adversary. The map, however, is not controllable by the adversary. The question is whether the searcher is able to explore the graph completely or find the treasure for all possible decisions of the adversary. We analyze these games in multiple settings, with and without costs on the edges, on directed and undirected graphs and with different constraints (allowing multiple visits to vertices or edges) on the solution. We prove PSPACE-completeness for most of these games. Additionally, we analyze the complexity of related problems that have additional constraints on the solution.

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arXiv2024

On the Complexity of Recoverable Robust Optimization in the Polynomial Hierarchy

Recoverable robust optimization is a popular multi-stage approach, in which it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We consider recoverable robust optimization in combination with discrete budgeted uncertainty. In this setting, it seems plausible that many problems become $Σ^p_3$-complete and therefore it is impossible to find compact IP formulations of them (unless the unlikely conjecture NP $= Σ^p_3$ holds). Even though this seems plausible, few concrete results of this kind are known. In this paper, we fill that gap of knowledge. We consider recoverable robust optimization for the nominal problems of Sat, 3Sat, vertex cover, dominating set, set cover, hitting set, feedback vertex set, feedback arc set, uncapacitated facility location, $p$-center, $p$-median, independent set, clique, subset sum, knapsack, partition, scheduling, Hamiltonian path/cycle (directed/undirected), TSP, $k$-disjoint path ($k \geq 2$), and Steiner tree. We show that for each of these problems, and for each of three widely used distance measures, the recoverable robust problem becomes $Σ^p_3$-complete. Concretely, we show that all these problems share a certain abstract property and prove that this property implies that their robust recoverable counterpart is $Σ^p_3$-complete. This reveals the insight that all the above problems are $Σ^p_3$-complete 'for the same reason'. Our result extends a recent framework by Grüne and Wulf.

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arXiv2023

The Complexity of Online Graph Games

Online computation is a concept to model uncertainty where not all information on a problem instance is known in advance. An online algorithm receives requests which reveal the instance piecewise and has to respond with irrevocable decisions. Often, an adversary is assumed that constructs the instance knowing the deterministic behavior of the algorithm. Thus, the adversary is able to tailor the input to any online algorithm. From a game theoretical point of view, the adversary and the online algorithm are players in an asymmetric two-player game. To overcome this asymmetry, the online algorithm is equipped with an isomorphic copy of the graph, which is referred to as unlabeled map. By applying the game theoretical perspective on online graph problems, where the solution is a subset of the vertices, we analyze the complexity of these online vertex subset games. For this, we introduce a framework for reducing online vertex subset games from TQBF. This framework is based on gadget reductions from 3-SATISFIABILITY to the corresponding offline problem. We further identify a set of rules for extending the 3-SATISFIABILITY-reduction and provide schemes for additional gadgets which assure that these rules are fulfilled. By extending the gadget reduction of the vertex subset problem with these additional gadgets, we obtain a reduction for the corresponding online vertex subset game. At last, we provide example reductions for online vertex subset games based on VERTEX COVER, INDEPENDENT SET, and DOMINATING SET, proving that they are PSPACE-complete. Thus, this paper establishes that the online version with a map of NP-complete vertex subset problems form a large class of PSPACE-complete problems.

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