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^p₂-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^p₂-complete for over 50 underlying problems whose decision versions are NP-complete, 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.

Many fundamental problems in computational geometry admit no algorithm running in o(n log n) time for n planar input points, via classical reductions from sorting. Prominent examples include the computation of convex hulls, quadtrees, onion layer decompositions, Euclidean minimum spanning trees, KD-trees, Voronoi diagrams, and decremental closest-pair. A classical result shows that, given n points sorted along a single direction, the convex hull can be constructed in linear time. Subsequent works established that for all of the other above problems, this information does not suffice. In 1989, Aggarwal, Guibas, Saxe, and Shor asked: Under which conditions can a Voronoi diagram be computed in o(n log n) time? Since then, the question of whether sorting along two directions enables a o(n log n)-time algorithm for such problems has remained open and has been repeatedly mentioned in the literature. In this paper, we introduce the Presort Hierarchy: A problem is 1-Presortable if, given a sorting along one axis, it permits a (possibly randomised) o(n log n)-time algorithm. It is 2-Presortable if sortings along both axes suffice. It is Presort-Hard otherwise. Our main result is that quadtrees, and by extension Delaunay triangulations, Voronoi diagrams, and Euclidean minimum spanning trees, are 2-Presortable: we present an algorithm with expected running time O(n √{log n}). This addresses the longstanding open problem posed by Aggarwal, Guibas, Saxe, and Shor (albeit randomised). We complement this result by showing that some of the other above geometric problems are also 2-Presortable or Presort-Hard.

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 Hamiltonian cycles of a given graph by forbidding only few edges; or 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 2 p . Such complexity insights are important because they imply that all these problems cannot be modelled by a compact integer program (unless the unlikely conjecture NP = Σ 2 p holds). Concretely, we prove Σ 2 p -completeness of the following interdiction problems: satisfiability, 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 ≥ 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 “for the same reason”. Our result extends a recent framework by Grüne and Wulf.

The Fréchet distance is a distance measure between trajectories in ℝ^d or walks in a graph G. Given constant-time shortest path queries, the Discrete Fréchet distance D_G(P, Q) between two walks P and Q can be computed in O(|P|⋅|Q|) time using a dynamic program. Driemel, van der Hoog, and Rotenberg [SoCG'22] show that for weighted planar graphs this approach is likely tight, as there can be no strongly-subquadratic algorithm to compute a 1.01-approximation of D_G(P, Q) unless the Orthogonal Vector Hypothesis (OVH) fails. Such quadratic-time conditional lower bounds are common to many Fréchet distance variants. However, they can be circumvented by assuming that the input comes from some well-behaved class: There exist (1+ε)-approximations, both in weighted graphs and in ℝ^d, that take near-linear time for c-packed or κ-straight walks in the graph. In ℝ^d there also exists a near-linear time algorithm to compute the Fréchet distance whenever all input edges are long compared to the distance. We consider computing the Fréchet distance in unweighted planar graphs. We show that there exist no strongly-subquadratic 1.25-approximations of the discrete Fréchet distance between two disjoint simple paths in an unweighted planar graph in strongly subquadratic time, unless OVH fails. This improves the previous lower bound, both in terms of generality and approximation factor. We subsequently show that adding graph structure circumvents this lower bound: If the graph is a regular tiling with unit-weighted edges, then there exists an Õ((|P|+|Q|)^{1.5})-time algorithm to compute D_G(P, Q). Our result has natural implications in the plane, as it allows us to define a new class of well-behaved curves that facilitate (1+ε)-approximations of their discrete Fréchet distance in subquadratic time.

The diameter of a polytope is a fundamental geometric parameter that plays a crucial role in understanding the efficiency of the simplex method. Despite its central nature, the computational complexity of computing the diameter of a given polytope is poorly understood. Already in 1994, Frieze and Teng [Comp. Compl.] recognized the possibility that this task could potentially be harder than NP-hard, and asked whether the corresponding decision problem is complete for the second level of the polynomial hierarchy, i.e. $\Pi_{2}^{p}$-complete. In the following years, partial results could be obtained. In a cornerstone result, Frieze and Teng themselves proved weak NP-hardness for a family of custom defined polytopes. Sanità [FOCS18] in a break-through result proved that already for the much simpler fractional matching polytope the problem is strongly NP-hard. Very recently, Steiner and Nöbel [SODA25] generalized this result to the even simpler bipartite perfect matching polytope and the circuit diameter. In this paper, we finally show that computing the diameter of the bipartite perfect matching polytope is $\Pi_{2}^{p}$ hard. Since the corresponding decision problem is also trivially contained in $\Pi_{2}^{p}$, this decidedly answers Frieze and Teng’s 30 year old question. Our results in particular hold even when the constraint matrix of the given polytope is totally unimodular. They also hold when the diameter is replaced by the circuit diameter. As our second main result, we prove that for some $\varepsilon\gt 0$ the (circuit) diameter of the bipartite perfect matching polytope cannot be approximated by a factor better than ($1+\varepsilon$). This answers a recent question by Nöbel and Steiner. It is the first known inapproximability result for the circuit diameter, and extends Sanità ’s inapproximability result of the diameter to the totally unimodular case.

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^p₃-complete and therefore it is impossible to find compact IP formulations of them (unless the unlikely conjecture NP = Σ^p₃ 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-directed disjoint path (k ≥ 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^p₃-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^p₃-complete. This reveals the insight that all the above problems are Σ^p₃-complete "for the same reason". Our result extends a recent framework by Grüne and Wulf.

ABSTRACT We introduce a new bilevel version of the classic shortest path problem and completely characterize its computational complexity with respect to several problem variants. In our problem, the leader and the follower each control a subset of the edges of a graph and together aim at building a path between two given vertices, while each of the two players minimizes the cost of the resulting path according to their own cost function. We investigate both directed and undirected graphs, as well as the special case of directed acyclic graphs. Moreover, we distinguish two versions of the follower's problem: Either they have to complete the edge set selected by the leader such that the joint solution is exactly a path or they have to complete the edge set selected by the leader such that the joint solution is a superset of a path. In general, the bilevel problem turns out to be much harder in the former case: We show that the follower's problem is already NP‐hard here and that the leader's problem is even hard for the second level of the polynomial hierarchy, while both problems are one level easier in the latter case. Interestingly, for directed acyclic graphs, this difference turns around, as we give a polynomial‐time algorithm for the first version of the bilevel problem, but it stays NP‐hard in the second case. Finally, we consider restrictions that render the problem tractable. We prove that, for a constant number of leader's edges, one of our problem variants is actually equivalent to the shortest‐‐cycle problem, which is a known combinatorial problem with partially unresolved complexity status. In particular, our problem admits a polynomial‐time randomized algorithm that can be derandomized if and only if the shortest‐‐cycle problem admits a deterministic polynomial‐time algorithm.

In 1982 Papadimitriou and Yannakakis introduced the Exact Matching problem, in which given a red and blue edge-colored graph $G$ and an integer $k$ one has to decide whether there exists a perfect matching in $G$ with exactly $k$ red edges. Even though a randomized polynomial-time algorithm for this problem was quickly found a few years later, it is still unknown today whether a deterministic polynomial-time algorithm exists. This makes the Exact Matching problem an important candidate to test the RP=P hypothesis. In this paper we focus on approximating Exact Matching. While there exists a simple algorithm that computes in deterministic polynomial-time an almost perfect matching with exactly $k$ red edges, not a lot of work focuses on computing perfect matchings with almost $k$ red edges. In fact such an algorithm for bipartite graphs running in deterministic polynomial-time was published only recently (STACS'23). It outputs a perfect matching with $k'$ red edges with the guarantee that $0.5k \leq k' \leq 1.5k$. In the present paper we aim at approximating the number of red edges without exceeding the limit of $k$ red edges. We construct a deterministic polynomial-time algorithm, which on bipartite graphs computes a perfect matching with $k'$ red edges such that $k/3 \leq k' \leq k$.

Given an integer $k$ and a graph where every edge is colored either red or blue, the goal of the exact matching problem is to find a perfect matching with the property that exactly $k$ of its edges are red. Soon after Papadimitriou and Yannakakis (JACM 1982) introduced the problem, a randomized polynomial-time algorithm solving the problem was described by Mulmuley et al. (Combinatorica 1987). Despite a lot of effort, it is still not known today whether a deterministic polynomial-time algorithm exists. This makes the exact matching problem an important candidate to test the popular conjecture that the complexity classes P and RP are equal. In a recent article (MFCS 2022), progress was made towards this goal by showing that for bipartite graphs of bounded bipartite independence number, a polynomial time algorithm exists. In terms of parameterized complexity, this algorithm was an XP-algorithm parameterized by the bipartite independence number. In this article, we introduce novel algorithmic techniques that allow us to obtain an FPT-algorithm. If the input is a general graph we show that one can at least compute a perfect matching $M$ which has the correct number of red edges modulo 2, in polynomial time. This is motivated by our last result, in which we prove that an FPT algorithm for general graphs, parameterized by the independence number, reduces to the problem of finding in polynomial time a perfect matching $M$ with at most $k$ red edges and the correct number of red edges modulo 2.

Abstract An instance of the non-preemptive tree packing problem consists of an undirected graph $$G=(V,E)$$ G = ( V , E ) together with a weight w(e) for every edge $$e\in E$$ e ∈ E . The goal is to activate every edge e for some time interval of length w(e), such that the activated edges keep G connected for the longest possible overall time. We derive a variety of results on this problem. The problem is strongly NP-hard even on graphs of treewidth 2, and it does not allow a polynomial time approximation scheme (unless P=NP). Furthermore, we discuss the performance of a simple greedy algorithm, and we construct and analyze a number of parameterized and exact algorithms.