arXiv subjects

Max Bannach

Publications and source records attributed to Max Bannach.

At least 19 recordsLinked to original sources

arXiv2026

The Keplerian Traveling Salesperson Problem

We address a fundamental challenge in space mission design and space logistics: planning interplanetary trajectories for missions that must rendezvous with multiple bodies. Such mission occur, for instance, in active debris removal, in-orbit servicing, or asteroid belt exploration. We model these problems as a variant of the Traveling salesperson problem (TSP), which we term the Keplerian TSP (KTSP). Unlike the well-studied TSP, the KTSP accounts for the motion of orbital targets, leading to time-dependent and asymmetric transfer costs that capture key real-world effects in astrodynamics. We provide a rigorous formalization of the KTSP and release a benchmark suite to support its study. Central to our approach is a time-unfolding technique that reformulates the continuous problem as a discrete optimization task in a time-expanded network. This representation makes the benchmark accessible to researchers in discrete optimization even without prior knowledge of celestial mechanics. We also develop an alternative encoding as an integer linear program using Interval-based Dynamic Discretization Discovery to handle the time-dependent nature of transfers. We leverage state-of-the-art ILP solvers to solve the KTSP instances, accompanied by a detailed computational study that highlights their strengths and limitations. We complement these exact methods with an initial solution heuristic, an improvement heuristic, and preprocessing routines that preserve optimality.

math.OC↗

arXiv2025

#P is Sandwiched by One and Two #2DNF Calls: Is Subtraction Stronger Than We Thought?

The canonical class in the realm of counting complexity is #P. It is well known that the problem of counting the models of a propositional formula in disjunctive normal form (#DNF) is complete for #P under Turing reductions. On the other hand, #DNF $\in$ spanL and spanL $\not\subseteq$ #P unless NL = NP. Hence, the class of functions logspace-reducible to #DNF is a strict subset of #P under plausible complexity-theoretic assumptions. By contrast, we show that two calls to a (restricted) #2DNF oracle suffice to capture gapP, namely, that the logspace many-one closure of the subtraction between the results of two #2DNF calls is gapP. Because #P $\not\subseteq$ gapP, #P is strictly contained between one and two #2DNF oracle calls. Surprisingly, the propositional formulas needed in both calls are linear-time computable, and the reduction preserves interesting structural as well as symmetry properties, leading to algorithmic applications. We show that a single subtraction suffices to compensate for the absence of negation while still capturing gapP, i.e., our results carry over to the monotone fragments of #2SAT and #2DNF. Since our reduction is linear-time, it preserves sparsity and, as a consequence we obtain a sparsification lemma for both #2SAT and #2DNF. This has only been known for kSAT with k $\geq$ 3 and respective counting versions. We further show that both #2DNF calls can be combined into a single call if we allow a little postprocessing (computable by AC0- or TC0-circuits). Consequently, we derive refined versions of Toda's Theorem: PH $\subseteq$ [#MON2SAT]$^{log}_{TC0}$ = [#MON2DNF]$^{log}_{TC0}$ and PH $\subseteq$ [#IMPL2SAT]$^{log}_{AC0}$. Our route to these results is via structure-aware reductions that preserve parameters like treewidth up to an additive overhead. The absence of multiplicative overhead indeed yields parameterized SETH-tight lower bounds.

cs.CC↗

arXiv2024

Strong Structural Bounds for MaxSAT: The Fine Details of Using Neuromorphic and Quantum Hardware Accelerators

Hardware accelerators like quantum annealers or neuromorphic chips are capable of finding the ground state of a Hamiltonian. A promising route in utilizing these devices is via methods from automated reasoning: The problem at hand is first encoded into MaxSAT; then MaxSAT is reduced to Max2SAT; and finally, Max2SAT is translated into a Hamiltonian. It was observed that different encodings can dramatically affect the efficiency of the hardware accelerators. Yet, previous studies were only concerned with the size of the encodings rather than with syntactic or structural properties. We establish structure-aware reductions between MaxSAT, Max2SAT, and the quadratic unconstrained binary optimization problem (QUBO) that underlies such hardware accelerators. All these problems turn out to be equivalent under linear-time, treewidth-preserving reductions. As a consequence, we obtain tight lower bounds under ETH and SETH for Max2SAT and QUBO, as well as a new time-optimal fixed-parameter algorithm for QUBO. While our results are tight up to a constant additive factor for the primal treewidth, we require a constant multiplicative factor for the incidence treewidth. To close the emerging gap, we supplement our results with novel time-optimal algorithms for fragments of MaxSAT based on model counting.

cs.LO↗

arXiv2024

Asteroid Mining: ACT&Friends' Results for the GTOC 12 Problem

In 2023, the 12th edition of Global Trajectory Competition was organised around the problem referred to as "Sustainable Asteroid Mining". This paper reports the developments that led to the solution proposed by ESA's Advanced Concepts Team. Beyond the fact that the proposed approach failed to rank higher than fourth in the final competition leader-board, several innovative fundamental methodologies were developed which have a broader application. In particular, new methods based on machine learning as well as on manipulating the fundamental laws of astrodynamics were developed and able to fill with remarkable accuracy the gap between full low-thrust trajectories and their representation as impulsive Lambert transfers. A novel technique was devised to formulate the challenge of optimal subset selection from a repository of pre-existing optimal mining trajectories as an integer linear programming problem. Finally, the fundamental problem of searching for single optimal mining trajectories (mining and collecting all resources), albeit ignoring the possibility of having intra-ship collaboration and thus sub-optimal in the case of the GTOC12 problem, was efficiently solved by means of a novel search based on a look-ahead score and thus making sure to select asteroids that had chances to be re-visited later on.

astro-ph.EP↗

arXiv2024

Certifying Guidance & Control Networks: Uncertainty Propagation to an Event Manifold

We perform uncertainty propagation on an event manifold for Guidance & Control Networks (G&CNETs), aiming to enhance the certification tools for neural networks in this field. This work utilizes three previously solved optimal control problems with varying levels of dynamics nonlinearity and event manifold complexity. The G&CNETs are trained to represent the optimal control policies of a time-optimal interplanetary transfer, a mass-optimal landing on an asteroid and energy-optimal drone racing, respectively. For each of these problems, we describe analytically the terminal conditions on an event manifold with respect to initial state uncertainties. Crucially, this expansion does not depend on time but solely on the initial conditions of the system, thereby making it possible to study the robustness of the G&CNET at any specific stage of a mission defined by the event manifold. Once this analytical expression is found, we provide confidence bounds by applying the Cauchy-Hadamard theorem and perform uncertainty propagation using moment generating functions. While Monte Carlo-based (MC) methods can yield the results we present, this work is driven by the recognition that MC simulations alone may be insufficient for future certification of neural networks in guidance and control applications.

eess.SY↗

arXiv2024

On the Descriptive Complexity of Vertex Deletion Problems

Vertex deletion problems for graphs are studied intensely in classical and parameterized complexity theory. They ask whether we can delete at most k vertices from an input graph such that the resulting graph has a certain property. Regarding k as the parameter, a dichotomy was recently shown based on the number of quantifier alternations of first-order formulas that describe the property. In this paper, we refine this classification by moving from quantifier alternations to individual quantifier patterns and from a dichotomy to a trichotomy, resulting in a complete classification of the complexity of vertex deletion problems based on their quantifier pattern. The more fine-grained approach uncovers new tractable fragments, which we show to not only lie in FPT, but even in parameterized constant-depth circuit complexity classes. On the other hand, we show that vertex deletion becomes intractable already for just one quantifier per alternation, that is, there is a formula of the form {\forall}x{\exists}y{\forall}z({\psi}), with {\psi} quantifier-free, for which the vertex deletion problem is W[1]-hard. The fine-grained analysis also allows us to uncover differences in the complexity landscape when we consider different kinds of graphs and more general structures: While basic graphs (undirected graphs without self-loops), undirected graphs, and directed graphs each have a different frontier of tractability, the frontier for arbitrary logical structures coincides with that of directed graphs.

cs.LO↗

arXiv2023

Structure-Guided Automated Reasoning

Algorithmic meta-theorems state that problems definable in a fixed logic can be solved efficiently on structures with certain properties. An example is Courcelle's Theorem, which states that all problems expressible in monadic second-order logic can be solved efficiently on structures of small treewidth. Such theorems are usually proven by algorithms for the model-checking problem of the logic, which is often complex and rarely leads to highly efficient solutions. Alternatively, we can solve the model-checking problem by grounding the given logic to propositional logic, for which dedicated solvers are available. Such encodings will, however, usually not preserve the input's treewidth. This paper investigates whether all problems definable in monadic second-order logic can efficiently be encoded into \Lang{sat} such that the input's treewidth bounds the treewidth of the resulting formula. We answer this in the affirmative and, hence, provide an alternative proof of Courcelle's Theorem. Our technique can naturally be extended: There are treewidth-aware reductions from the optimization version of Courcelle's Theorem to MaxSAT and from the counting version of the theorem to \#SAT. By using encodings to SAT, we obtain, ignoring polynomial factors, the same running time for the model-checking problem as we would with dedicated algorithms. Another immediate consequence is a treewidth-preserving reduction from the model-checking problem of monadic second-order logic to integer linear programming (ILP). We complement our upper bounds with new lower bounds based on ETH; and we show that the block size of the input's formula and the treewidth of the input's structure are tightly linked.

cs.LO↗

arXiv2023

Existential Second-Order Logic Over Graphs: Parameterized Complexity

By Fagin's Theorem, NP contains precisely those problems that can be described by formulas starting with an existential second-order quantifier, followed by only first-order quantifiers (ESO formulas). Subsequent research refined this result, culminating in powerful theorems that characterize for each possible sequence of first-order quantifiers how difficult the described problem can be. We transfer this line of inquiry to the parameterized setting, where the size of the set quantified by the second-order quantifier is the parameter. Many natural parameterized problems can be described in this way using simple sequences of first-order quantifiers: For the clique or vertex cover problems, two universal first-order quantifiers suffice ("for all pairs of vertices ... must hold"); for the dominating set problem, a universal followed by an existential quantifier suffice ("for all vertices, there is a vertex such that ..."); and so on. We present a complete characterization that states for each possible sequence of first-order quantifiers how high the parameterized complexity of the described problems can be. The uncovered dividing line between quantifier sequences that lead to tractable versus intractable problems is distinct from that known from the classical setting, and it depends on whether the parameter is a lower bound on, an upper bound on, or equal to the size of the quantified set.

cs.LO↗

arXiv2023

Efficient Enumeration of Markov Equivalent DAGs

Enumerating the directed acyclic graphs (DAGs) of a Markov equivalence class (MEC) is an important primitive in causal analysis. The central resource from the perspective of computational complexity is the delay, that is, the time an algorithm that lists all members of the class requires between two consecutive outputs. Commonly used algorithms for this task utilize the rules proposed by Meek (1995) or the transformational characterization by Chickering (1995), both resulting in superlinear delay. In this paper, we present the first linear-time delay algorithm. On the theoretical side, we show that our algorithm can be generalized to enumerate DAGs represented by models that incorporate background knowledge, such as MPDAGs; on the practical side, we provide an efficient implementation and evaluate it in a series of experiments. Complementary to the linear-time delay algorithm, we also provide intriguing insights into Markov equivalence itself: All members of an MEC can be enumerated such that two successive DAGs have structural Hamming distance at most three.

cs.AI↗

arXiv2022

On the Parallel Parameterized Complexity of MaxSAT Variants

In the maximum satisfiability problem (MAX-SAT) we are given a propositional formula in conjunctive normal form and have to find an assignment that satisfies as many clauses as possible. We study the parallel parameterized complexity of various versions of MAX-SAT and provide the first constant-time algorithms parameterized either by the solution size or by the allowed excess relative to some guarantee ("above guarantee" versions). For the dual parameterized version where the parameter is the number of clauses we are allowed to leave unsatisfied, we present the first parallel algorithm for MAX-2SAT (known as ALMOST-2SAT). The difficulty in solving ALMOST-2SAT in parallel comes from the fact that the iterative compression method, originally developed to prove that the problem is fixed-parameter tractable at all, is inherently sequential. We observe that a graph flow whose value is a parameter can be computed in parallel and use this fact to develop a parallel algorithm for the vertex cover problem parameterized above the size of a given matching. Finally, we study the parallel complexity of MAX-SAT parameterized by the vertex cover number, the treedepth, the feedback vertex set number, and the treewidth of the input's incidence graph. While MAX-SAT is fixed-parameter tractable for all of these parameters, we show that they allow different degrees of possible parallelization. For all four we develop dedicated parallel algorithms that are constructive, meaning that they output an optimal assignment - in contrast to results that can be obtained by parallel meta-theorems, which often only solve the decision version.

cs.CC↗

arXiv2022

Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs with Applications

Counting and sampling directed acyclic graphs from a Markov equivalence class are fundamental tasks in graphical causal analysis. In this paper we show that these tasks can be performed in polynomial time, solving a long-standing open problem in this area. Our algorithms are effective and easily implementable. As we show in experiments, these breakthroughs make thought-to-be-infeasible strategies in active learning of causal structures and causal effect identification with regard to a Markov equivalence class practically applicable.

cs.LG↗

arXiv2022

MaxSAT with Absolute Value Functions: A Parameterized Perspective

The natural generalization of the Boolean satisfiability problem to optimization problems is the task of determining the maximum number of clauses that can simultaneously be satisfied in a propositional formula in conjunctive normal form. In the weighted maximum satisfiability problem each clause has a positive weight and one seeks an assignment of maximum weight. The literature almost solely considers the case of positive weights. While the general case of the problem is only restricted slightly by this constraint, many special cases become trivial in the absence of negative weights. In this work we study the problem with negative weights and observe that the problem becomes computationally harder - which we formalize from a parameterized perspective in the sense that various variations of the problem become W[1]-hard if negative weights are present. Allowing negative weights also introduces new variants of the problem: Instead of maximizing the sum of weights of satisfied clauses, we can maximize the absolute value of that sum. This turns out to be surprisingly expressive even restricted to monotone formulas in disjunctive normal form with at most two literals per clause. In contrast to the versions without the absolute value, however, we prove that these variants are fixed-parameter tractable. As technical contribution we present a kernelization for an auxiliary problem on hypergraphs in which we seek, given an edge-weighted hypergraph, an induced subgraph that maximizes the absolute value of the sum of edge-weights.

cs.CC↗

arXiv2020

Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs

Counting and uniform sampling of directed acyclic graphs (DAGs) from a Markov equivalence class are fundamental tasks in graphical causal analysis. In this paper, we show that these tasks can be performed in polynomial time, solving a long-standing open problem in this area. Our algorithms are effective and easily implementable. Experimental results show that the algorithms significantly outperform state-of-the-art methods.

cs.LG↗

arXiv2020

Solving Packing Problems with Few Small Items Using Rainbow Matchings

An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number $k$ of small items. Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by $k$. The algorithms are randomized with one-sided error and run in time $4^{k} \cdot k! \cdot n^{O(1)}$. To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. We also present a deterministic fixed-parameter for Bin Packing with run time $(k!)^{2}\cdot k \cdot 2^{k}\cdot n\cdot \log(n)$.

cs.DS↗

arXiv2020

Complete Edge-Colored Permutation Graphs

We introduce the concept of complete edge-colored permutation graphs as complete graphs that are the edge-disjoint union of "classical" permutation graphs. We show that a graph $G=(V,E)$ is a complete edge-colored permutation graph if and only if each monochromatic subgraph of $G$ is a "classical" permutation graph and $G$ does not contain a triangle with~$3$ different colors. Using the modular decomposition as a framework we demonstrate that complete edge-colored permutation graphs are characterized in terms of their strong prime modules, which induce also complete edge-colored permutation graphs. This leads to an $\mathcal{O}(|V|^2)$-time recognition algorithm. We show, moreover, that complete edge-colored permutation graphs form a superclass of so-called symbolic ultrametrics and that the coloring of such graphs is always a Gallai coloring.

math.CO↗

arXiv2019

Positive-Instance Driven Dynamic Programming for Graph Searching

Research on the similarity of a graph to being a tree - called the treewidth of the graph - has seen an enormous rise within the last decade, but a practically fast algorithm for this task has been discovered only recently by Tamaki (ESA 2017). It is based on dynamic programming and makes use of the fact that the number of positive subinstances is typically substantially smaller than the number of all subinstances. Algorithms producing only such subinstances are called positive-instance driven (PID). We give an alternative and intuitive view on this algorithm from the perspective of the corresponding configuration graphs in certain two-player games. This allows us to develop PID-algorithms for a wide range of important graph parameters such as treewidth, pathwidth, and treedepth. We analyse the worst case behaviour of the approach on some well-known graph classes and perform an experimental evaluation on real world and random graphs.

cs.DS↗

arXiv2019

Towards Work-Efficient Parallel Parameterized Algorithms

Parallel parameterized complexity theory studies how fixed-parameter tractable (fpt) problems can be solved in parallel. Previous theoretical work focused on parallel algorithms that are very fast in principle, but did not take into account that when we only have a small number of processors (between 2 and, say, 1024), it is more important that the parallel algorithms are work-efficient. In the present paper we investigate how work-efficient fpt algorithms can be designed. We review standard methods from fpt theory, like kernelization, search trees, and interleaving, and prove trade-offs for them between work efficiency and runtime improvements. This results in a toolbox for developing work-efficient parallel fpt algorithms.

cs.DS↗

arXiv2019

On the Descriptive Complexity of Color Coding

Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing -- purely in terms of the syntactic structure of describing formulas -- when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntactic properties of first-order quantifiers that can be eliminated from formulas describing parameterized problems. The result applies to many packing and embedding problems, but also to the long path problem. Together with a new result on the parameterized complexity of formula families involving only a fixed number of variables, we get that many problems lie in FPT just because of the way they are commonly described using logical formulas.

cs.CC↗