arXiv subjects

Christian Nöbel

Publications and source records attributed to Christian Nöbel.

7 recordsLinked to original sources

arXiv2026

On the Complexity of the Odd-Red Bipartite Perfect Matching Polytope

The odd-red bipartite perfect matching problem asks to find a perfect matching containing an odd number of red edges in a given red-blue edge-colored bipartite graph. While this problem lies in $\mathsf{P}$, its polyhedral structure remains elusive, despite renewed attention to achieving better polyhedral understanding, nurtured by recent advances from two complementary angles. Apart from being a special case of bimodular integer programs, whose polyhedral structure is also badly understood, it is related to one of the most notorious open derandomization questions in theoretical computer science: whether there is a deterministic efficient algorithm for the exact bipartite perfect matching problem, which asks to find a perfect matching with exactly $k$ red edges. Recent progress towards deterministic algorithms for this problem crucially relies on a good polyhedral understanding. Motivated by this, Jia, Svensson, and Yuan show that the extension complexity of the exact bipartite perfect matching polytope is exponential in general. Interestingly, their result is true even for the easier odd-red bipartite perfect matching problem. For this problem, they introduce an exponential-size relaxation and leave open whether it is an exact description. Apart from showing that this description is not exact and even hard to separate over, we show, more importantly, that the red-odd bipartite perfect matching polytope exhibits complex facet structure: any exact description needs constraints with large and diverse coefficients. This rules out classical relaxations based on constraints with all coefficients in $\{0,\pm1\}$, such as the above-mentioned one, and suggests that significant deviations from prior approaches may be needed to obtain an exact description. More generally, we obtain that also polytopes corresponding to bimodular integer programs have complex facet structure.

cs.DS↗

arXiv2025

Toward Optimal Approximations for Resource-Minimization for Fire Containment on Trees and Non-Uniform k-Center

One of the most elementary spreading models on graphs can be described by a fire spreading from a burning vertex in discrete time steps. At each step, all neighbors of burning vertices catch fire. A well-studied extension to model fire containment is to allow for fireproofing a number $B$ of non-burning vertices at each step. Interestingly, basic computational questions about this model are computationally hard even on trees. One of the most prominent such examples is Resource Minimization for Fire Containment (RMFC), which asks how small $B$ can be chosen so that a given subset of vertices will never catch fire. Despite recent progress on RMFC on trees, prior work left a significant gap in terms of its approximability. We close this gap by providing an optimal $2$-approximation and an asymptotic PTAS, resolving two open questions in the literature. Both results are obtained in a unified way, by first designing a PTAS for a smooth variant of RMFC, which is obtained through a careful LP-guided enumeration procedure. Moreover, we show that our new techniques, with several additional ingredients, carry over to the non-uniform $k$-center problem (NUkC), by exploiting a link between RMFC on trees and NUkC established by Chakrabarty, Goyal, and Krishnaswamy. This leads to the first approximation algorithm for NUkC that is optimal in terms of the number of additional centers that have to be opened.

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arXiv2025

Short circuit walks in fixed dimension

Circuit augmentation schemes are a family of combinatorial algorithms for linear programming that generalize the simplex method. To solve the linear program, they construct a so-called monotone circuit walk: They start at an initial vertex of the feasible region and traverse a discrete sequence of points on the boundary, while moving along certain allowed directions (circuits) and improving the objective function at each step until reaching an optimum. Since the existence of short circuit walks has been conjectured (Circuit Diameter Conjecture), several works have investigated how well one can efficiently approximate shortest monotone circuit walks towards an optimum. A first result addressing this question was given by De Loera, Kafer, and Sanità [SIAM J. Opt., 2022], who showed that given as input an LP and the starting vertex, finding a $2$-approximation for this problem is NP-hard. Cardinal and the third author [Math. Prog. 2023] gave a stronger lower bound assuming the exponential time hypothesis, showing that even an approximation factor of $O(\frac{\log m}{\log \log m})$ is intractable for LPs defined by $m$ inequalities. Both of these results were based on reductions from highly degenerate polytopes in combinatorial optimization with high dimension. In this paper, we significantly strengthen the aforementioned hardness results by showing that for every fixed $\varepsilon>0$ approximating the problem on polygons with $m$ edges to within a factor of $O(m^{1-\varepsilon})$ is NP-hard. This result is essentially best-possible, as it cannot be improved beyond $o(m)$. In particular, this implies hardness for simple polytopes and in fixed dimension.

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arXiv2025

Complexity of polytope diameters via perfect matchings

The Circuit diameter of polytopes was introduced by Borgwardt, Finhold and Hemmecke as a fundamental tool for the study of circuit augmentation schemes for linear programming and for estimating combinatorial diameters. Determining the complexity of computing the circuit diameter of polytopes was posed as an open problem by Sanità as well as by Kafer, and was recently reiterated by Borgwardt, Grewe, Kafer, Lee and Sanità. In this paper, we solve this problem by showing that computing the circuit diameter of a polytope given in halfspace-description is strongly NP-hard. To prove this result, we show that computing the combinatorial diameter of the perfect matching polytope of a bipartite graph is NP-hard. This complements a result by Sanità (FOCS 2018) on the NP-hardness of computing the diameter of fractional matching polytopes and implies the new result that computing the diameter of a $\{0,1\}$-polytope is strongly NP-hard, which may be of independent interest. In our second main result, we give a precise graph-theoretic description of the monotone diameter of perfect matching polytopes and use this description to prove that computing the monotone (circuit) diameter of a given input polytope is strongly NP-hard as well.

math.OC↗

arXiv2024

Bike network planning in limited urban space

The lack of cycling infrastructure in urban environments hinders the adoption of cycling as a viable mode for commuting, despite the evident benefits of (e-)bikes as sustainable, efficient, and health-promoting transportation modes. Bike network planning is a tedious process, relying on heuristic computational methods that frequently overlook the broader implications of introducing new cycling infrastructure, in particular the necessity to repurpose car lanes. In this work, we call for optimizing the trade-off between bike and car networks, effectively pushing for Pareto optimality. This shift in perspective gives rise to a novel linear programming formulation towards optimal bike network allocation. Our experiments, conducted using both real-world and synthetic data, testify the effectiveness and superiority of this optimization approach compared to heuristic methods. In particular, the framework provides stakeholders with a range of lane reallocation scenarios, illustrating potential bike network enhancements and their implications for car infrastructure. Crucially, our approach is adaptable to various bikeability and car accessibility evaluation criteria, making our tool a highly flexible and scalable resource for urban planning. This paper presents an advanced decision-support framework that can significantly aid urban planners in making informed decisions on cycling infrastructure development.

math.OC↗

arXiv2023

Totally odd depth-graded multiple zeta values and period polynomials

Inspired by a paper of Tasaka, we study the relations between totally odd, motivic depth-graded multiple zeta values. Our main objective is to determine the rank of the matrix $C_{N,r}$ defined by Brown. We will give new proofs for (conjecturally optimal) upper bounds on the rank of $C_{N,3}$ and $C_{N,4}$, which were first obtained by Tasaka. Finally, we present a recursive approach to the general problem, which reduces evaluating the rank of $C_{N,r}$ to an isomorphism conjecture.

math.NT↗

arXiv2023

Advances on Strictly $Δ$-Modular IPs

There has been significant work recently on integer programs (IPs) $\min\{c^\top x \colon Ax\leq b,\,x\in \mathbb{Z}^n\}$ with a constraint marix $A$ with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant $Δ\in \mathbb{Z}_{>0}$, $Δ$-modular IPs are efficiently solvable, which are IPs where the constraint matrix $A\in \mathbb{Z}^{m\times n}$ has full column rank and all $n\times n$ minors of $A$ are within $\{-Δ, \dots, Δ\}$. Previous progress on this question, in particular for $Δ=2$, relies on algorithms that solve an important special case, namely strictly $Δ$-modular IPs, which further restrict the $n\times n$ minors of $A$ to be within $\{-Δ, 0, Δ\}$. Even for $Δ=2$, such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly $Δ$-modular IPs. Prior advances were restricted to prime $Δ$, which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly $Δ$-modular IPs in strongly polynomial time if $Δ\leq4$.

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