arXiv subjects

Steffen Borgwardt

Publications and source records attributed to Steffen Borgwardt.

At least 19 recordsLinked to original sources

arXiv2026

A Canceling Heuristic for the Directed Traveling Salesman Problem

The Traveling Salesman Problem (TSP) is one of the classic and hard problems in combinatorial optimization. We develop a new heuristic that uses a connection between Minimum Cost Flow Problems and the TSP to improve on a given suboptimal tour, such as a local optimum found using a classic heuristic. Minimum Cost Flow Problems can be solved efficiently through linear programming or combinatorial algorithms based on cycle canceling. We investigate the potential of flow-canceling in the context of the TSP. Through a restriction of the search space to cycles and circulations that alternate between arcs in- and outside of the tour, practical results exhibit that only a low number of subtours is created, and a lightweight patching step suffices for a high success rate and gap closure towards an optimum.

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arXiv2025

On Circuit Imbalance and 0/1 Circuits for Coloring and Spanning Forest Problems

Circuits are fundamental objects in linear programming and oriented matroid theory, representing the elementary difference vectors of a polyhedron between points in its affine space. A recent concept introduced by Ekbatani, Natura, and Végh, the circuit imbalance, serves as a complexity measure relevant to iteration bounds for circuit-based augmentation and circuit diameters, as well as the general interpretability of circuits in terms of the underlying application. In this paper, we analyze linear programming formulations of relaxed combinatorial optimization problems to prove two contrasting types of results related to the circuit imbalance. On one hand, we identify simple and common constraint structures, in particular arising in graph-theoretic problems, that inherently lead to an exponential circuit imbalance. These constructions show that, in quite general situations, working with the entire set of circuits poses significant challenges for an application of circuit augmentation or the study of circuit diameters. On the other hand, through a case study of two classic graph-theoretic problems with exponential imbalance, the vertex graph coloring problem and the maximum weight forest problem, we exhibit the existence of sets and subsets of highly interpretable circuits of (best-case) imbalance 1. These sets correspond to the recoloring of vertices or to the addition or removal of edges, respectively, for example generalizing classic concepts of Kempe dynamics in coloring. Their interpretability in terms of the underlying application facilitates a study of circuit walks in the corresponding polytopes. We prove that a restriction of circuit walks to these sets suffices to not only guarantee reachability of the integral extreme-points of the skeleton, but leads to linear and constant circuit diameter bounds, respectively.

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arXiv2025

Reconstruction of C_4-free graphs from the set of closed neighborhoods and digital convexity

Fomin, Kratochvíl, Lokshtanov, Mancini, and Telle showed that every $C_{4}$-free graph is reconstructible from the \emph{multiset} of closed neighborhoods. We strengthen their result proving that every $C_{4}$-free graph is reconstructible from the \emph{set} of closed neighborhoods. This extends the work of Lafrance et al.\ by showing that all $C_{4}$-free graphs, and hence all graphs of girth at least five, are reconstructible from their digitally convex sets. A subset $S$ of vertices in a graph $G$ is digitally convex if, for every vertex $v \notin S$, there is a private neighbor of $v$. We establish that reconstruction from digitally convex sets is equivalent to reconstruction from the set of closed neighborhoods.

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arXiv2025

Improved Decomposition Bounds for Partition Polytopes and Odd-Covers

The assignments of a set of $m$ items into $n$ clusters of prescribed sizes $k_1,\dots,k_n$ can be encoded as the vertices of the partition polytope $\mathrm{PP}(k_1,\dots,k_n)$. We prove that, if $K = \max\{k_1,\dots,k_n\}$, then the combinatorial diameter of $\mathrm{PP}(k_1,\dots,k_n)$ is at most $\lceil 3K/2\rceil$. This improves the previously known upper bound of $2K$. A cycle (or path) odd-cover of a graph $G$ is a set of cycles (or paths) with symmetric difference $G$. We prove that every Eulerian graph $G$ with maximum degree $Δ$ admits a cycle odd-cover and a path odd-cover, each of size at most $\lceil 3Δ/4\rceil$. This improves the previously known upper bound of $Δ$. The two proofs share many similarities and are both based on the proof of Akiyama, Exoo, and Harary that every graph with maximum degree 4 has linear arboricity at most 3.

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arXiv2025

Evacuation Planning on Time-Expanded Networks with Integrated Wildfire Information

We study the problem of evacuation planning for natural disasters, focusing on wildfire evacuations. By creating pre-planned evacuation routes that can be updated based on real-time data, we provide an easily adjustable approach to evacuation planning and implementation. Our method uses publicly available data and can be tailored for a particular region or circumstance. We formulate large-scale evacuations as maximum flow problems on time-expanded networks, in which we integrate hazard information given in the form of a shapefile. An initial flow and evacuation plan is found based on a predicted fire, and is then updated based on revised fire information received during the evacuation. We provide a proof of concept on three locations with historic deadly fires using data available through OpenStreetMaps, a basemap for a geographic information system (GIS), on a NetworkX Python script. The results validate viable running times and quality of information for application in practice. Particular strengths are the scalability and modularity of our approach and accompanying software package.

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arXiv2024

On the Circuit Diameter Conjecture for Counterexamples to the Hirsch Conjecture

Circuit diameters of polyhedra are a fundamental tool for studying the complexity of circuit augmentation schemes for linear programming and for finding lower bounds on combinatorial diameters. The main open problem in this area is the circuit diameter conjecture, the analogue of the Hirsch conjecture in the circuit setting. A natural question is whether the well-known counterexamples to the Hirsch conjecture carry over. Previously, Stephen and Yusun showed that the Klee-Walkup counterexample to the unbounded Hirsch conjecture does not transfer to the circuit setting. Our main contribution is to show that the original counterexamples for the other variants, for bounded polytopes and using monotone walks, also do not transfer. Our results rely on new observations on structural properties of these counterexamples. To resolve the bounded case, we exploit the geometry of certain $2$-faces of the polytopes underlying all known bounded Hirsch counterexamples in Santos' work. For Todd's monotone Hirsch counterexample, we provide two alternative approaches. The first one uses sign-compatible circuit walks, and the second one uses the observation that Todd's polytope is anti-blocking. Along the way, we enumerate all linear programs over the polytope and find four new orientations that contradict the monotone Hirsch conjecture, while the remaining $7107$ satisfy the bound.

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arXiv2024

On the Diameter of a 2-Sum of Polyhedra

The study of the combinatorial diameter of a polyhedron is a classical topic in linear-programming theory due to its close connection with the possibility of a polynomial simplex-method pivot rule. The 2-sum operation is a classical operation for graphs, matrices, and matroids; we extend this definition to polyhedra. We analyze the diameters of 2-sum polyhedra, which are those polyhedra that arise from this operation. These polyhedra appear in matroid and integer-programming theory as a natural way to link two systems in a joint model with a single shared constraint and the 2-sum also appears as a key operation in Seymour's decomposition theorem for totally-unimodular matrices. We show that the diameter of a 2-sum polyhedron is quadratic in the diameters of its summands. The methods transfer to a linear bound for the addition of a unit column to an equality system, or equivalently, to the relaxation of an equality constraint to an inequality constraint. Further, we use our methods to analyze the distance between vertices on certain faces of a 3-sum polyhedron.

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arXiv2024

Combinatorics of generalized parking-function polytopes

For $\mathbf{b}=(b_1,\dots,b_n)\in \mathbb{Z}_{>0}^n$, a $\mathbf{b}$-parking function is defined to be a sequence $(β_1,\dots,β_n)$ of positive integers whose nondecreasing rearrangement $β'_1\leq β'_2\leq \cdots \leq β'_n$ satisfies $β'_i\leq b_1+\cdots + b_i$. The $\mathbf{b}$-parking-function polytope $\mathfrak{X}_n(\mathbf{b})$ is the convex hull of all $\mathbf{b}$-parking functions of length $n$ in $\mathbb{R}^n$. Geometric properties of $\mathfrak{X}_n(\mathbf{b})$ were previously explored in the specific case where $\mathbf{b}=(a,b,b,\dots,b)$ and were shown to generalize those of the classical parking-function polytope. In this work, we study $\mathfrak{X}_n(\mathbf{b})$ in full generality. We present a minimal inequality and vertex description for $\mathfrak{X}_n(\mathbf{b})$, prove it is a generalized permutahedron, and study its $h$-polynomial. Furthermore, we investigate $\mathfrak{X}_n(\mathbf{b})$ through the perspectives of building sets and polymatroids, allowing us to identify its combinatorial types and obtain bounds on its combinatorial and circuit diameters.

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arXiv2024

On the Hardness of Short and Sign-Compatible Circuit Walks

The circuits of a polyhedron are a superset of its edge directions. Circuit walks, a sequence of steps along circuits, generalize edge walks and are "short" if they have few steps or small total length. Both interpretations of short are relevant to the theory and application of linear programming. We study the hardness of several problems relating to the construction of short circuit walks. We establish that for a pair of vertices of a $0/1$-network-flow polytope, it is NP-complete to determine the length of a shortest circuit walk, even if we add the requirement that the walk must be sign-compatible. Our results also imply that determining the minimal number of circuits needed for a sign-compatible decomposition is NP-complete. Further, we show that it is NP-complete to determine the smallest total length (for $p$-norms $\lVert \cdot \rVert_p$, $1 < p \leq \infty$) of a circuit walk between a pair of vertices. One method to construct a short circuit walk is to pick up a correct facet at each step, which generalizes a non-revisiting walk. We prove that it is NP-complete to determine if there is a circuit direction that picks up a correct facet; in contrast, this problem can be solved in polynomial time for TU polyhedra.

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arXiv2023

On Combinatorial Network Flows Algorithms and Circuit Augmentation for Pseudoflows

There is a wealth of combinatorial algorithms for classical min-cost flow problems and their simpler variants like max flow or shortest path problems. It is well-known that many of these algorithms are related to the Simplex method and the more general circuit augmentation schemes: prime examples are the network Simplex method, a refinement of the primal Simplex method, and min-mean cycle canceling, which corresponds to a steepest-descent circuit augmentation scheme. We are interested in a deeper understanding of the relationship between circuit augmentation and combinatorial network flows algorithms. To this end, we generalize from the consideration of primal or dual flows to so-called pseudoflows, which adhere to arc capacities but allow for a violation of flow balance. We introduce `pseudoflow polyhedra,' in which slack variables are used to quantify this violation, and characterize their circuits. This enables the study of combinatorial network flows algorithms in view of the walks that they trace in these polyhedra, and in view of the pivot rules for the steps. In doing so, we provide an `umbrella,' a general framework, that captures several algorithms. We show that the Successive Shortest Path Algorithm for min-cost flow problems, the Shortest Augmenting Path Algorithm for max flow problems, and the Preflow-Push algorithm for max flow problems lead to (non-edge) circuit walks in these polyhedra. The former two are replicated by circuit augmentation schemes for simple pivot rules. Further, we show that the Hungarian Method leads to an edge walk and is replicated, equivalently, as a circuit augmentation scheme or a primal Simplex run for a simple pivot rule.

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arXiv2023

Circuits in Extended Formulations

Circuits and extended formulations are classical concepts in linear programming theory. The circuits of a polyhedron are the elementary difference vectors between feasible points and include all edge directions. We study the connection between the circuits of a polyhedron $P$ and those of an extended formulation of $P$, i.e., a description of a polyhedron $Q$ that linearly projects onto $P$. It is well known that the edge directions of $P$ are images of edge directions of $Q$. We show that this `inheritance' under taking projections does not extend to the set of circuits. We provide counterexamples with a provably minimal number of facets, vertices, and extreme rays, including relevant polytopes from clustering, and show that the difference in the number of circuits that are inherited and those that are not can be exponentially large in the dimension. We further prove that counterexamples exist for any fixed linear projection map, unless the map is injective. Finally, we characterize those polyhedra $P$ whose circuits are inherited from all polyhedra $Q$ that linearly project onto $P$. Conversely, we prove that every polyhedron $Q$ satisfying mild assumptions can be projected in such a way that the image polyhedron $P$ has a circuit with no preimage among the circuits of $Q$. Our proofs build on standard constructions such as homogenization and disjunctive programming.

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arXiv2023

Path Odd-Covers of Graphs

We introduce and study "path odd-covers", a weakening of Gallai's path decomposition problem and a strengthening of the linear arboricity problem. The "path odd-cover number" $p_2(G)$ of a graph $G$ is the minimum cardinality of a collection of paths whose vertex sets are contained in $V(G)$ and whose symmetric difference of edge sets is $E(G)$. We prove an upper bound on $p_2(G)$ in terms of the maximum degree $Δ$ and the number of odd-degree vertices $v_{\text{odd}}$ of the form $\max\left\{v_{\text{odd}}/{2}, 2\left\lceil Δ/{2}\right \rceil\right\}$. This bound is only a factor of $2$ from a rather immediate lower bound of the form $\max \left\{ {v_{\text{odd}} }/{2} , \left\lceil Δ/{2}\right\rceil \right\}$. We also investigate some natural relaxations of the problem which highlight the connection between the path odd-cover number and other well-known graph parameters. For example, when allowing for subdivisions of $G$, the previously mentioned lower bound is always tight except in some trivial cases. Further, a relaxation that allows for the addition of isolated vertices to $G$ leads to a match with the linear arboricity when $G$ is Eulerian. Finally, we transfer our observations to establish analogous results for cycle odd-covers.

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arXiv2023

On the Combinatorial Diameters of Parallel and Series Connections

The investigation of combinatorial diameters of polyhedra is a classical topic in linear programming due to its connection with the possibility of an efficient pivot rule for the simplex method. We are interested in the diameters of polyhedra formed from the so-called parallel or series connection of oriented matroids: oriented matroids are the natural way to connect representable matroid theory with the combinatorics of linear programming, and these connections are fundamental operations for the construction of more complicated matroids from elementary matroid blocks. We prove that, for polyhedra whose combinatorial diameter satisfies the Hirsch-conjecture bound regardless of the right-hand sides in a standard-form description, the diameters of their parallel or series connections remain small in the Hirsch-conjecture bound. These results are a substantial step toward devising a diameter bound for all polyhedra defined through totally-unimodular matrices based on Seymour's famous decomposition theorem. Our proof techniques and results exhibit a number of interesting features. While the parallel connection leads to a bound that adds just a constant, for the series connection one has to linearly take into account the maximal value in a specific coordinate of any vertex. Our proofs also require a careful treatment of non-revisiting edge walks in degenerate polyhedra, as well as the construction of edge walks that may take a `detour' to facets that satisfy the non-revisiting conjecture when the underlying polyhedron may not.

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arXiv2022

An Integer Program for Pricing Support Points of Exact Barycenters

The computation of exact barycenters for a set of discrete measures is of interest in applications where sparse solutions are desired, and to assess the quality of solutions returned by approximate algorithms and heuristics. The task is known to be NP-hard for growing dimension and, even in low dimensions, extremely challenging in practice due to an exponential scaling of the linear programming formulations associated with the search for sparse solutions. A common approach to facilitate practical computations is an approximation based on the choice of a small, fixed set $S_0$ of support points, or a fixed set $S^*_0$ of combinations of support points from the measures, that may be assigned mass. Through a combination of linear and integer programming techniques, we model an integer program to compute additional combinations, and in turn support points, that, when added to $S^*_0$ or $S_0$, allow for a better approximation of the underlying exact barycenter problem. The approach improves on the scalability of a classical column generation approach: instead of a pricing problem that has to evaluate exponentially many reduced cost values, we solve a mixed-integer program of quadratic size. The properties of the model, and practical computations, reveal a tailored branch-and-bound routine as a good solution strategy.

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arXiv2022

A Column Generation Approach to the Discrete Barycenter Problem

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of discrete probability measures. Although an exact barycenter is computable through linear programming, the underlying linear program can be extremely large. For worst-case input, a best known linear programming formulation is exponential in the number of variables, but has a low number of constraints, making it an interesting candidate for column generation. In this paper, we devise and study two column generation strategies: a natural one based on a simplified computation of reduced costs, and one through a Dantzig-Wolfe decomposition. For the latter, we produce efficiently solvable subproblems, namely, a pricing problem in the form of a classical transportation problem. The two strategies begin with an efficient computation of an initial feasible solution. While the structure of the constraints leads to the computation of the reduced costs of all remaining variables for setup, both approaches may outperform a computation using the full program in speed, and dramatically so in memory requirement. In our computational experiments, we exhibit that, depending on the input, either strategy can become a best choice.

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arXiv2022

On the Computational Complexity of Finding a Sparse Wasserstein Barycenter

The discrete Wasserstein barycenter problem is a minimum-cost mass transport problem for a set of probability measures with finite support. In this paper, we show that finding a barycenter of sparse support is hard, even in dimension 2 and for only 3 measures. We prove this claim by showing that a special case of an intimately related decision problem SCMP -- does there exist a measure with a non-mass-splitting transport cost and support size below prescribed bounds? -- is NP-hard for all rational data. Our proof is based on a reduction from planar 3-dimensional matching and follows a strategy laid out by Spieksma and Woeginger (1996) for a reduction to planar, minimum circumference 3-dimensional matching. While we closely mirror the actual steps of their proof, the arguments themselves differ fundamentally due to the complex nature of the discrete barycenter problem. Containment of SCMP in NP will remain open. We prove that, for a given measure, sparsity and cost of an optimal transport to a set of measures can be verified in polynomial time in the size of a bit encoding of the measure. However, the encoding size of a barycenter may be exponential in the encoding size of the underlying measures.

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arXiv2022

An Algorithm for the Separation-Preserving Transition of Clusterings

The separability of clusters is one of the most desired properties in clustering. There is a wide range of settings in which different clusterings of the same data set appear. We are interested in applications where there is a need for an explicit, gradual transition of one separable clustering into another one. This transition should be a sequence of simple, natural steps that upholds separability of the clusters throughout. We design an algorithm for such a transition. We exploit the intimate connection of separability and linear programming over bounded-shape partition and transportation polytopes: separable clusterings lie on the boundary of partition polytopes, form a subset of the vertices of the corresponding transportation polytopes, and circuits of both polytopes are readily interpreted as sequential or cyclical exchanges of items between clusters. This allows for a natural approach to achieve the desired transition through a combination of two walks: an edge walk between two so-called radial clusterings in a transportation polytope, computed through an adaptation of classical tools of sensitivity analysis and parametric programming; and a walk from a separable clustering to a corresponding radial clustering, computed through a tailored, iterative routine updating cluster sizes and re-optimizing the cluster assignment of items.

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arXiv2021

A Note on the Approximability of Deepest-Descent Circuit Steps

Linear programs (LPs) can be solved by polynomially many moves along the circuit direction improving the objective the most, so-called deepest-descent steps (dd-steps). Computing these steps is NP-hard (De Loera et al., arXiv, 2019), a consequence of the hardness of deciding the existence of an optimal circuit-neighbor (OCNP) on LPs with non-unique optima. We prove OCNP is easy under the promise of unique optima, but already $O(n^{1-\varepsilon})$-approximating dd-steps remains hard even for totally unimodular $n$-dimensional 0/1-LPs with a unique optimum. We provide a matching $n$-approximation.

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