The peaceable queens problem asks to determine the maximum number such that there is a placement of white queens and black queens on an chessboard so that no queen can capture any queen of the opposite color. In this paper, we consider the peaceable queens problem and its variant on the toroidal board. For the regular board, we show that , for all sufficiently large . This improves on the bound of van Bommel and MacEachern [16]. For the toroidal board, we provide new upper and lower bounds. Somewhat surprisingly, our bounds show that there is a sharp contrast in behaviour between the odd torus and the even torus. Our lower bounds are given by explicit constructions. For the upper bounds, we formulate the problem as a non-linear optimization problem with at most 100 variables, regardless of the size of the board. We solve our non-linear program exactly using modern optimization software. We also provide a local search algorithm and a software implementation which converges very rapidly to solutions which appear optimal. Our algorithm is sufficiently robust that it works on both the regular and toroidal boards. For example, for the regular board, the algorithm quickly finds the so-called Ainley construction. Thus, our work provides some further evidence that the Ainley construction is indeed optimal. *Matthew Drescher was supported by the National Science Foundation under Grant #2127309 to the Computing Research Association for the CIFellows 2021 Project. This paper has been awarded the “Code and Data Available” and “Results Reproduced” badges as recognition that the author(s) have followed reproducibility principles. Code and data that allow readers to reproduce the results in this paper are available at https://doi.org/10.5281/zenodo.13787471. Participation in the ALENEX artifact evaluation phase was optional and performed at the request of the author(s).

An induced packing of cycles in a graph is a set of vertex-disjoint cycles with no edges between them. We generalise the classic Erdős-Pósa theorem to induced packings of cycles. More specifically, we show that there exists a function f (k) = O (k log k ) such that for every positive integer k, every graph G contains either an induced packing of k cycles or a set X of at most f (k ) vertices such that the closed neighbourhood of X intersects all cycles in G. Our proof is constructive and yields a polynomial-time algorithm finding either the induced packing of cycles or the set X. Furthermore, we show that for every positive integer d, if a graph G does not contain two cycles at distance more than d, then G contains sets X1, X2 ⊆ V (G ) with |X1| ≤ 12(d + 1) and |X2| ≤ 12 such that, after removing the ball of radius 2d around X1 or the ball of radius 3d around X2, the resulting graphs are forests.

Abstract. We give an approximate Menger-type theorem for the case when a graph [Formula: see text] contains two [Formula: see text] paths [Formula: see text] and [Formula: see text] such that [Formula: see text] is an induced subgraph of [Formula: see text]. More generally, we prove that there exists a function [Formula: see text], such that for every graph [Formula: see text] and [Formula: see text], either there exist two [Formula: see text] paths [Formula: see text] and [Formula: see text] such that the distance between [Formula: see text] and [Formula: see text] is at least [Formula: see text], or there exists [Formula: see text] such that the ball of radius [Formula: see text] centered at [Formula: see text] intersects every [Formula: see text] path.

We study the feedback vertex set problem in tournaments from the polyhedral point of view, and in particular we show that performing just one round of the Sherali–Adams hierarchy gives a relaxation with integrality gap 7/3. This allows us to derive a 7/3-approximation algorithm for the feedback vertex set problem in tournaments that matches the best deterministic approximation guarantee due to Mnich, Williams, and Végh, and is a simplification and runtime improvement of their approach.

A metric graph is a pair (G, d), where G is a graph and d: E(G) → R≥ 0 is a distance function. Let p∈ [1 ,∞] be fixed. An isometric embedding of the metric graph (G, d) in ℓpk=(Rk,dp) is a map ϕ: V(G) → Rk such that dp(ϕ(v) ,ϕ(w)) = d(vw) for all edges vw∈ E(G). The ℓp-dimension of G is the least integer k such that there exists an isometric embedding of (G, d) in ℓpk for all distance functions d such that (G, d) has an isometric embedding in ℓpK for some K. It is easy to show that ℓp-dimension is a minor-monotone property. In this paper, we characterize the minor-closed graph classes C with bounded ℓp-dimension, for p∈ { 2 ,∞}. For p= 2 ,we give a simple proof that C has bounded ℓ2-dimension if and only if C has bounded treewidth. In this sense, the ℓ2-dimension of a graph is ‘tied’ to its treewidth. For p= ∞, the situation is completely different. Our main result states that a minor-closed class C has bounded ℓ∞-dimension if and only if C excludes a graph obtained by joining copies of K4 using the 2-sum operation, or excludes a Möbius ladder with one ‘horizontal edge’ removed.

A clutter is k-wise intersecting if every k members have a common element, yet no element belongs to all members. We conjecture that every 4-wise intersecting clutter is non-ideal. As evidence for our conjecture, we prove it in the binary case. Two key ingredients for our proof are Jaeger’s 8-flow theorem for graphs, and Seymour’s characterization of the binary matroids with the sums of circuits property. As further evidence for our conjecture, we also note that it follows from an unpublished conjecture of Seymour from 1975.

Consider the family of graphs without k node-disjoint odd cycles, where k is a constant. Determining the complexity of the stable set problem for such graphs G is a long-standing problem. We give a polynomial-time algorithm for the case that G can be further embedded in a (possibly nonorientable) surface of bounded genus. Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that 2-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This extends the fact that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface (Kawarabayashi & Nakamoto, 2007). Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost nonnegative integer circulation of a certain homology class, which turns out to be efficiently solvable in our case.

We prove that for every proper minor-closed class $\mathcal{M}$ of $\mathbb{F}_p$-representable matroids, there exists an $O(1)$-competitive algorithm for the matroid secretary problem on $\mathcal{M}$. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards, and Whittle. We also note that, for asymptotically almost all matroids, the matroid secretary algorithm that selects a random basis, ignoring weights, is $(2+o(1))$-competitive. In fact, assuming the conjecture that almost all matroids are paving, there is a $(1+o(1))$-competitive algorithm for almost all matroids.

Let H be a planar graph. By a classical result of Robertson and Seymour, there is a function f : ℕ → ℝ such that for all k ∊ ℕ and all graphs G, either G contains k vertex-disjoint subgraphs each containing H as a minor, or there is a subset X of at most f(k) vertices such that G–X has no H-minor. We prove that this remains true with f(k) = ck log k for some constant c = c(H). This bound is best possible, up to the value of c, and improves upon a recent result of Chekuri and Chuzhoy [STOC 2013], who established this with f(k) = ck logd k for some universal constant d. The proof is constructive and yields a polynomial-time O(log OPT)-approximation algorithm for packing subgraphs containing an H-minor.

Abstract Flip graphs are a ubiquitous class of graphs, which encode relations on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called $$\alpha$$ α -orientations of a graph G, in which every vertex v has a specified outdegree $$\alpha (v)$$ α ( v ) , and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two $$\alpha$$ α -orientations of a planar graph G is at most two is -complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance between graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang et al. (Acta Math Sin Engl Ser 35(4):569–576, 2019).

Let $W_t$ denote the wheel on t+1 vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k \geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as a minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.

Let $G$ be a graph and $p \in [1, \infty]$. The parameter $f_p(G)$ is the least integer $k$ such that for all $m$ and all vectors $(r_v)_{v \in V(G)} \subseteq \mathbb{R}^m$, there exist vectors $(q_v)_{v \in V(G)} \subseteq \mathbb{R}^k$ satisfying $\|r_v-r_w\|_p=\|q_v-q_w\|_p$ for all $vw\in E(G).$ It is easy to check that $f_p(G)$ is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour [J. Combin. Theory Ser. B, 92 (2004), pp. 325--357], there are a finite number of excluded minors for the property $f_p(G) \leq k$. In this paper, we determine the complete set of excluded minors for $f_\infty(G) \leq 2$. The two excluded minors are the wheel on five vertices and the graph obtained by gluing two copies of $K_4$ along an edge and then deleting that edge. We also show that the same two graphs are the complete set of excluded minors for $f_1(G) \leq 2$. In addition, we give a family of examples that show that $f_\infty$ is unbounded on the class of planar graphs and $f_\infty$ is not bounded as a function of tree-width.

Let $M$ be a representable matroid and $Q, R, S, T$ subsets of the ground set such that the smallest separation that separates $Q$ from $R$ has order $k$, and the smallest separation that separates $S$ from $T$ has order $l$. We prove that if $M$ is sufficiently large, then there is an element $e$ such that in one of $M\backslash e$ and $M\!/e$ both connectivities are preserved. For matroids representable over a finite field we prove a stronger result: we show that we can remove $e$ such that both a connectivity and a minor of $M$ are preserved. (A corrected version is attached.)