An $n$-Venn diagram is a diagram in the plane consisting of $n$ simple closed curves that intersect only finitely many times such that each of the $2^n$ possible intersections is represented by a single connected region. An $n$-Venn diagram has at most $2^n-2$ crossings, and if this maximum number of crossings is attained, then only two curves intersect in every crossing. To complement this, Bultena and Ruskey considered $n$-Venn diagrams that minimize the number of crossings, which implies that many curves intersect in every crossing. Specifically, they proved that the total number of crossings in any $n$-Venn diagram is at least $L_n:=\lceil\frac{2^n-2}{n-1}\rceil$, and if this lower bound is attained then essentially all $n$ curves intersect in every crossing. Diagrams achieving this bound are called minimum Venn diagrams, and are known only for $n\leq 7$. Bultena and Ruskey conjectured that they exist for all $n\geq 8$. In this work, we establish an asympototic version of their conjecture. For $n=8$ we construct a diagram with 40 crossings, only 3 more than the lower bound $L_8=37$. Furthermore, for every $n$ of the form $n=2^k$ for some integer $k\geq 4$, we construct an $n$-Venn diagram with at most $(1+\frac{33}{8n})L_n=(1+o(1))L_n$ many crossings. Via a doubling trick this also gives $(n+m)$-Venn diagrams for all $0\leq m

In 1984, Winkler conjectured that every simple Venn diagram with n curves can be extended to a simple Venn diagram with n+1 curves. This conjecture is equivalent to the statement that the dual graph of any simple Venn diagram has a Hamilton cycle. In this work, we construct counterexamples to Winkler’s conjecture for all n ≥ 6. As part of this proof, we computed all 3.430.404 simple Venn diagrams with n = 6 curves (even their number was not previously known), among which we found 72 counterexamples. We also disprove another conjecture about the Hamiltonicity of the arrangement graph of a Venn diagram. Specifically, while working on Winkler’s conjecture, Pruesse and Ruskey proved that this graph has a Hamilton cycle for every simple Venn diagram with n curves, and conjectured that this also holds for non-simple diagrams. We construct counterexamples to this conjecture for all n ≥ 4.

For an arrangement \(\mathcal{H}\) of hyperplanes in \(\mathbb{R}^n\) through the origin, a region is a connected subset of \(\mathbb{R}^n \setminus \mathcal{H}\). The graph of regions \(G(\mathcal{H})\) has a vertex for every region, and an edge between any two vertices whose corresponding regions are separated by a single hyperplane from \(\mathcal{H}\). We aim to compute a Hamiltonian path or cycle in the graph \(G(\mathcal{H})\), i.e., a path or cycle that visits every vertex (=region) exactly once. Our first main result is that if \(\mathcal{H}\) is a supersolvable arrangement, then the graph of regions \(G(\mathcal{H})\) has a Hamiltonian cycle. More generally, we consider quotients of lattice congruences of the poset of regions \(\mathsf{P}(\mathcal{H}, R_0)\), obtained by orienting the graph \(G(\mathcal{H})\) away from a particular base region \(R_0\). Our second main result is that if \(\mathcal{H}\) is supersolvable and \(R_0\) is a canonical base region, then for any lattice congruence \(\equiv\) on \(\mathsf{P}(\mathcal{H}, R_0) =: L\), the cover graph of the quotient lattice \(L/{\equiv}\) has a Hamiltonian path.

Symmetry reduction is crucial for solving many interesting SAT instances in practice. Numerous approaches have been proposed, which try to strike a balance between symmetry reduction and computational overhead. Arguably the most readily applicable method is the computation of static symmetry breaking constraints: a constraint restricting the search-space to non-symmetrical solutions is added to a given SAT instance. A distinct advantage of static symmetry breaking is that the SAT solver itself is not modified. A disadvantage is that the strength of symmetry reduction is usually limited. In order to boost symmetry reduction, the state-of-the-art tool BreakID (Devriendt et al., 2016) pioneered the identification and tailored breaking of a particular substructure of symmetries, the so-called row interchangeability groups. In this paper, we propose a new symmetry breaking tool called satsuma. The core principle of our tool is to exploit more diverse but frequently occurring symmetry structures. This is enabled by new practical detection algorithms for row interchangeability, row-column symmetry, Johnson symmetry, and various combinations. Based on the resulting structural description, we then produce symmetry breaking constraints. We provide benchmarks testing the effectiveness of our new implementation in conjunction with the state-of-the-art SAT solver Kissat. To this end, we compare satsuma, BreakID, and using no symmetry breaking. We find that satsuma successfully speeds up Kissat on last year’s SAT competition instances. Compared to BreakID, we observe significantly better breaking performance on instances with Johnson symmetry, and lower computational overhead across all tested families.

We study the problem of reconfiguring odd matchings, that is, matchings that cover all but a single vertex. Our reconfiguration operation is a so-called flip where the unmatched vertex of the first matching gets matched, while consequently another vertex becomes unmatched. We consider two distinct settings: the geometric setting, in which the vertices are points embedded in the plane and all occurring odd matchings are crossing-free, and a combinatorial setting, in which we consider odd matchings in general graphs. For the latter setting, we provide a complete polynomial time checkable characterization of graphs in which any two odd matchings can be reconfigured into each another. This complements the previously known result that the flip graph is always connected in the geometric setting [Oswin Aichholzer et al., 2025]. In the combinatorial setting, we prove that the diameter of the flip graph, if connected, is linear in the number of vertices. Furthermore, we establish that deciding whether there exists a flip sequence of length k transforming one given matching into another is NP-complete in both the combinatorial and the geometric settings. To prove the latter, we introduce a framework that allows us to transform partial order types into general position with only polynomial overhead. Finally, we demonstrate that when parameterized by the flip distance k, the problem is fixed-parameter tractable (FPT) in the geometric setting when restricted to convex point sets.

We initiate the study of Hamiltonian cycles up to symmetries of the underlying graph. Our focus lies on the extremal case of Hamiltonian-transitive graphs, i.e., Hamiltonian graphs where, for every pair of Hamiltonian cycles, there is a graph automorphism mapping one cycle to the other. This generalizes the extensively studied uniquely Hamiltonian graphs. In this paper, we show that Cayley graphs of abelian groups are not Hamiltonian-transitive (under some mild conditions and some non-surprising exceptions), i.e., they contain at least two structurally different Hamiltonian cycles. To show this, we reduce Hamiltonian-transitivity to properties of the prime factors of a Cartesian product decomposition, which we believe is interesting in its own right. We complement our results by constructing infinite families of regular Hamiltonian-transitive graphs and take a look at the opposite extremal case by constructing a family with many different Hamiltonian cycles up to symmetry.

Symmetry breaking is a widely popular approach to enhance solvers in constraint programming, such as those for SAT or MIP. Symmetry breaking predicates (SBPs) typically impose an order on variables and single out the lexicographic leader (lex-leader) in each orbit of assignments. Although it is NP-hard to find complete lex-leader SBPs, incomplete lex-leader SBPs are widely used in practice. In this paper, we investigate the complexity of computing complete SBPs, lex-leader or otherwise, for SAT. Our main result proves a natural barrier for efficiently computing SBPs: efficient certification of graph non-isomorphism. Our results explain the difficulty of obtaining short SBPs for important CP problems, such as matrix-models with row-column symmetries and graph generation problems. Our results hold even when SBPs are allowed to introduce additional variables. We show polynomial upper bounds for breaking certain symmetry groups, namely automorphism groups of trees and wreath products of groups with efficient SBPs.

Symmetry reduction is crucial for solving many interesting SAT instances in practice. Numerous approaches have been proposed, which try to strike a balance between symmetry reduction and computational overhead. Arguably the most readily applicable method is the computation of static symmetry breaking constraints: a constraint restricting the search-space to non-symmetrical solutions is added to a given SAT instance. A distinct advantage of static symmetry breaking is that the SAT solver itself is not modified. A disadvantage is that the strength of symmetry reduction is usually limited. In order to boost symmetry reduction, the state-of-the-art tool BreakID [Devriendt et. al] pioneered the identification and tailored breaking of a particular substructure of symmetries, the so-called row interchangeability groups. In this paper, we propose a new symmetry breaking tool called satsuma. The core principle of our tool is to exploit more diverse but frequently occurring symmetry structures. This is enabled by new practical detection algorithms for row interchangeability, row-column symmetry, Johnson symmetry, and various combinations. Based on the resulting structural description, we then produce symmetry breaking constraints. We compare this new approach to BreakID on a range of instance families exhibiting symmetry. Our benchmarks suggest improved symmetry reduction in the presence of Johnson symmetry and comparable performance in the presence of row-column symmetry. Moreover, our implementation runs significantly faster, even though it identifies more diverse structures.