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 Sanita [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.

We present several algorithms to generate tables for asymmetric numeral systems and prove that they are optimal in terms of discrepancy. In turn, this gives rise to the strongest proven bound on entropy loss. We further give improved theoretical bounds for the entropy loss in tabled asymmetric numeral systems and a brief empirical evaluation of the stream variant.

The (monotone) diameter of a polytope is a fundamental parameter with important connections to the efficiency of the simplex method. Despite the central role played by this parameter in discrete and linear optimization, determining the precise complexity of computing the diameter of an input polytope remains a long-standing open problem. In 1984 Frieze and Teng [FT94] proved the first cornerstone result in this direction by establishing that computing the diameter of an input polytope is weakly NP-hard. In a recent breakthrough- paper, Sanita (FOCS 2018, [San18]) studied the diameter of a special class of graph-based polytopes, known as fractional matching polytopes, and showed that determining their diameters is NP-hard, thus establishing strong NP-hardness of computing the diameter of polytopes.

Abstract A set $${\mathcal {G}}$$ G of planar graphs on the same number n of vertices is called simultaneously embeddable if there exists a set P of n points in the plane such that every graph $$G \in {\mathcal {G}}$$ G ∈ G admits a (crossing-free) straight-line embedding with vertices placed at points of P. A conflict collection is a set of planar graphs of the same order with no simultaneous embedding. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether there exists a conflict collection of size 2. While this remains widely open, we give a short proof that for sufficiently large n there exists a conflict collection consisting of at most $$(3+o(1))\log _2(n)$$ ( 3 + o ( 1 ) ) log 2 ( n ) planar graphs on n vertices. This constitutes a double-exponential improvement over the previously best known bound of $$O(n\cdot 4^{n/11})$$ O ( n · 4 n / 11 ) for the same problem by Goenka et al. (Graphs Combin 39:100, 2023). Using our method we also provide a computer-free proof that for every integer $$n\in [107,193]$$ n ∈ [ 107 , 193 ] there exists a conflict collection of 30 planar n-vertex graphs, improving upon the previously smallest known conflict collection consisting of 49 graphs of order 11, which was found using heavy computer assistance. While the construction by Goenka et al. was explicit, our construction of a conflict collection of size $$O(\log n)$$ O ( log n ) is based on the probabilistic method and is thus only implicit. Motivated by this, for every large enough n we give a different, fully explicit construction of a collection of less than $$n^6$$ n 6 planar n-vertex graphs with no simultaneous embedding.

Abstract. The cochromatic number [Formula: see text] of a graph [Formula: see text] is the smallest number of colors in a vertex-coloring of [Formula: see text] such that every color class forms an independent set or a clique. In three papers written around 1990, Erdős, Gimbel, and collaborators raised several open problems regarding the relationship of the chromatic and cochromatic number of a graph. In this short note, we address several of these problems; in particular, we disprove a 36-year-old conjecture of Erdős, Gimbel, and Straight, answer negatively a problem posed by Erdős and Gimbel in 1993, and give positive evidence for a prize money question of Erdős and Gimbel.

A finite set $P$ of points in the plane is $n$-universal with respect to a class $\mathcal{C}$ of planar graphs if every $n$-vertex graph in $\mathcal{C}$ admits a crossing-free straight-line drawing with vertices at points of $P$. For the class of all planar graphs the best known upper bound on the size of a universal point set is quadratic and the best known lower bound is linear in $n$. Some classes of planar graphs are known to admit universal point sets of near linear size, however, there are no truly linear bounds for interesting classes beyond outerplanar graphs. In this paper, we show that there is a universal point set of size $2n-2$ for the class of bipartite planar graphs with $n$ vertices. The same point set is also universal for the class of $n$-vertex planar graphs of maximum degree $3$. The point set used for the results is what we call an exploding double chain, and we prove that this point set allows planar straight-line embeddings of many more planar graphs, namely of all subgraphs of planar graphs admitting a one-sided Hamiltonian cycle. The result for bipartite graphs also implies that every $n$-vertex plane graph has a $1$-bend drawing all whose bends and vertices are contained in a specific point set of size $4n-6$, this improves a bound of $6n-10$ for the same problem by Löffler and Tóth.

Given an integer $k$ and a graph where every edge is colored either red or blue, the goal of the exact matching problem is to find a perfect matching with the property that exactly $k$ of its edges are red. Soon after Papadimitriou and Yannakakis (JACM 1982) introduced the problem, a randomized polynomial-time algorithm solving the problem was described by Mulmuley et al. (Combinatorica 1987). Despite a lot of effort, it is still not known today whether a deterministic polynomial-time algorithm exists. This makes the exact matching problem an important candidate to test the popular conjecture that the complexity classes P and RP are equal. In a recent article (MFCS 2022), progress was made towards this goal by showing that for bipartite graphs of bounded bipartite independence number, a polynomial time algorithm exists. In terms of parameterized complexity, this algorithm was an XP-algorithm parameterized by the bipartite independence number. In this article, we introduce novel algorithmic techniques that allow us to obtain an FPT-algorithm. If the input is a general graph we show that one can at least compute a perfect matching $M$ which has the correct number of red edges modulo 2, in polynomial time. This is motivated by our last result, in which we prove that an FPT algorithm for general graphs, parameterized by the independence number, reduces to the problem of finding in polynomial time a perfect matching $M$ with at most $k$ red edges and the correct number of red edges modulo 2.

Abstract. Assume [Formula: see text] players are placed on [Formula: see text] vertices of a graph [Formula: see text]. The following game was introduced by Winkler: An adversary puts a hat on each player, where each hat has a color out of [Formula: see text] available colors. The players can see the hat of each of their neighbors in [Formula: see text] but cannot see their own hats. Using a predetermined guessing strategy, the players then simultaneously guess the color of their hats. The players win if at least one of them guesses correctly; otherwise, the adversary wins. The largest integer [Formula: see text] such that there is a winning strategy for the players is denoted by [Formula: see text], and this is called the hat guessing number of [Formula: see text]. Although this game has received much attention in recent years, not much is known about how the hat guessing number relates to other graph parameters. For instance, a natural open question is whether the hat guessing number can be bounded from above in terms of degeneracy. In this paper, we prove that the hat guessing number of a graph can be bounded from above in terms of a related notion, which we call strong degeneracy. We further give an exact characterization of graphs with bounded strong degeneracy. As a consequence, we significantly improve the best known upper bound on the hat guessing number of outerplanar graphs from [Formula: see text] to 40 and further derive upper bounds on the hat guessing number for any class of [Formula: see text]-free graphs with bounded expansion, such as the class of [Formula: see text]-free planar graphs; more generally, for [Formula: see text]-free graphs with bounded Hadwiger number or without a [Formula: see text]-subdivision; and for Erdős–Rényi random graphs with constant average degree.

In this paper, we disprove the long-standing conjecture that any complete geometric graph on 2n vertices can be partitioned into n plane spanning trees. Our construction is based on so-called bumpy wheel sets. We fully characterize which bumpy wheels can and in particular which cannot be partitioned into plane spanning trees (or even into arbitrary plane subgraphs). Furthermore, we show a sufficient condition for generalized wheels to not admit a partition into plane spanning trees, and give a complete characterization when they admit a partition into plane spanning double stars. Finally, we initiate the study of partitions into beyond planar subgraphs, namely into k-planar and k-quasi-planar subgraphs and obtain first bounds on the number of subgraphs required in this setting.

In the Exact Matching Problem (EM), we are given a graph equipped with a fixed coloring of its edges with two colors (red and blue), as well as a positive integer $k$. The task is then to decide whether the given graph contains a perfect matching exactly $k$ of whose edges have color red. EM generalizes several important algorithmic problems such as perfect matching and restricted minimum weight spanning tree problems. When introducing the problem in 1982, Papadimitriou and Yannakakis conjectured EM to be $\textbf{NP}$-complete. Later however, Mulmuley et al.~presented a randomized polynomial time algorithm for EM, which puts EM in $\textbf{RP}$. Given that to decide whether or not $\textbf{RP}=\textbf{P}$ represents a big open challenge in complexity theory, this makes it unlikely for EM to be $\textbf{NP}$-complete, and in fact indicates the possibility of a deterministic polynomial time algorithm. EM remains one of the few natural combinatorial problems in $\textbf{RP}$ which are not known to be contained in $\textbf{P}$, making it an interesting instance for testing the hypothesis $\textbf{RP}=\textbf{P}$. Despite EM being quite well-known, attempts to devise deterministic polynomial algorithms have remained illusive during the last 40 years and progress has been lacking even for very restrictive classes of input graphs. In this paper we finally push the frontier of positive results forward by proving that EM can be solved in deterministic polynomial time for input graphs of bounded independence number, and for bipartite input graphs of bounded bipartite independence number. This generalizes previous positive results for complete (bipartite) graphs which were the only known results for EM on dense graphs.

We characterize all digraphs $H$ such that orientations of chordal graphs with no induced copy of $H$ have bounded dichromatic number.

A conjecture by Lichiardopol [ SIAM J. Discrete Math., 28 (2014), pp. 1618--1627] states that for every $k \ge 1$ there exists an integer $g(k)$ such that every digraph of minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. Motivated by Lichiardopol's conjecture, we study the existence of vertex-disjoint directed cycles satisfying length constraints in digraphs of large connectivity or large minimum degree. Our main result is that for every $k \in \mathbb{N}$, there exists $s(k) \in \mathbb{N}$ such that every strongly $s(k)$-connected digraph contains $k$ vertex-disjoint directed cycles of pairwise distinct lengths. In contrast, for every $k \in \mathbb{N}$ we construct a strongly $k$-connected digraph containing no two vertex- or arc-disjoint directed cycles of the same length. It is an open problem whether $g(3)$ exists. Here we prove the existence of an integer $K$ such that every digraph of minimum out- and in-degree at least $K$ contains 3 vertex-disjoint directed cycles of pairwise distinct lengths.

We investigate which planar point sets allow simultaneous straight-line embeddings of all planar graphs on a fixed number of vertices. We first show that at least $(1.293-o(1))n$ points are required to find a straight-line drawing of each $n$-vertex planar graph (vertices are drawn as the given points); this improves the previous best constant $1.235$ by Kurowski (2004). Our second main result is based on exhaustive computer search: We show that no set of 11 points exists, on which all planar 11-vertex graphs can be simultaneously drawn plane straight-line. This strengthens the result by Cardinal, Hoffmann, and Kusters (2015), that all planar graphs on $n \le 10$ vertices can be simultaneously drawn on particular $n$-universal sets of $n$ points while there are no $n$-universal sets of size $n$ for $n \ge 15$. We also provide 49 planar 11-vertex graphs which cannot be simultaneously drawn on any set of 11 points. This, in fact, is another step towards a (negative) answer of the question, whether every two planar graphs can be drawn simultaneously - a question raised by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw, and Mitchell (2007).

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).