arXiv subjects

Jan Kynčl

Publications and source records attributed to Jan Kynčl.

35 records · Page 2Linked to original sources

arXiv2018

Zeros of the Möbius function of permutations

We show that if a permutation $π$ contains two intervals of length 2, where one interval is an ascent and the other a descent, then the Möbius function $μ[π]$ of the interval $[1,π]$ is zero. As a consequence, we show that the proportion of permutations of length $n$ with principal Möbius function equal to zero is asymptotically bounded below by $(1-1/e)^2\ge 0.3995$. This is the first result determining the value of $μ[1,π]$ for an asymptotically positive proportion of permutations $π$. We also show that if a permutation $ϕ$ can be expressed as a direct sum of the form $α\oplus 1 \oplus β$, then any permutation $π$ containing an interval order-isomorphic to $ϕ$ has $μ[1, π]=0$; we deduce this from a more general result showing that $μ[σ, π]=0$ whenever $π$ contains an interval of a certain form. Finally, we show that if a permutation $π$ contains intervals isomorphic to certain pairs of permutations, or to certain permutations of length six, then $μ[1, π] = 0$.

math.CO↗

arXiv2017

Induced Ramsey-type results and binary predicates for point sets

Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Nešetřil and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey. As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Nešetřil and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $Γ$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $Γ$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.

math.CO↗

arXiv2017

Peeling potatoes near-optimally in near-linear time

We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon $P$ with $n$ vertices. We give a randomized near-linear-time $(1-\varepsilon)$-approximation algorithm for this problem: in $O(n( \log^2 n + (1/\varepsilon^3) \log n + 1/\varepsilon^4))$ time we find a convex polygon contained in $P$ that, with probability at least $2/3$, has area at least $(1-\varepsilon)$ times the area of an optimal solution. We also obtain similar results for the variant of computing a convex polygon inside $P$ with maximum perimeter. To achieve these results we provide new results in geometric probability. The first result is a bound relating the probability that two points chosen uniformly at random inside $P$ are mutually visible and the area of the largest convex body inside $P$. The second result is a bound on the expected value of the difference between the perimeter of any planar convex body $K$ and the perimeter of the convex hull of a uniform random sample inside $K$.

cs.CG↗

arXiv2017

Irreversible 2-conversion set in graphs of bounded degree

An irreversible $k$-threshold process (also a $k$-neighbor bootstrap percolation) is a dynamic process on a graph where vertices change color from white to black if they have at least $k$ black neighbors. An irreversible $k$-conversion set of a graph $G$ is a subset $S$ of vertices of $G$ such that the irreversible $k$-threshold process starting with $S$ black eventually changes all vertices of $G$ to black. We show that deciding the existence of an irreversible 2-conversion set of a given size is NP-complete, even for graphs of maximum degree 4, which answers a question of Dreyer and Roberts. Conversely, we show that for graphs of maximum degree 3, the minimum size of an irreversible 2-conversion set can be computed in polynomial time. Moreover, we find an optimal irreversible 3-conversion set for the toroidal grid, simplifying constructions of Pike and Zou.

cs.DM↗

arXiv2017

Unified Hanani-Tutte theorem

We introduce a common generalization of the strong Hanani-Tutte theorem and the weak Hanani-Tutte theorem: if a graph $G$ has a drawing $D$ in the plane where every pair of independent edges crosses an even number of times, then $G$ has a planar drawing preserving the rotation of each vertex whose incident edges cross each other evenly in $D$. The theorem is implicit in the proof of the strong Hanani-Tutte theorem by Pelsmajer, Schaefer and Štefankovič. We give a new, somewhat simpler proof.

math.CO↗

arXiv2017

Near equipartitions of colored point sets

Suppose that $nk$ points in general position in the plane are colored red and blue, with at least $n$ points of each color. We show that then there exist $n$ pairwise disjoint convex sets, each of them containing $k$ of the points, and each of them containing points of both colors. We also show that if $P$ is a set of $n(d+1)$ points in general position in $\mathbb{R}^d$ colored by $d$ colors with at least $n$ points of each color, then there exist $n$ pairwise disjoint $d$-dimensional simplices with vertices in $P$, each of them containing a point of every color. These results can be viewed as a step towards a common generalization of several previously known geometric partitioning results regarding colored point sets.

math.CO↗

arXiv2016

Hardness of Permutation Pattern Matching

Permutation Pattern Matching (or PPM) is a decision problem whose input is a pair of permutations $π$ and $τ$, represented as sequences of integers, and the task is to determine whether $τ$ contains a subsequence order-isomorphic to $π$. Bose, Buss and Lubiw proved that PPM is NP-complete on general inputs. We show that PPM is NP-complete even when $π$ has no decreasing subsequence of length 3 and $τ$ has no decreasing subsequence of length 4. This provides the first known example of PPM being hard when one or both of $π$ and $σ$ are restricted to a proper hereditary class of permutations. This hardness result is tight in the sense that PPM is known to be polynomial when both $π$ and $τ$ avoid a decreasing subsequence of length 3, as well as when $π$ avoids a decreasing subsequence of length 2. The result is also tight in another sense: we will show that for any hereditary proper subclass C of the class of permutations avoiding a decreasing sequence of length 3, there is a polynomial algorithm solving PPM instances where $π$ is from C and $τ$ is arbitrary. We also obtain analogous hardness and tractability results for the class of so-called skew-merged patterns. From these results, we deduce a complexity dichotomy for the PPM problem restricted to $π$ belonging to $Av(ρ)$, where $Av(ρ)$ denotes the class of permutations avoiding a permutation $ρ$. Specifically, we show that the problem is polynomial when $ρ$ is in the set {1, 12, 21, 132, 213, 231, 312}, and it is NP-complete for any other $ρ$.

math.CO↗

arXiv2016

On the nonexistence of $k$-reptile simplices in $\mathbb R^3$ and $\mathbb R^4$

A $d$-dimensional simplex $S$ is called a $k$-reptile (or a $k$-reptile simplex) if it can be tiled by $k$ simplices with disjoint interiors that are all mutually congruent and similar to $S$. For $d=2$, triangular $k$-reptiles exist for all $k$ of the form $a^2, 3a^2$ or $a^2 + b^2$ and they have been completely characterized by Snover, Waiveris, and Williams. On the other hand, the only $k$-reptile simplices that are known for $d \ge 3$, have $k = m^d$, where $m$ is a positive integer. We substantially simplify the proof by Matoušek and the second author that for $d=3$, $k$-reptile tetrahedra can exist only for $k=m^3$. We then prove a weaker analogue of this result for $d=4$ by showing that four-dimensional $k$-reptile simplices can exist only for $k=m^2$.

math.CO↗

arXiv2015

On three measures of non-convexity

The invisibility graph $I(X)$ of a set $X \subseteq \mathbb{R}^d$ is a (possibly infinite) graph whose vertices are the points of $X$ and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in $X$. We consider the following three parameters of a set $X$: the clique number $ω(I(X))$, the chromatic number $χ(I(X))$ and the convexity number $γ(X)$, which is the minimum number of convex subsets of $X$ that cover $X$. We settle a conjecture of Matoušek and Valtr claiming that for every planar set $X$, $γ(X)$ can be bounded in terms of $χ(I(X))$. As a part of the proof we show that a disc with $n$ one-point holes near its boundary has $χ(I(X)) \ge \log\log(n)$ but $ω(I(X))=3$. We also find sets $X$ in $\mathbb{R}^5$ with $χ(X)=2$, but $γ(X)$ arbitrarily large.

math.MG↗

arXiv2015

Clustered planarity testing revisited

The Hanani--Tutte theorem is a classical result proved for the first time in the 1930s that characterizes planar graphs as graphs that admit a drawing in the plane in which every pair of edges not sharing a vertex cross an even number of times. We generalize this result to clustered graphs with two disjoint clusters, and show that a straightforward extension to flat clustered graphs with three or more disjoint clusters is not possible. For general clustered graphs we show a variant of the Hanani--Tutte theorem in the case when each cluster induces a connected subgraph. Di Battista and Frati proved that clustered planarity of embedded clustered graphs whose every face is incident with at most five vertices can be tested in polynomial time. We give a new and short proof of this result, using the matroid intersection algorithm.

cs.CG↗

arXiv2015

Bounds for Pach's selection theorem and for the minimum solid angle in a simplex

We estimate the selection constant in the following geometric selection theorem by Pach: For every positive integer $d$ there is a constant $c_d > 0$ such that whenever $X_1,..., X_{d+1}$ are $n$-element subsets of $\mathbb{R}^d$, then we can find a point $\mathbf{p} \in \mathbb{R}^d$ and subsets $Y_i \subseteq X_i$ for every $i \in [d+1]$, each of size at least $c_d n$, such that $\mathbf{p}$ belongs to all {\em rainbow} $d$-simplices determined by $Y_1,..., Y_{d+1}$, that is, simplices with one vertex in each $Y_i$. We show a super-exponentially decreasing upper bound $c_d\leq e^{-(1/2-o(1))(d \ln d)}$. The ideas used in the proof of the upper bound also help us prove Pach's theorem with $c_d \geq 2^{-2^{d^2 + O(d)}}$, which is a lower bound doubly exponentially decreasing in $d$ (up to some polynomial in the exponent). For comparison, Pach's original approach yields a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and Suk recently obtained a hypergraph density result implying a proof of Pach's theorem with $c_d \geq2^{-O(d^2\log d)}$. In our construction for the upper bound, we use the fact that the minimum solid angle of every $d$-simplex is super-exponentially small. This fact was previously unknown and might be of independent interest. For the lower bound, we improve the "separation" part of the argument by showing that in one of the key steps only $d+1$ separations are necessary, compared to $2^d$ separations in the original proof. We also provide a measure version of Pach's theorem.

math.MG↗

arXiv2014

Crossing numbers and combinatorial characterization of monotone drawings of $K_n$

In 1958, Hill conjectured that the minimum number of crossings in a drawing of $K_n$ is exactly $Z(n) = \frac{1}{4} \lfloor\frac{n}{2}\rfloor \left\lfloor\frac{n-1}{2}\right\rfloor \left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor$. Generalizing the result by Ábrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by $x$-monotone curves. In fact, our proof shows that the conjecture remains true for $x$-monotone drawings of $K_n$ in which adjacent edges may cross an even number of times, and instead of the crossing number we count the pairs of edges which cross an odd number of times. We further discuss a generalization of this result to shellable drawings, a notion introduced by Ábrego et al. We also give a combinatorial characterization of several classes of $x$-monotone drawings of complete graphs using a small set of forbidden configurations. For a similar local characterization of shellable drawings, we generalize Carathéodory's theorem to simple drawings of complete graphs.

math.CO↗

arXiv2014

Saturated simple and $k$-simple topological graphs

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with $n$ vertices and $O(n)$ edges. For every $k>1$, we give similar constructions for $k$-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most $k$ points in common. We show that in any $k$-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most $2k$ times. Another construction shows that the bound $2k$ cannot be improved. Several other related problems are also considered.

math.CO↗

arXiv2012

Graph sharing games: complexity and connectivity

We study the following combinatorial game played by two players, Alice and Bob, which generalizes the Pizza game considered by Brown, Winkler and others. Given a connected graph G with nonnegative weights assigned to its vertices, the players alternately take one vertex of G in each turn. The first turn is Alice's. The vertices are to be taken according to one (or both) of the following two rules: (T) the subgraph of G induced by the taken vertices is connected during the whole game, (R) the subgraph of G induced by the remaining vertices is connected during the whole game. We show that if rules (T) and/or (R) are required then for every epsilon > 0 and for every positive integer k there is a k-connected graph G for which Bob has a strategy to obtain (1-epsilon) of the total weight of the vertices. This contrasts with the original Pizza game played on a cycle, where Alice is known to have a strategy to obtain 4/9 of the total weight. We show that the problem of deciding whether Alice has a winning strategy (i.e., a strategy to obtain more than half of the total weight) is PSPACE-complete if condition (R) or both conditions (T) and (R) are required. We also consider a game played on connected graphs (without weights) where the first player who violates condition (T) or (R) loses the game. We show that deciding who has the winning strategy is PSPACE-complete.

cs.DM↗

arXiv2012

Improvement on the decay of crossing numbers

We prove that the crossing number of a graph decays in a continuous fashion in the following sense. For any epsilon>0 there is a delta>0 such that for a sufficiently large n, every graph G with n vertices and m > n^{1+epsilon} edges, has a subgraph G' of at most (1-delta)m edges and crossing number at least (1-epsilon)cr(G). This generalizes the result of J. Fox and Cs. Toth.

math.CO↗

arXiv2010

Ramsey-type constructions for arrangements of segments

Improving a result of Károlyi, Pach and Tóth, we construct an arrangement of $n$ segments in the plane with at most $n^{\log{8} / \log{169}}$ pairwise crossing or pairwise disjoint segments. We use the recursive method based on flattenable arrangements which was established by Larman, Matoušek, Pach and Törőcsik. We also show that not every arrangement can be flattened, by constructing an intersection graph of segments which cannot be realized by an arrangement of segments crossing a common line. Moreover, we also construct an intersection graph of segments crossing a common line which cannot be realized by a flattenable arrangement.

math.CO↗

arXiv2008

Solution of Peter Winkler's Pizza Problem

Bob cuts a pizza into slices of not necessarily equal size and shares it with Alice by alternately taking turns. One slice is taken in each turn. The first turn is Alice's. She may choose any of the slices. In all other turns only those slices can be chosen that have a neighbor slice already eaten. We prove a conjecture of Peter Winkler by showing that Alice has a strategy for obtaining 4/9 of the pizza. This is best possible, that is, there is a cutting and a strategy for Bob to get 5/9 of the pizza. We also give a characterization of Alice's best possible gain depending on the number of slices. For a given cutting of the pizza, we describe a linear time algorithm that computes Alice's strategy gaining at least 4/9 of the pizza and another algorithm that computes the optimal strategy for both players in any possible position of the game in quadratic time. We distinguish two types of turns, shifts and jumps. We prove that Alice can gain 4/9, 7/16 and 1/3 of the pizza if she is allowed to make at most two jumps, at most one jump and no jump, respectively, and the three constants are the best possible.

cs.DM↗