Monotone Paths Research Articles

Monotone Paths in Line Arrangements with a Small Number of Directions

We give subquadratic bounds on the maximum length of an x-monotone path in an arrangement of n lines with at most C log log n directions, where C is a suitable constant. For instance, the maximum length of an x-monotone path in an arrangement of n lines having at most ten slopes is O(n67/34). In particular, we get tight estimates for the case of lines having at most five directions, by showing that previous constructions—Ω(n3/2) for arrangements with four slopes and Ω(n5/3) for arrangements with five slopes—due to Sharir and Matousek, respectively, are (asymptotically) best possible.

Monotone paths in line arrangements

We show that for any n there is an arrangement of n lines which contains an x-monotone path of length Ω(n 7/4) .

Monotone Paths on Zonotopes and Oriented Matroids

AbstractMonotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip. It is proved that any monotone path on a $d$-dimensional zonotope with $n$ generators admits at least $\left\lceil 2n/\left( n-d+2 \right) \right\rceil -1$ flips for all $n\ge d+2\ge 4$ and that for any fixed value of $n-d$, this lower bound is sharp for infinitely many values of $n$. In particular, monotone paths on zonotopes which admit only three flips are constructed in each dimension $d\ge 3$. Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented matroid. An application in the context of Coxeter groups of a result known to be valid for monotone paths on simple zonotopes is included.

Large Monotone Paths in Graphs with Bounded Degree

We prove that for every e>0 and positive integer r, there exists Δ0=Δ0(e) such that if Δ>Δ0 and n>n(Δ,e,r) then there exists a packing of K n with ⌊(n−1)/Δ⌋ graphs, each having maximum degree at most Δ and girth at least r, where at most en 2 edges are unpacked. This result is used to prove the following: Let f be an assignment of real numbers to the edges of a graph G. Let α(G,f) denote the maximum length of a monotone simple path of G with respect to f. Let α(G) be the minimum of α(G,f), ranging over all possible assignments. Now let αΔ be the maximum of α(G) ranging over all graphs with maximum degree at most Δ. We prove that Δ+1≥αΔ≥Δ(1−o(1)). This extends some results of Graham and Kleitman [6] and of Calderbank et al. [4] who considered α(K n ).

Comparative Analysis of the Contribution to a System Failure on a Busy Interval of Monotone and Nonmonotone Paths

Two fast simulation methods are proposed to investigate the contribution of a system of monotone and nonmonotone paths to a failure (loss of a customer). This contribution is analyzed numerically. Examples of systems are given where the principle of monotone failures is not fulfilled.

Monotone paths in edge-ordered sparse graphs

An edge-ordered graph is an ordered pair ( G, f), where G= G( V, E) is a graph and f is a bijective function, f: E( G)→{1,2,…,| E( G)|}. f is called an edge ordering of G. A monotone path of length k in ( G, f) is a simple path P k+1 : v 1,v 2,…,v k+1 in G such that either, f(( v i , v i+1 ))< f(( v i+1 , v i+2 )) or f(( v i , v i+1 ))> f(( v i+1 , v i+2 )) for i=1,2,…, k−1. Given an undirected graph G, denote by α( G) the minimum over all edge orderings of the maximum length of a monotone path. In this paper we give bounds on α( G) for various families of sparse graphs, including trees, planar graphs and graphs with bounded arboricity.

Monotone paths on polytopes

We investigate the vertex-connectivity of the graph of f -mono- tone paths on a d-polytope P with respect to a generic functional f . The third author has conjectured that this graph is always (d − 1)-connected. We resolve this conjecture positively for simple polytopes and show that the graph is 2-connected for any d-polytope with d ≥ 3. However,we disprove the conjecture in general by exhibiting counterexamples for each d ≥ 4 in which the graph has a vertex of degree two. We also re-examine the Baues problem for cellular strings on polytopes, solved by Billera,Kapranov and Sturmfels. Our analysis shows that their positive result is a direct consequence of shellability of polytopes and is therefore less related to convexity than is at first apparent.

Paths on polymatroids

This paper establishes bounds on the length of certain strictly monotone paths, relative to a given linear objective function, on a polymatroid or on the base of a polymatroid. Specialized bounds are given for a strict polymatroid and for a matroid polyhedron. Bounds on the diameter are included.

Lower bounds on the length of monotone paths in arrangements

We show that the maximal number of turns of anx-monotone path in an arrangement ofn lines is Ω(n 5/3) and the maximal number of turns of anx-monotone path in arrangement ofn pseudolines is Ω(n 2/logn).

A note on monotone paths in labeled graphs

We characterize countable graphs with the property that every one-to-one labeling of edges by positive integers admits a monotone path.

On the First Passage Time Distribution for a Class of Markov Chains

Abstract : Consider a stochastically monotone Markov chain with monotone paths on a partially ordered countable set S. Let C be an increasing subset of S with finite complement. Then the first passage time from an element of S to C is shown to be IFRA (increasing failure rate on the average). Several applications are presented including coherent systems, shockmodels, and convolutions of IFRA distributions. (Author)

Monotone paths in ordered graphs

LetV fin andE fin resp. denote the classes of graphsG with the property that no matter how we label the vertices (edges, resp.) ofG by members of a linearly ordered set, there will exist paths of arbitrary finite lengths with monotonically increasing labels. The classesV inf andE inf are defined similarly by requiring the existence of an infinite path with increasing labels. We proveE inf ⫋V inf ⫋V fin ⫋E fin. Finally we consider labellings by positive integers and characterize the class corresponding toV inf.

Long Monotone Paths in Abstract Polytopes

As is now well known, the simplex method, under its various pivoting rules, is not a “good algorithm” in the sense of J. Edmonds, i.e., the number of pivots needed to solve a given linear programming problem by this method cannot be bounded by a polynomial function of the number of rows and columns defining it, Klee, Minty and Jerosolow have developed methods for constructing examples of such linear programs on polytopes. The aim of this paper is to extend these constructions to abstract polytopes.

On Matroid Theorems of Edmonds and Rado

Journal of the London Mathematical SocietyVolume s2-2, Issue 2 p. 251-256 Notes and papers On Matroid Theorems of Edmonds and Rado D. J. A. Welsh, D. J. A. Welsh Merton College, Oxford, University of MichiganSearch for more papers by this author D. J. A. Welsh, D. J. A. Welsh Merton College, Oxford, University of MichiganSearch for more papers by this author First published: April 1970 https://doi.org/10.1112/jlms/s2-2.2.251Citations: 26AboutRelatedInformationPDFPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessClose modalShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Citing Literature Volumes2-2, Issue2April 1970Pages 251-256 RelatedInformation RecommendedRado Partition Theorem for Random Subsets of IntegersF Rödl, A Ruciński, Proceedings of the London Mathematical SocietyErdős–Szekeres‐type theorems for monotone paths and convex bodiesJacob Fox, János Pach, Benny Sudakov, Andrew Suk, Proceedings of the London Mathematical SocietyReal phase structures on matroid fans and matroid orientationsJohannes Rau, Arthur Renaudineau, Kris Shaw, Journal of the London Mathematical SocietyOn a Conjecture of R. RadoS. Todorčević, Journal of the London Mathematical SocietyRemarks on Some Theorems of Rado on Universal GraphsB. Rotman, Journal of the London Mathematical Society

Monotone paths in dense edge-ordered graphs

The altitude of a graph $G$, denoted $f(G)$, is the largest integer $k$ such that under each ordering of $E(G)$, there exists a path of length $k$ which traverses edges in increasing order. In 1971, Chv\'atal and Koml\'os asked for $f(K_n)$, where $K_n$ is the complete graph on $n$ vertices. In 1973, Graham and Kleitman proved that $f(K_n) \ge \sqrt{n - 3/4} - 1/2$ and in 1984, Calderbank, Chung, and Sturtevant proved that $f(K_n) \le (\frac{1}{2} + o(1))n$. We show that $f(K_n) \ge (\frac{1}{20} - o(1))(n/\lg n)^{2/3}$.