A note on hop-constrained walk polytopes

Journal of Combinatorial Theory, Series A, 2009

This paper discusses properties of the graphs of 2-way and 3-way transportation polytopes, in particular, their possible numbers of vertices and their diameters. Our main results include a quadratic bound on the diameter of axial 3-way transportation polytopes and a catalogue of non-degenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by . It also allowed us to discover some new results. For example, we prove that the number of vertices of an m × n transportation polytope is a multiple of the greatest common divisor of m and n.

2022

Pyramidal tours with step-backs are Hamiltonian tours of a special kind: the salesperson starts in city 1, then visits some cities in ascending order, reaches city $n$, and returns to city 1 visiting the remaining cities in descending order. However, in the ascending and descending direction, the order of neighboring cities can be inverted (a step-back). It is known that on pyramidal tours with step-backs the traveling salesperson problem can be solved by dynamic programming in polynomial time. We define the polytope of pyramidal tours with step-backs $\operatorname{PSB}(n)$ as the convex hull of the characteristic vectors of all possible pyramidal tours with step-backs in a complete directed graph. The 1-skeleton of $\operatorname{PSB}(n)$ is the graph whose vertex set is the vertex set of the polytope, and the edge set is the set of geometric edges or one-dimensional faces of the polytope. We present a linear-time algorithm to verify vertex adjacencies in 1-skeleton of the polytop...

Discrete Optimization, 2013

The pedigree is a combinatorial object defined over the cartesian product of certain subsets of edges in a complete graph. There is a 1-1 correspondence between the pedigrees and Hamiltonian cycles (tours) in a complete graph. Linear optimization over Hamiltonian cycles, also known as the symmetric traveling salesman problem (STSP) has several 0-1 integer and mixed integer formulations. The Multistage Insertion formulation (MI-formulation) is one such 0-1 integer formulation of the STSP. Any solution to the MI-formulation is a pedigree and vice versa. However the polytope corresponding to the pedigrees has properties not shared by the STSP polytope. For instance, (i) the pedigree polytope is a combinatorial polytope, in the sense, given any two nonadjacent vertices of the polytope W 1 , W 2 , we can find two other nonadjacent vertices, W 3 , W 4 , such that W 1 + W 2 = W 3 + W 4 and (ii) testing the nonadjacency of tours is an NP-complete problem, while the corresponding problem for the pedigrees is strongly polynomial. In this paper we demonstrate how the study of the nonadjacency structure is useful in understanding that of the tour polytope. We prove that a sufficiency condition for nonadjacency in the tour polytope is nonadjacency of the corresponding pedigrees in the pedigree polytope. This proof makes explicit use of properties of both the pedigree polytope and the MI-relaxation problem.

Mathematics and Statistics, 2023

In this paper, we consider the polytope P(G) of all elementary dicycles of a digraph G. Dicycles problem, in graph theory and combinatorial optimization, solved by polyhedral approaches has been extensively studied in literature. Therefore cutting plane and branch and cut algorithms are unavoidable to exactly solve such a combinatorial optimization problem. For this purpose, we introduce a new family of valid inequalities called s − t alternating 3-arc path inequalities for the polytope of elementary dicycles P(G). Indeed, these inequalities can be used in cutting plane and branch and cut algorithms to construct strengthened relaxations of a linear formulation of the dicycle problem. To prove the facetness of s − t alternating 3-arc path inequalities, in opposite to what is usually done that consists basically to determine the affine subspace of a linear description of the considered polytope, we resort to constructive algorithms. Given the set of arcs of the digraph G, algorithms devised and introduced are based on the fact that from a first elementary dicycle, all other dicycles are iteratively generated by replacing some arcs of previously generated dicycles by others such that the current elementary dicycle contains an arc that does not belong to any other previously generated dicycles. These algorithms generate dicyles with affinely independent incidence vectors that satisfy s − t alternating 3-arc path inequalities with equality. It can easily be verified that all these devised algorithms are polynomial from time complexity point of view.

Discrete & Computational Geometry, 2009

The curvature of a polytope, defined as the largest possible total curvature of the associated central path, can be regarded as the continuous analogue of its diameter. We prove the analogue of the result of Klee and Walkup. Namely, we show that if the order of the curvature is less than the dimension d for all polytope defined by 2d inequalities and for all d, then the order of the curvature is less that the number of inequalities for all polytopes.

Annals of the New York Academy of Sciences, 1989

Journal of Al-Qadisiyah for Computer Science and Mathematics

Transportation polytopes of non- negative m × n matrices compose of two vectors: a and b which row sums are equal to fixed constant and column sums are equal to different constant. The transportation polytopes are denoted by T (a, b) and these two vectors are called margins. An open problem that the 2-way transportation polytopes are Hamiltonian is proved in this paper with application of optimization.

Discrete Mathematics, 2009

We suggest defining the structure of an unoriented graph R d on the set of reflexive polytopes of a fixed dimension d. The edges are induced by easy mutations of the polytopes to create the possibility of walks along connected components inside this graph. For this, we consider two types of mutations: Those provided by performing duality via nef-partitions, and those arising from varying the lattice. Then for d ≤ 3, we identify the flow polytopes among the reflexive polytopes of each single component of the graph R d . For this, we present for any dimension d ≥ 2 an explicit finite list of quivers giving all d-dimensional reflexive flow polytopes up to lattice isomorphism. We deduce as an application that any such polytope has at most 6(d − 1) facets.

Aapp Physical Mathematical and Natural Sciences, 2013

The fact that linear optimization over a polytope can be done in polynomial time in the input size of the instance, has created renewed interest in studying 0−1 polytopes corresponding to combinatorial optimization problems. Studying their polyhedral structure has resulted in new algorithms to solve very large instances of some difficult problems like the symmetric traveling salesman problem. The multistage insertion formulation (MI) given by the author, in 1982, for the symmetric traveling salesman problem (STSP), gives rise to a combinatorial object called the pedigree. The pedigrees are in one-to-one correspondence with Hamiltonian cycles. Given n, the convex hull of all the pedigrees is called the corresponding pedigree polytope. In this article we bring together the research done a little over a decade by the author and his doctoral students, on the pedigree polytope, its structure, membership problem and properties of the MI formulation for the STSP. In addition we summarise some of the computational and other peripheral results relating to pedigree approach to solve the STSP. The pedigree polytope possesses properties not shared by the STSP (tour) polytope, which makes it interesting to study the pedigrees, both from theoretical and algorithmic perspectives.

Mathematische Zeitschrift, 2000

We investigate the vertex-connectivity of the graph of f -monotone 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.

2002

The concept of perfection of a polytope was introduced by S. A. Robertson. Intuitively speaking, a polytope P is perfect if and only if it cannot be deformed to a polytope of different shape without changing the action of its symmetry group G(P ) on its face-lattice F (P ). By Rostami's conjecture, the perfect 4-polytopes form a particular set of Wythoffian polytopes. In the present paper first this known set is briefly surveyed. In the rest of the paper two new classes of perfect 4-polytopes are constructed and discussed, hence Rostami's conjecture is disproved. It is emphasized that in contrast to an existing opinion in the literature, the classification of perfect 4-polytopes is not complete as yet.

Journal of Applied and Industrial Mathematics, 2018

We consider the skeleton of the pyramidal tours polytope. Hamiltonian tour is called pyramidal if the salesperson starts in city 1, then visits some cities in increasing order, reaches city n and returns to city 1, visiting the remaining cities in decreasing order. The polytope PYR(n) is defined as the convex hull of characteristic vectors of all pyramidal tours in the complete graph Kn. The skeleton of the polytope PYR(n) is the graph whose vertex set is the vertex set of PYR(n) and edge set is the set of geometric edges or one-dimensional faces of PYR(n). We describe the necessary and sufficient condition for the adjacency of vertices of the polytope PYR(n). On this basis we developed an algorithm to check the vertex adjacency with a linear complexity. We establish that the diameter of PYR(n) skeleton equals 2, and the asymptotically exact estimate of PYR(n) skeleton's clique number is Θ(n 2). It is known that this value characterizes the time complexity in a broad class of algorithms based on linear comparisons.

Contemporary Mathematics, 2006

We describe a construction for d-polytopes generalising the well known stacking operation. The construction is applied to produce 2-simplicial and 2-simple 4-polytopes with g 2 = 0 on any number of n ≥ 13 vertices. In particular, this implies that the ray ℓ 1 , described by Bayer , is fully contained in the convex hull of all flag vectors of 4-polytopes. Especially interesting examples on 9, 10 and 11 vertices are presented. 2000 Mathematics Subject Classification. Primary 52B05;52B12. 1 2 ANDREAS PAFFENHOLZ AND AXEL WERNER Theorem 4.3. Elementary 2-simplicial, 2-simple 4-polytopes with k vertices exist for k = 5, 9, 10, 11 and k ≥ 13. This implies (using the notation of Bayer [Bay87]): Corollary 4.4. The ray ℓ 1 is contained in the convex hull of all flag vectors of 4-polytopes. Additionally, we briefly analyse the consequences of the various recent polytope constructions in [PZ04] and [Zie04] for the flag vector cone.

Journal of Graph Theory

Every generic linear functional on a convex polytope induces an orientation on the graph of . From the resulting directed graph one can define a notion of ‐arborescence and ‐monotone path on , as well as a natural graph structure on the vertex set of ‐monotone paths. These concepts are important in geometric combinatorics and optimization. This paper bounds the number of ‐arborescences, the number of ‐monotone paths, and the diameter of the graph of ‐monotone paths for polytopes in terms of their dimension and number of vertices or facets.

Discrete Applied Mathematics, 1995

Using a refined version of Chernikova's algorithm we determined a complete and irredundant linear description of small asymmetric traveling salesman polytopes. We present such a description for the monotone polytope on 5 nodes consisting of 7615 facet-defining inequalities, and we present 319 015 facet-defining inequalities which, together with 11 equations, fully describe the (non-monotone) asymmetric polytope for 6 nodes.