Monotone paths on polytopes

SIAM Journal on Discrete Mathematics

2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that f0(P)f d−1 (P) ≤ d2 d+1 for a large collection of families of such polytopes P. Here f0(P) (resp. f d−1 (P)) is the number of vertices (resp. facets) of P , and d is its dimension. Whether this holds for all 2-level polytopes was asked in [7], and experimental results from [16] showed it true for d ≤ 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching polytopes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.

In this paper we prove that each convex 3-polytope contains a path on three vertices with restricted degrees which is one of the ten types. This result strengthens a theorem by Kotzig that each convex 3-polytope has an edge with the degree sum of its end vertices at most 13.

Journal of Combinatorial Theory, Series B, 1975

Journal of Combinatorial Theory, Series B, 1971

In polyhedral combinatorics one often has to analyze the facial structure of less than full dimensional polyhedra. The presence of implicit or explicit equations in the linear system defining such a polyhedron leads to technical difficulties when analyzing its facial structure. It is therefore customary to approach the study of such a polytope P through the study of one of its (full dimensional) relaxations (monotonizations) known as the submissive and the dominant of P. Finding sufficient conditions for an inequality that induces a facet of the submissive or the dominant of a polyhedron to also induce a facet of the polyhedron itself has been posed in the literature as an important research problem. Our paper goes a long way towards solving this problem. We address the problem in the framework of a generalized monotonization of a polyhedron P, g-mort(P), that subsumes both the submissive and the dominant, and give a sufficient condition for an inequality that defines a facet of g-mon(P) to define a facet of P. For the important cases of the traveling salesman (TS) polytope in both its symmetric and asymmetric variants, and of the linear ordering polytope, we give sufficient conditions trivially easy to verify, for a facet of the monotone completion to define a facet of the original polytope itself.

Monatshefte f�r Mathematik, 1990

We call a convex subset N of a convex d-polytope P c E d a k-nucleus of P if N meets every k-face of P, where 0 < k < d. We note that P has disjoint k-nuclei if and only if there exists a hyperplane in E d which bisects the (relative) interior of every k-face of P, and that this is possible only if/~-/~< k ~< d-1.

2006

The k edge-disjoint 2-hop-constrained paths problem consists in finding a minimum cost subgraph such that between two given nodes s and t there exist at least k edge-disjoint paths of at most 2 edges. We give an integer programming formulation for this problem and characterize the associated polytope.

Monotone 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 ip. It is proved that any monotone path on a d-dimensional zonotope with n generators admits at least d2n=(n ? d + 2)e ? 1 ips for all n d+2 4 and that for any xed value of n?d, this lower bound is sharp for in nitely many values of n. In particular, monotone paths on zonotopes which admit only three ips are constructed in each dimension d 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.

Operations Research Letters, 2004

A hop-constrained walk is a walk with at most H arcs. The cases H 6 3 have been addressed previously. Here, we consider the case H = 4. We present an extended formulation for 4-walks and use the projection theorem of Balas and Pulleyblank to derive a complete linear description of the 4-walk polytope.

Biointerface Research in Applied Chemistry, 2020

The neighborhood M-polynomial is effective in recovering neighborhood degree sum based topological indices that predict different physicochemical properties and biological activities of molecular structures. Topological indices can transform the information found in molecular graphs and networks into numerical characteristics and thus make a major contribution to the study of structure-property and structure-activity relationships. In this work, the neighborhood M-polynomial of the para-line graph of some convex polytopes is obtained. From the neighborhood M-polynomial, some neighborhood degree-based topological indices are recovered. Applications of the work are described. In addition, a quantitative and graphical comparison is made.

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.

Lecture Notes in Computer Science, 2016

2-level polytopes naturally appear in several areas of mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. We investigate upper bounds on the product of the number of facets f d−1 (P) and the number of vertices f0(P), where d is the dimension of a 2-level polytope P. This question was first posed in [3], where experimental results showed f0(P)f d−1 (P) ≤ d2 d+1 up to d = 6. We show that this bound holds for all known (to the best of our knowledge) 2-level polytopes coming from combinatorial settings, including stable set polytopes of perfect graphs and all 2-level base polytopes of matroids. For the latter family, we also give a simple description of the facet-defining inequalities. These results are achieved by an investigation of related combinatorial objects, that could be of independent interest.

Honam Mathematical Journal, 2015

We have calculated the volume of the graph polytopes associated with several types of graphs.

Israel Journal of Mathematics, 1984

Certain construction theorems are represented, which facilitate an inductive combinatorial construction of polytopes. That is, applying the constructions to a d-polytope with n vertices, given combinatorially, one gets many combinatorial d-polytopes-and polytopes only-with n + I vertices. The constructions are strong enough to yield from the 4-simplex all the 1330 4-polytopes with up to 8 vertices.

Polytopes: Abstract, Convex and Computational, 1994

Convex and Computational NATO ASI Series Advanced Science Institutes Series A Series presenting the results of activities sponsored by the NATO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities.