Combinatorics and Geometry of Transportation Polytopes: An Update

A transportation polytope consists of all multidimensional arrays or tables of non-negative real numbers that satisfy certain sum conditions on subsets of the entries. They arise naturally in optimization and statistics, and also have interest for discrete mathematics because permutation matrices, latin squares, and magic squares appear naturally as lattice points of these polytopes. In this paper we survey advances on the understanding of the combinatorics and geometry of these polyhedra and include some recent unpublished results on the diameter of graphs of these polytopes. In particular, this is a thirty-year update on the status of a list of open questions last visited in the 1984 book by Yemelichev, Kovalev and Kravtsov and the 1986 survey paper of Vlach.Comment: 35 pages, 13 figure

Graphs of Transportation Polytopes

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 Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$.Comment: 29 pages, 7 figures. Final version. Improvements to the exposition of several lemmas and the upper bound in Theorem 1.1 is improved by a factor of tw

Graphs of Transportation Polytopes

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 Yemelichev et al. (1984). It also allowed us to discover some new results. For example, we prove that the number of vertices of an $m\times n$ transportation polytope is a multiple of the greatest common divisor of $m$ and $n$

Graphs of Transportation Polytopes

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 nondegenerate transportation polytopes of small sizes. The catalogue disproves five conjectures about these polyhedra stated in the monograph by Yemelichev et al. (1984). It also allowed 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

A Text Messaging Intervention to Support Latinx Family Caregivers of Individuals With Dementia (CuidaTEXT): Development and Usability Study.

BACKGROUND: Latinx family caregivers of individuals with dementia face many barriers to caregiver support access. Interventions to alleviate these barriers are urgently needed. OBJECTIVE: This study aimed to describe the development of CuidaTEXT, a tailored SMS text messaging intervention to support Latinx family caregivers of individuals with dementia. METHODS: CuidaTEXT is informed by the stress process framework and social cognitive theory. We developed and refined CuidaTEXT using a mixed methods approach that included thematic analysis and descriptive statistics. We followed 6 user-centered design stages, namely, the selection of design principles, software vendor collaboration, evidence-based foundation, caregiver and research and clinical advisory board guidance, sketching and prototyping, and usability testing of the prototype of CuidaTEXT among 5 Latinx caregivers. RESULTS: CuidaTEXT is a bilingual 6-month-long SMS text messaging-based intervention tailored to caregiver needs that includes 1-3 daily automatic messages (n=244) about logistics, dementia education, self-care, social support, end of life, care of the person with dementia, behavioral symptoms, and problem-solving strategies; 783 keyword-driven text messages for further help with the aforementioned topics; live chat interaction with a coach for further help; and a 19-page reference booklet summarizing the purpose and functions of the intervention. The 5 Latinx caregivers who used the prototype of CuidaTEXT scored an average of 97 out of 100 on the System Usability Scale. CONCLUSIONS: CuidaTEXTs prototype demonstrated high usability among Latinx caregivers. CuidaTEXTs feasibility is ready to be tested

A Text Messaging Intervention to Support Latinx Family Caregivers of Individuals With Dementia (CuidaTEXT): Development and Usability Study

Background: Latinx family caregivers of individuals with dementia face many barriers to caregiver support access. Interventions to alleviate these barriers are urgently needed. Objective: This study aimed to describe the development of CuidaTEXT, a tailored SMS text messaging intervention to support Latinx family caregivers of individuals with dementia. Methods: CuidaTEXT is informed by the stress process framework and social cognitive theory. We developed and refined CuidaTEXT using a mixed methods approach that included thematic analysis and descriptive statistics. We followed 6 user-centered design stages, namely, the selection of design principles, software vendor collaboration, evidence-based foundation, caregiver and research and clinical advisory board guidance, sketching and prototyping, and usability testing of the prototype of CuidaTEXT among 5 Latinx caregivers. Results: CuidaTEXT is a bilingual 6-month-long SMS text messaging–based intervention tailored to caregiver needs that includes 1-3 daily automatic messages (n=244) about logistics, dementia education, self-care, social support, end of life, care of the person with dementia, behavioral symptoms, and problem-solving strategies; 783 keyword-driven text messages for further help with the aforementioned topics; live chat interaction with a coach for further help; and a 19-page reference booklet summarizing the purpose and functions of the intervention. The 5 Latinx caregivers who used the prototype of CuidaTEXT scored an average of 97 out of 100 on the System Usability Scale. Conclusions: CuidaTEXT’s prototype demonstrated high usability among Latinx caregivers. CuidaTEXT’s feasibility is ready to be tested