Some lower bounds on sparse outer approximations of polytopes

Motivated by the need to better understand the properties of sparse cutting-planes used in mixed integer programming solvers, the paper [2] studied the idealized problem of how well a polytope is approximated by the use of sparse valid inequalities. As an extension to this work, we study the following less idealized questions in this paper: (1) Are there integer programs, such that sparse inequalities do not approximate the integer hull well even when added to a linear programming relaxation? (2) Are there polytopes, where the quality of approximation by sparse inequalities cannot be significantly improved by adding a budgeted number of arbitrary (possibly dense) valid inequalities? (3) Are there polytopes that are difficult to approximate under every rotation? (4) Are there polytopes that are difficult to approximate in all directions using sparse inequalities? We answer each of the above questions in the positive

Mixed-integer Quadratic Programming is in NP

Mixed-integer quadratic programming is the problem of optimizing a quadratic function over points in a polyhedral set where some of the components are restricted to be integral. In this paper, we prove that the decision version of mixed-integer quadratic programming is in NP, thereby showing that it is NP-complete. This is established by showing that if the decision version of mixed-integer quadratic programming is feasible, then there exists a solution of polynomial size. This result generalizes and unifies classical results that quadratic programming is in NP and integer linear programming is in NP

Impact of Internet gambling on problem gambling among adolescents in Italy: Findings from a large-scale nationally representative survey

Aims: The primary aim of the present study was to understand the impact of online gambling on gambling problems in a large-scale nationally representative sample of Italian youth, and to identify and then further examine a subgroup of online gamblers who reported higher rates of gambling problems. Design: Data from the ESPAD®Italia2013 (European School Survey Project on Alcohol and Other Drugs) Study were used for analyses of adolescent Internet gambling. Setting: Self-administered questionnaires were completed by a representative sample of high school students, aged 15–19 years. Participants: A total of 14,778 adolescent students. Measurements: Respondents’ problem gambling severity; gambling behavior (participation in eight different gambling activities, the number of gambling occasions and the number of online gambling occasions, monthly gambling expenditure); Socio-demographics (e.g., family structure and financial status); and control variables were measured individually (i.e., use of the Internet for leisure activities and playing video games). Findings: Rates of problem gambling were five times higher among online gamblers than non-online gamblers. In addition, factors that increased the risk of becoming a problem online gambler included living with non-birth parents, having a higher perception of financial family status, being more involved with gambling, and the medium preferences of remote gamblers (e.g., Internet cafes, digital television, and video game console). Conclusions: The online gambling environment may pose significantly greater risk to vulnerable players. Family characteristics and contextual elements concerning youth Internet gambling (e.g., remote mediums) may play a key role in explaining problem online gambling among adolescents