An Integer Programming Quiz Picture Round

Abstract

Pub quizzes have been a popular weekday evening activity in the United Kingdom for many years. These quizzes consist of a number of teams competing for a cash prize over several rounds of questions. Typical rounds may include general knowledge, music, science, sports, and current affairs. More recently, because of surreptitious use of smart phones it has become a challenge to quiz setters to devise questions that will confound the would-be cheat. As a result, rounds of picture questions have become increasingly prevalent and in this article I would like to present a particularly novel type of picture round based on the traveling salesman problem (TSP) or, more specifically, the TSP art pioneered by Bob Bosch and others. The TSP may be formulated using integer programming (IP) and is known to be NP-complete. The P versus NP problem is considered to be of such importance that the Clay Mathematics Institute included it as one of the millennium problems, offering a $1 million prize for its resolution. For an official description of the problem together with a video see http://www.claymath.org/millennium/P_vs_NP/. The TSP has generated a great deal of attention over the past few decades and researchers in this field have taken from and hugely contributed to the more general IP literature. William J. Cook’s terrific book In Pursuit of the Traveling Salesman (2012) gives a very detailed but accessible coverage of the problem and is essential reading for anyone with more than a passing interest in optimization. Cook (2012) includes a useful introduction to TSP art in a consciousness expanding chapter entitled Aesthetics. The basic idea is to generate TSP solutions that will approximate target images, usually portraits, when viewed at an appropriate distance. These are sometimes referred to as single line drawings and may be at the very least eye-catching and in some instances quite beautiful. In 2010 a sculpture called Embrace based on this idea gained Bosch the mathematical art exhibition first prize awarded by the American Mathematical Society and the Mathematics Association of America. This exquisite piece of mathematical artwork may be viewed at http://www .ams.org/mathimagery/displayimage.php?pid=274. Bosch, together with Craig Kaplan, has developed many sophisticated methods to enhance the aesthetic appeal of TSP images, see for example Kaplan and Bosch (2005) and their respective Web pages at http://www.oberlin.edu/math/faculty/bosch/tspartpage.html and http://www.cgl.uwaterloo.ca/~csk/ projects/tsp/. In this article I use the basic ideas to render images of iconic figures. My objective was to produce a range of images of varying clarity to be used as a basis for a quiz picture round. The process begins with a target image in the form of a black and white digital photograph as with the portrait of Marilyn Monroe in Figure 1. This target image is divided into rectangular panels and the average grayscale value of each panel is computed. The panels of the original photograph are each replaced by rectangles containing a number of points proportional to the grayscale value of the respective panel. An R language function (TspDotMap.r) to achieve this “stippling,” as it is known, is included as a supplementary file and I will refer to the output of this function as a dot map. The function stores the dot map in standard TSPLIB format, which is described in detail at http://plato.asu.edu/tsplib.pdf. An example (marilyn.tsp) is included with this article. Minor editing of the R function output is necessary as can be seen from inspection of this example. Figure 2 shows the stippled image generated from the target image in Figure 1. The TSP portrait may then be rendered using the Concorde software downloadable at http://www .tsp.gatech.edu/concorde.html. Searching for optimal tours in large problems can be time consuming