Constant-factor, polynomial-time approximation algorithms are presented for two variations of the traveling salesman problem with time windows. In the first variation, the traveling repairman problem, the goal is to find a path that visits the maximum possible number of locations during their time windows. In the second variation, the speeding deliveryman problem, the goal is to find a path that uses the minimum possible speedup to visit all locations during their time windows. For both variations, the time windows are of unit length, and the distance metric is based on a weighted, undirected graph. Algorithms with improved approximation ratios are given for the case when the input is defined on a tree rather than a general graph. The algorithms are also extended to handle time windows whose lengths fall in any bounded range.
If a bat and a ball cost $1.10 in total and the bat costs $1.00 more than the ball, how much does the ball cost? Many people trained in logical reasoning answer this question incorrectly. Is the kind of logical trap posed by this question similar to the logical traps in computer science? This paper examines the similarity between computer science programming problems with intuitive yet incorrect "lure" answers and logical problems from psychology and economics that share this characteristic. We find that there are fundamental similarities between these kinds of problems and that these problems can even be used as predictors of grades in introductory programming courses. Furthermore, we demonstrate that certain cognitive styles identified in recent psychological literature perform better on such problems.
A bicriteria approximation algorithm is presented for the unrooted traveling repairman problem, realizing increased profit in return for increased speedup of repairman motion. The algorithm generalizes previous results from the case in which all time windows are the same length to the case in which their lengths can range between l and 2. This analysis can extend to any range of time window lengths, following our earlier techniques [11]. This relationship between repairman profit and speedup is applicable over a range of values that is dependent on the cost of putting the input in an especially desirable form, involving what are called "trimmed windows." For time windows with lengths between 1 and 2, the range of values for speedup s for which our analysis holds is 1 ≤ s ≤ 6. In this range, we establish an approximation ratio that is constant for any specific value of s.
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