The Directed Multicut (DM) problem is: given a simple directed graph G = (V , E) with positive capacities u e on the edges, and a set K ⊆ V × V of ordered pairs of nodes of G, find a minimum capacity K-multicut; C ⊆ E is a K-multicut if in G − C there is no (s, t)-path for any (s, t) ∈ K. In the uncapacitated case (UDM) the goal is to find a minimum size K-multicut. The best approximation ratio known for DM is O(min{ √ n, opt}) by Gupta, where n = |V |, and opt is the optimal solution value. All known nontrivial approximation algorithms for the problem solve large linear programs. We give the first combinatorial approximation algorithms for the problem. Our main result is anÕ(n 2/3 /opt 1/3 )-approximation algorithm for UDM, which improves the O(min{opt, √ n})-approximation for opt = (n 1/2+ε ). Combined with the article of Gupta, we get that UDM can be approximated within better than O( √ n), unless opt =˜ ( √ n). We also give a simple and fast O(n 2/3 )-approximation algorithm for DM.
Any deterministic algorithm can be viewed as a game between the algorithm player and the input player. A randomized algorithm can be viewed as a mixed strategy for the first player, used to minimize the disadvantage of being the first to reveal its move. We suggest a simple and accessible guessing game that can serve as both a way to explain notions in algorithms (like worst case input) to students and also to illustrate the power of randomization, presented in an intuitive way.
Probability theory is branch of mathematics that plays one of the central roles, not only in computer science, but also in science at large. We present a way to integrate non-intuitive probability experiments into introductory programming courses. We consider a set of the probability problems in which the intuitive approach leads to the wrong solution. These problems are based on interesting scenarios, attractive to students, and could be successfully "translated" into programming assignments. We present and discuss numerical simulations and verification of the correct solution through the computational approach. The proposed enrichment provides opportunity to engage students in experimental problem solving. Surprising computational results enhance students' curiosity and interest. This component promotes active involvement in the course. In addition, these problems provide an opportunity to make a connection between mathematics and computer science topics. Such non-intuitive answers may be remembered by students and may promote better understanding in the basic probability course they take later.
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