Olympiads in Informatics, 2016, Vol. 10, 87–98 87 © 2016 IOI, Vilnius University DOI: 10.15388/ioi.2016.06 Understanding Unsolvable Problem Jonathan Irvin GUNAWAN Undergraduate Student School of Computing, National University of Singapore Computing 1, 13 Computing Drive, Singapore 117417 e-mail: [email protected] Abstract. In recent IOIs, there are several problems that seem unsolvable, until we realise that there is a special case to the problem that makes it tractable. In IOI 2014, the problem ‘Friend’ appears to be a standard NP-hard Maximum Independent Set problem. However, the graph is gen- erated in a very special way, hence there is a way to solve the problem in polynomial time. There were several contestants who didn’t identify the special case in this problem, and hence were stuck at the problem. In this paper, we will study a well-known technique called reduction to show that a problem we are currently tackling is intractable. In addition, we introduce techniques to identify special cases such that contestants will be prepared to tackle these problems. Keywords: special case, unsolvable, NP-hard. 1. Introduction The problem ‘Friend’ in IOI 2014 required contestants to find a set of vertices with maxi- mum total weight, such that no two vertices in the set are sharing a common edge. This is a classical Weighted Maximum Independent Set problem. We can show that Weight- ed Maximum Independent Set problem is NP-hard by reduction from 3-SAT (Cormen et al., 2009). Since the formulation of NP-completeness 4 decades ago, no one has been able to propose a solution to any NP-hard problem in polynomial time. Clearly, it is not expected that a high school student can solve the problem in 5 hours. None of the Indo- nesian IOI 2014 team solved this problem during the contest. After returning from the competition, I asked the Indonesian team about this problem. None of the team members were aware of the fact that Maximum Independent Set is an NP-hard problem, and thus were stuck trying to solve a general Maximum Independent Set problem. A similar problem also occurred in IOI 2008. The problem ‘Island’ required contes- tants to find a longest path in a graph with 1,000,000 vertices. The longest path problem is a classic NP-hard problem which can be reduced from the Hamiltonian path problem. If a contestant is not aware that the longest path problem is difficult to solve, the con- testant may spend a lot of his/her time just to tackle the general longest path problem, without realising that there is a special case to the given graph. 88 J.I. Gunawan Generally, some contestants spend too much thinking time trying to solve something that is believed to be unsolvable. If only they realise that their attempt is intractable, they may try a different approach and find a special case of this problem. In section 2 of this paper, we will introduce a classic reduction technique often used in theoretical computer science research. In the context of competitive programming, we may find out that a problem which we are attempting is unlikely to be solvable. After realizing that a problem is intractable, we are going to discuss how to proceed to solve the problem in section 3. Finally, in section 4 we will take a look at some common special cases in competitive programming that can be used to solve these kind of problems. Fig. 1. The result of Indonesian team in IOI 2014, taken from http://stats.ioinformat- ics.org. The red squared column highlights the ‘Friend’ problem. Fig. 2. IOI 2014 tasks statistics, taken from http://stats.ioinformatics.org. ‘Friend’ problem is the second least accepted problem in IOI 2014. It may be because some contes- tants (at least all the Indonesians) were stuck at trying to solve a general case of Maximum Independent Set. Fig. 3. The result of Indonesian team in IOI 2008, taken from http://stats.ioinformat- ics.org. The red squared column highlights the ‘Island’ problem. Understanding Unsolvable Problem 89 2. 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