Nayak, Bhagirathi and Aravind, N R
(2018)
Hardness of Games and Graph Sampling.
Masters thesis, Indian Institute of Technology Hyderabad.
Abstract
The work presented in this document is divided into two parts. The �rst part presents the hardness of games and
the second part presents Graph sampling. Non-deterministic constraint logic[1] is used to prove the hardness of
games. The games which are considered in this work is Reversi (2 player bounded game), Peg Solitaire (single
player bounded game), Badland (single player bounded game). It also contains a theoretical study of peg
solitaire on special graph classes. Reversi is proved to be PSPACE-Complete using Bounded 2CL, Peg Solitaire
is proved to be NP-Complete using Bounded NCL. Badland is proved to be NP-Complete by a reduction from
3-SAT. The objective of study of peg solitaire of special graph classes is to �nd the maximum number of marbles
we can remove from a fully �lled board, if the player is given the privilege to remove a marble from any cell
initially, then following the rules after the initial move.
The second part of the work is dedicated to graph sampling. Given a graph G, we try to sample a represen-
tative subgraph Gs which is similar to the original graph G. The properties that are being studied are Degree
Distribution, Clustering Coefficient, Average Shortest Path Length, Largest Connected Component Size. To
measure the similarity between the original graph and sample we use the metrics Kolmogorov - Smirnov test
and Kullback - Leibler divergence test. Tightly Induced Edge Sampling performs well on general graphs but
it's performance decreases when the graph is a tree. Overall TIBFS and KARGER produces a sample which
closely matches the distribution of original graphs.
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