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Chapter 0: Q23P (page 1)
Let s an undirected graph having a complete subgraph with at least nodes, where m is the number of nodes in. Show that HALF-CLIQUE is NP-complete.
The clique is a clique of , and hence
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Is the statement a member of Th? Why or why not? What about the statement ?
This problem is inspired by the single-player game Minesweeper, generalized to an arbitrary graph. Let be an undirected graph, where each node either contains a single, hidden mine or is empty. The player chooses nodes, one by one. If the player chooses a node containing a mine, the player loses. If the player chooses an empty node, the player learns the number of neighboring nodes containing mines. (A neighboring node is one connected to the chosen node by an edge.) The player wins if and when all empty nodes have been so chosen.
In the mine consistency problem, you are given a graph along with numbers labeling some of 鈥檚 nodes. You must determine whether a placement of mines on the remaining nodes is possible, so that any node v that is labeled m has exactly m neighboring nodes containing mines. Formulate this problem as a language and show that it is.
Write a formal description of the following graph.
Show that for any two languages , a language J exists, where
Let . Show that AMBIGCFG is undecidable. (Hint: Use a reduction from PCP. Given an instance
of the Post Correspondence Problem, construct a CFG Gwith the rules
where a1,...,ak are new terminal symbols. Prove that this reduction works.)
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