Algorithms Q&A - Recent questions in NP-Completeness

Snow White and 7 Conference Rooms

Sun, 18 Dec 2022 10:55:44 +0000

You are the chief algorithm officer for Snow White Inc. Because of your algorithmic abilities, Snow White Inc. is very successful and has hundreds of thousands of investors. You are planning the next investor conference. You want to make sure that all investors who know each other attend the meeting in different rooms. You are trying to decide if the 7 conference rooms in the Snow White Inc. head quarters will be sufficient or not. Prove that this decision problem is NP-complete by using one of the known NP-complete problems (CLIQUE, 3-SAT, Hamiltonian Path, Hamiltonian Cycle, Coloring, Independent Set, etc.)

Prove that Sub Graph Isomorphism is Hard

Sat, 21 Dec 2019 00:38:58 +0000

SI: Given graphs G, H: does graph G contain graph H as a subgraph?

Prove that this decision problem is NP-complete.

Long chain of friends

Mon, 07 May 2018 00:13:29 +0000

You are given a list of people, and statements of the form “x knows y”. You are asked to find, is there a chain of k people, such as x₁ knows x₂, x₂ knows x₃, and x_k-1 knows x_k. Prove that this problem is NP-complete by using one of the known NP-complete problems (CLIQUE, 3-SAT, Hamiltonian Path, Hamiltonian Cycle, Independent Set, etc.)

Prove that 3-Coloring is NP Hard (starting with SAT as known NP hard problem)

Wed, 29 Nov 2017 18:55:07 +0000

Prove that 3-Coloring is NP Hard (starting with SAT as known NP hard problem)

What's the time complexity for solving Sudoku with backtrack method?

Sat, 29 Apr 2017 02:36:50 +0000

We have a 9x9 Sudoku like this

8       0       0       0       0       0       0       0       0

0       0       3       6       0       0       0       0       0

0       7       0       0       9       0       2       0       0

0       5       0       0       0       7       0       0       0

0       0       0       0       4       5       7       0       0

0       0       0       1       0       0       0       3       0

0       0       1       0       0       0       0       6       8

0       0       8       5       0       0       0       1       0

0       9       0       0       0       0       4       0       0

Zero means empty, which needs to be filled.

We may guess the number from 1 to 9 in Matrix[i][j] , and we also need to check if there is any repeated number in i row and j column. If it is not qualified at [i][j], we need to back trace to the previous empty. So we can guess the whole Sudoku with this method. What's the time complexity? How to prove it?

Cook-Levin Theorem

Tue, 06 Dec 2016 14:32:10 +0000

In section 9.4.1 in our textbook, as well as in the following slide, we encountered this definition:

I could not understand why is the length of the CNF formula is O(f(n)^3), where f(n) is the steps for the NTM to decide a string of length n.

Can anyone explain it? Maybe some examples or illustration would be super helpful.

Thanks a bunch!

k-degree constrained spanning tree

Fri, 02 Dec 2016 22:08:36 +0000

Finding a spanning tree is an easily solvable polynomial time problem. Consider a "k-degree constrained spanning tree", wherein we have to find a spanning tree such that no vertex in the spanning tree has degree more than k. Show that k-degree constrained spanning tree problem is NP-complete.

Prove that Vertex Cover is NP-hard problem

Mon, 21 Nov 2016 01:30:43 +0000

Vertex Cover problem is defined as follows:

Given a Graph G and integer k.

Does G have a subset S of vertices, such that: |S| = k, and every edge in G has at least one of the end points in S.

Prove that Vertex Cover is an NP-hard problem by showing a reduction from one of the known NP-complete problems.