高级算法 (Fall 2021)/Problem Set 4: Difference between revisions

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Suppose we have graphs <math>G=(V,E)</math> and <math>H=(V,F)</math> on the same vertex set.
Suppose we have graphs <math>G=(V,E)</math> and <math>H=(V,F)</math> on the same vertex set.
We wish to partition <math>V</math> into clusters <math>V_1,V_2,\cdots</math> so as to maximise:
We wish to partition <math>V</math> into clusters <math>V_1,V_2,\cdots</math> so as to maximise:
:<math>(#)</math>

Revision as of 07:48, 20 December 2021

  • 每道题目的解答都要有完整的解题过程。中英文不限。

Problem 1

Problem 2

A [math]\displaystyle{ k }[/math]-uniform hypergraph is an ordered pair [math]\displaystyle{ G=(V,E) }[/math], where [math]\displaystyle{ V }[/math] denotes the set of vertices and [math]\displaystyle{ E }[/math] denotes the set of edges. Moreover, each edge in [math]\displaystyle{ E }[/math] now contains [math]\displaystyle{ k }[/math] distinct vertices, instead of [math]\displaystyle{ 2 }[/math] (so a [math]\displaystyle{ 2 }[/math]-uniform hypergraph is just what we normally call a graph). A hypergraph is [math]\displaystyle{ k }[/math]-regular if all vertices have degree [math]\displaystyle{ k }[/math]; that is, each vertex is exactly contained within [math]\displaystyle{ k }[/math] hypergraph edges.

Show that for sufficiently large [math]\displaystyle{ k }[/math], the vertices of a [math]\displaystyle{ k }[/math]-uniform, [math]\displaystyle{ k }[/math]-regular hypergraph can be [math]\displaystyle{ 2 }[/math]-colored so that no edge is monochromatic. What's the smallest value of [math]\displaystyle{ k }[/math] you can achieve?

Problem 3

Suppose we have graphs [math]\displaystyle{ G=(V,E) }[/math] and [math]\displaystyle{ H=(V,F) }[/math] on the same vertex set. We wish to partition [math]\displaystyle{ V }[/math] into clusters [math]\displaystyle{ V_1,V_2,\cdots }[/math] so as to maximise:

[math]\displaystyle{ (#) }[/math]