README.md

MATH6010-homework

• Dependency
• Homework 1: Minimum Dominating Set
  • Solution
• Homework 2: Monochromatic K4
  • Solution
• Homework 3: 0-1 Knapsack problem
  • Solution
• Homework 3: Steiner Triple Systems
  • Solution
• Homework 3: Uniform Graph Partition
  • Solution

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This repo contains my assignments for MATH6010 (2022 spring) at SJTU.

Follow the commands below to start.

git clone https://github.com/sszxc/MATH6010-homework.git
cd MATH6010-homework/
python -m pip install -r requirements.txt

Dependency

NetworkX for the creation, manipulation, and study of the structure, dynamics, and functions of complex networks.

matplotlib for creating static, animated, and interactive visualizations.

tqdm for a smart progress meter.

Homework 1: Minimum Dominating Set

Definition: A dominating set for a graph $G=(V,E)$ is a subset $D$ of $V$ such that every vertex not in $D$ is adjacent to at least one member of $D$.

Theorem: Let $G=(V,E)$ be a graph on $n$ vertices, with minimum degree $\delta >1$. Then $G$ has a dominating set of at most $n[1+\ln (\delta +1)]/(\delta +1)$ vertices.

Task: Design a greedy algorithm to find a dominating set that satisfies the theorem.

Solution

python hw1_MinDomSet\min_dom_set.py

For example, the following is a solution for a given graph, which satisfies the theorem.

Homework 2: Monochromatic K4

Definition: $K_n$ is a complete graph (a clique) with $n$ nodes.

Theorem: There is a two-coloring of $K_n$ with at most $C_n^4{2}^{-5}$ monochromatic $K_4$.

Task: Design an algorithm based on derandomization to find a method of coloring the edges that satisfies the theorem.

Solution

python hw2_MonochromaticK4\monochromatic_k4.py

The algorithm is specially optimized to take only 23 seconds for a problem of 100 vertices.

For example, the following is a two-coloring solution for $K_{10}$. It has zero $K_4$ which is below the theoretical bound 6.56.

Homework 3: 0-1 Knapsack problem

Given weights and values of n items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack.

Task: Design an algorithm based on simulated annealing to solve the 0-1 knapsack problem.

Solution

python hw3_knapsack\SimulatedAnnealing.py

For example, the following is the solving process for a given knapsack problem.

Homework 3: Steiner Triple Systems

Definition: A Steiner triple system is a set system $(V,B)$ in which every block has size three, and every pair of points from $V$ is contained in a unique block. If $|V|=v$, then we denote such a system as an STS(v).

Task: Design a hill-climbing algorithm to construct random STS(v).

Solution

python hw3_STS\STS_HillClimbing.py

For example, the following gif is the solving process for STS(9). Here, the STS is presented as a hypergraph: the big green circles are the points, and the small gray circles are the blocks that connect three points.

Homework 3: Uniform Graph Partition

Define a complete graph on $2n$ vertices; a cost function $E\to Z^{+}\cup 0$. Try to find the minimum value of $c([X_0,X_1])=\sum _{u,v\in E,u\in X_0,v\in X_1}\cos t(u,v)$ subject to $V=X_0\cup X_1$ and $|X_0|=|X_1|=n$.

Solution

python hw3_UniformGraphPartition\UniformGraphPartition.py