User: 

You are given access to make searches on the Internet. The format for making a search is:
1. After the user sends a message, you will decide whether to make a search. If you choose to make a search, you will output a single message, starting with the word <web_search>, followed by ONLY your web search query. After your search query, you must output the token </web_search> to denote the end of the search query. Keep your search query concise.
2. The system will then append the web search results at the end of the user message, starting with the token <web_search_results>. Take note that the user CANNOT SEE the content after <web_search_results> - YOU should use these results appropriately to improve your response. Additionally, If the web search results are already present in the user's message, then you MUST NOT request another web search.

IMPORTANT! You should choose to search the web ONLY if ANY of the following circumstances are met:
1. User is asking about current events or something that requires real-time information (news, sports scores, 'latest' information, etc.)
2. User is asking about some term you are unfamiliar with (e.g. if it's a little-known term, if it's a person you don't know well, etc.)
3. User is asking about anything that involves numerical facts (e.g. distances between places, population count, sizes, date and time, etc.)
4. User explicitly asks you to search or provide links to references.
If the user's query does NOT require a web search, YOU MUST NEVER run one for no reason. However, you must not refuse to search the web if the user asks you to do so.
If you are highly unsure about whether to make a search (e.g. if the user's query is vague or if you think you are unfamiliar with the query to a certain extent), you may directly ask the user for clarification.

When using web search results, you are encouraged to cite references where appropriate, using inline links in standard markdown format.
You are also allowed to make use of web search results from previous messages, if they are STRONGLY RELEVANT to the current message.

When you are not running a web search, respond completely as per normal - there is NO NEED to mention that you're not searching. 
Remember that you MUST NOT make up any false information.
Assistant: Understood. I will strictly follow these instructions in this conversation.
User: How do you prove a problem is no complete? There's some 2 steps that I can't seem to remember
Assistant: To prove that a problem is **NP-complete**, you typically follow two main steps:

1. **Show that the problem is in NP**:
   - You need to demonstrate that given a solution (often called a "certificate"), you can verify whether it is correct or not in polynomial time. This often involves providing a polynomial-time algorithm that can check the validity of a proposed solution.

2. **Reduce an existing NP-complete problem to your problem**:
   - Take a known NP-complete problem and show that you can transform (reduce) it into your problem in polynomial time. This shows that your problem is at least as hard as the known NP-complete problem. If you can do this, then you can conclude that if your problem can be solved in polynomial time, then so can all NP problems.

Note: If you want to prove that a problem is not NP-complete (or to prove it's in a lower complexity class), you can focus on showing that:
- It is in P (i.e., solvable in polynomial time).
- It's decidable in a way that doesn't meet the criteria for NP-completeness, possibly by showing it can be solved deterministically in polynomial time or showing it has a polynomial-time algorithm.

In summary, proving a problem is NP-complete consists of demonstrating it is in NP and then establishing a polynomial-time reduction from a known NP-complete problem to your problem. If you aim to show your problem is not NP-complete, focus on finding a polynomial-time algorithm or another way to distinguish it from
User: how do you show problem is in np?
Assistant: To show that a problem is in NP, you need to demonstrate that if you are given a candidate solution (also called a certificate), you can verify whether this solution is correct in polynomial time. Here are the steps to do this:

1. **Define the Problem**: Clearly state what the problem is and how a solution would look. This often involves specifying the inputs and the expected outputs.

2. **Identify a Certificate**: A certificate is a potential solution to the problem. For example, if you're dealing with a decision problem, the certificate would be a specific instance of the solution that you want to verify.

3. **Design a Verification Algorithm**:
   - You must create a polynomial-time algorithm that takes two inputs: 
     - The original problem instance (input).
     - The proposed certificate (solution).
   - This algorithm should check whether the certificate is indeed a valid solution to the problem.

4. **Analyze the Time Complexity**: Ensure that the time taken by the verification algorithm to check the certificate is polynomial in relation to the size of the input. The verification procedure should not take exponential time.

If you can successfully outline these steps and establish that there exists a polynomial-time verification process, then you have shown that the problem is in NP.
User: how do you do this for independent set?
Assistant: