# Questions tagged [np-complete]

Questions about the hardest problems in NP, i.e. of those that can be solved in polynomial time by nondeterministic Turing machines.

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### What is practical difference between NP and PSPACE-complete?

Here's something that has puzzled me lately, and perhaps someone can explain what I'm missing. Problems in NP are those that can be solved on a NDTM in polynomial time. Now assuming P$\,\neq\,$NP, ...
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### Subgraph isomorphism reduction from the Clique problem

I was trying to understand the Wikipedia proof for NP-completeness of subgraph isomorphism by reduction from the clique problem. It's really just one sentence: Let $H$ be the complete graph $K_k$; ...
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### To prove 4-SAT CNF is NP-complete [closed]

To answer the question below, 4-SAT: Given a formula in Conjunctive Normal Form, where each clause contains exactly 4 literals, does it have a satisfying truth assignment? I was going to prove that ...
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### complexity of decision problems vs computing functions [closed]

This is an area that admittedly I've always found subtle about CS and occasionally trips me up, and clearly others. recently on tcs.se a user asked an apparently innocuous question about N-Queens ...
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### Independent set on cubic triangle-free graphs

I know that maximum independent set on cubic triangle-free graphs is NP-complete. Is it still NP-complete in case we require the independent set to be of size exactly $|V|/2$? Basiclly, YES ...
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### Is this classic puzzle book game NP-complete?

There is a classic puzzle book game very similar to a crossword puzzle, except a list of words is given and then a $N \times N$ square board made up of unit squares is given, with some squares blacked ...
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### Is it intuitive to see that finding a Hamiltonian path is not in P while finding Euler path is?

I am not sure I see it. From what I understand, edges and vertices are complements for each other and it is quite surprising that this difference exists. Is there a good / quick / easy way to see ...
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### Reduce Vertex cover to SAT

I need to reduce the vertex cover problem to a SAT problem, or rather tell whether a vertex cover of size k exists for a given graph, after solving with a SAT solver. I know how to reduce a 3-SAT ...
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### Understanding reductions: Would a polynomial time algorithm for one NP-complete problem mean a polynomial time algorithm for all NP-complete problems?

To prove that some decision problem $A$ is NP-complete, my understanding is that it suffices to show that the problem is in NP (i.e. that one can verify or reject all statements in polynomial time), ...
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### How does the problem of having a coffee-machine close relate to vertex cover?

Meeting rooms on university campuses may or may not contain coffee machines. We would like to ensure that every meeting room either has a coffee machine or is close enough to a meeting room that ...
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### Pseudo-polynomial time algorithm for NP-Complete Problems

For problems like Knapsack there is a pseudo-polynomial time algorithm and it is NP-complete. So we reduce every other problem in NP in polytime to Knapsack. But why don't we have then a pseudo-...
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### Is set cover still NP-complete if you have a given k?

Set cover is NP-complete given an arbitrary set $U$, a set $S$ of subsets of $U$, and an integer $k$. However, what if $k$ is always a constant 3? Is that problem still NP-complete?
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### maximum flow with all or nothing through each edge

Consider a maximum flow problem, where each edge has a small integer capacity. Now, I want a solution that for each edge uses the entire capacity, or no flow through that edge at all. To avoid the ...
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### Why are there no complete problems for NP ∩ coNP?

The fact that PPAD is a subclass of TFNP seems to be taken as evidence that PPAD cannot be shown complete (or hard) for classes of independent interest like NP ∩ coNP. Slightly confusing, it even ...
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### Relativization of NP-completeness [duplicate]

This is actually exercise 3.7 from "Computational Complexity: A Modern Approach". I need to prove that the NP-Completeness of 3-sat does not relativize, i.e. I need to show that that exists some ...
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### Proof of NP-completeness of graph isomorphism through edge contractions that reduce a metric [duplicate]

Duplicate: NP-completeness of graph isomorphism through edge contractions with an edge validity condition I know that graph contractability is $NP$-complete. To be specific given $G=(V_1,E_1)$ ...
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### Will the Traveling Salesman Problem (TSP) become easier if the simple path constraint is omitted?

Will the Traveling Salesman Problem (TSP) become easier if the simple path constraint is omitted? That is, each node in the topological graph should be visited once at least (instead of exactly). Is ...
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### Showing a problem on a specific graph is NP-hard

Maybe there is a trivial answer to my question or maybe there is not any. But I ask it here and appreciate if anyone can give me a clear answer. We know that a set of problems like minimum clique ...
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### Complexity of Independent Set on Triangle-Free Planar Cubic Graphs

I know that IS (is there independent set of size at least $k$?) on planar cubic graphs is NP-Complete, and IS on triangle-free graphs is also NP-Complete. But how about IS on triangle-free planar ...
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### How to prove the NP-completeness of the Exact-3D-Matching problem by reducing the 3-Partition problem to it?

Background: The Exact-3D-Matching problem is defined as follows (The definition is from Jeff's lecture note: Lecture 29: NP-Hard Problems. You can also refer to 3-...
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### Is it allowed to do a binary search with an oracle when proving NP-completeness?

In https://cs.stackexchange.com/a/45524/28999, they do a binary search using an oracle for an NP-Complete problem. They show that the original problem can be reduced to that NP-Complete problem, ...
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### Check if K-Sum Variation is NP-Complete

Problem Given a range of integers $\{a,a+1,...,b-1,b\}$, find a subset of size $k$ such that the sum is equal to $s$. Question This problem came from evaluating some scheduling algorithms that I am ...
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### Transforming 3 SAT to Simultaneous In-Congruence Problem?

Garey and Jhonson mentions that a 3-SAT Problem can be transformed to another NP-Complete Problem - Simultaneous incongruences (AN2): Given a collection $[(a_1,b_1),…,(a_n,b_n)]$ of ordered pairs of ...
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### A Query regarding Quadratic Residuocity Problem

In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that: $$x^2\equiv q \pmod{n}.$$...
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### Direction of restriction for NP hard proves

I have a very silly question, as I am reading through all the proofs showing a problem is NP hard, one of the techniques is by showing an already-proven NP complete problem is a special case for that ...
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### If an NP-complete problem is shown to have a non-polynomial lower bound, would that prove that P != NP?

I understand that the Cook-Levin theorem proved that any NP problem is reducible to an NP-complete problem, which signifies that if a polynomial-time algorithm for an NP-complete problem is found, it ...
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### Which of the following problems can be reduced to the Hamiltonian path problem?

I'm taking the Algorithms: Design and Analysis II class, one of the questions asks: Assume that P ≠ NP. Consider undirected graphs with nonnegative edge lengths. Which of the following problems ...
I am trying to reduce a Knapsack problem to a problem I need to solve, and I am suspicious of its NP-Completness. The problem recieve an array of elements $v_1,v_2,...,v_n$ sorted in some order from ...
We are given $n$ tasks and $m$ resources. Each task $i$ requires a set $S_i$ of resources to be active, and each resource can be used by at most one task. The Resource Allocation problem asks: given \$...