# Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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### Hamilton Decomposition Decision Problem

Let $G=(V,E)$ be an undirected graph. A decomposition of $V$ into disjoint subsets $V_i$ is called a Hamilton decomposition of $G$ if the subgraph induced by each set $V_i$ is either a Hamilton graph ...
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### What is known about the hardness of the chromatic index for restricted graph classes?

There is a nice paper from 1991 that contains three diagrams about different graphclass-families showing what is known about the hardness of determining the chromatic index for them. Are there any ...
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### Does 1-in-3 SAT remain NP-hard even if every variable occurs both positively and negatively?

The standard problem 1-in-3 SAT (or XSAT or X3SAT) is: Instance: a CNF formula with every clause containing exactly 3 literals Question: is there a satisfying assignment setting precisely 1 literal ...
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### Complexity of finding the maximal number of pair-wise disjoint sets

Assume that I have $P$ sets with elements taken from $r$ possible ones. Each set is of size $n$ ($n<r$), where the sets can overlap. I want to determine whether the following two problems are NP-...
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### Edge-partitioning into rainbow triangles

I'm wondering if the following problem is NP-hard. Input: $G = (V,E)$ a simple graph, and a coloring $f : E \to \{1,2,3\}$ of the edges ($f$ does not verify any specific property). Question:...
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### Maximum weight “fair” matching

I'm interested in a variant of the maximum weight matching in a graph, which I call "Maximum Fair Matching". Assume that the graph is full (i.e. $E=V\times V$), has even number of vertices, and that ...
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### Is Deolalikar's 2010 proof that $P \ne NP$ correct?

There was recently a claimed proof that $P \ne NP$. Not long after its publication there were raised some issues with this proof. So ... is the proof correct or not ? (Please only answer this if you ...
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### P vs NP: Instructive example of when Brute Force search can be avoided

To be able to explain the P vs NP problem to non-mathematicians I would like to have a pedagogical example of when Brute Force-search can be avoided. The problem should ideally be immediately ...
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### Complexity of marriage matching problem?

Suppose you have $n$ males and $n$ females. Each person has $m$ attributes. Each person indicates a set of attributes that a possible candidate should have. A matching is a set of pairs. Each pair ...
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### Complexity of the $(3,2)_s$ SAT problem?

Let define the $(3,2)_s$ SAT problem : Given $F_3$, a satisfiable 3-CNF formula, and $F_2$, a 2-CNF formula ($F_3$ and $F_2$ are defined on the same variables). Is $F_3 \wedge F_2$ satisfiable? What ...
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### What is complexity of this max-edge subgraph problem?

While discussing the question I had asked here, @NealYoung and I encounter another problem, which is to judge complexity of the problem below: Given a connected undirected graph, finding a maximum-...
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### Does $NP$-hardness of $c$-approximation (for some $c>1$) imply $APX$-hardness?

Assume that for a given minimization problem with only integer solutions, it is $NP$-hard to decide if the optimal solution is 5 or 6. I.e., a polynomial-time algorithm with an approximation ratio ...
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### Minimum Triangle Covers

Given a graph $G$, what is the minimum number of edges of $G$ that we need to delete to make the graph triangle free? To my untrained eye, this appears to be a difficult problem. Is this problem ...
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### A partition problem in which some numbers may be cut

In the standard partition problem, we are given some numbers whose sum is $2s$ and have to decide whether they can be partitioned into two subset whose sum is $s$. It is known to be NP-hard. However,...
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### NP-hardness on Cayley graphs

What is known about complexity of NP-hard problems on Cayley graphs? Suppose that the graph is given explicitly as the multiplication table of the group and the list of generators. So the input ...
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### What are good approximation algorithms for the subset sum problem so far?

By "good", I mean either the algorithm provides a relatively tight bound or it has a relatively fast running time. Any reference is welcome.
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### Fitting minimum number of rectangles of width/height 1 into a 2D grid

Consider a problem in which you are given a 2D grid (e.g. a chessboard) where certain squares are occupied and you need to put the minimum number of non-overlapping rectangles of any size w x h where ...
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