# Questions tagged [np-hardness]

Questions related to NP-hardness and NP-completeness.

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### A $TM$ definition on approximation solutions to optimization problems

It looks like we are defining approximation solutions as a witness specifying program. We want to find a witness which agrees within some approximation of optimal. Is it possible to specify as '...
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### Is the traveling salesman problem still NP-hard if all edges need to be covered as well?

If we formulate the travelling salesman problem with an added edge-covering constraint as follows, is it still NP-hard? Given a graph G with non-negative edge weights, is there a circular walk in G ...
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### How hard is deciding the existence of a polygonization with prescribed perimeter?

Polygonization problem of a set of points in the Euclidean plane (2D lattice) is to find a simple polygon that passes through all points. Deciding the existence of a polygonization with minimum (or ...
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### Why does Dinur's proof of the PCP theorem fail to work for unique games?

What is the critical step where things go wrong if one attempts to use Dinur's proof the PCP theorem to prove the unique games conjecture by starting from a unique label cover instance and doing gap ...
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### Parametrized complexity of sparse optimization

Optimization problems of the type: minimize $c^T x$ subject to [maybe some linear constraints and] $||x||_0\le k$ are known to be NP-hard. [Actually, I just realized that I don't have a reference, so ...
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### Complexity of a scheduling problem with a fixed left bound of jobs

Consider the following scheduling problem. We have a finite set of jobs $\mathcal{J}= \{j_1, j_2, ..., j_n\}$ to do. Every job $j_i$ has its own value $c_i$ (the amount of money we are paid for ...
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### NP-complete decision problems on deterministic automata

Do you know any NP-complete decision problems on deterministic automata? Most NP-complete problems that come to my mind are either (see, or here) graph theoretical, or involve some string rewriting or ...
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### Complexity of constructing minimum depth decision trees

I am interested in the computational complexity of Problem 1: Given a finite, non-empty set $J$, given $A, B \subseteq \{0,1\}^J$ such that $A \cap B = \emptyset$, and given $n \in \mathbb{N}$, does ...
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