# Questions tagged [approximation-hardness]

Hardness of approximation, aka inapproximability.

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### Are runtime bounds in P decidable? (answer: no)

The question asked is whether the following question is decidable: Problem  Given an integer $k$ and Turing machine $M$ promised to be in P, is the runtime of $M$ ${O}(n^k)$ with respect ...
2k views

### Hardness of approximation - additive error

There is a rich literature and at least one very good book setting out the known hardness of approximation results for NP-hard problems in the context of multiplicative error (e.g. 2-approximation for ...
508 views

### Smoothed analysis of approximation algorithms

Smoothed analysis has been applied many times to understand the runtime of exact algorithms for many problems like linear programming and k-means. There are fairly general results in this realm, for ...
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### Is Gap-3SAT NP-complete even for 3CNF formulas where no pair of variables appears in significantly more clauses than the average?

In this question, a 3CNF formula means a CNF formula where each clause involves exactly three distinct variables. For a constant 0<s<1, Gap-3SATs is the following promise problem: Gap-3SATs ...
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### Compendium of the Best Approximation and Hardness Results for NP optimization problems

Do you know any up-to-date wiki dedicated to NP optimization problems with their best approximation and hardness result? Based on the feedback, it seems that it is safe to assume there is not such a ...
855 views

### Hardness of approximation assuming NP != coNP

Two of the common assumptions for proving hardness of approximation results are $P \neq NP$ and Unique Games Conjecture. Are there any hardness of approximation results assuming $NP \neq coNP$ ? I am ...
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### Polynomial time approximation algorithms for machine scheduling: how many open problems are left?

In 1999, Petra Schuurman and Gerhard J. Woeginger published the paper "Polynomial time approximation algorithms for machine scheduling: Ten open problems". Since then, to the best of my knowledge, ...
3k views

### Hardness of approximation without the PCP theorem

An important application of the PCP theorem is that it yields "hardness of approximation" type results. In some relatively simpler cases one can prove such hardness without PCP. Is there, however, any ...
348 views

### Gap hardness of Multi-Dimensional Cover

Given a finite set $X$ and a collection $F$ of subsets of $X$, we define a cover of $X$ in $F$ as a subset of $F$ whose union is equal to $X$. A cover $C$ of $X$ in $F$ is said to be exact if the ...
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### Approximating #P-hard problems

Consider the classical #P-complete problem #3SAT, i.e., to count the number of valuations to make a 3CNF with $n$ variables satisfiable. I am interested in the additive approximability. Clearly, there ...
491 views

### Is the following graph optimization problem approximable within a constant factor?

Let $G=(V,E)$ be an undirected graph, and let $\pi$ be a permutation of the vertices in $V$. For a node $v\in V$, we denote by $\text{pred}_{\pi}(v)$ (respectively $\text{succ}_{\pi}(v)$) the set of ...
699 views

### Is MAX CUT approximation resistant?

CSP optimization problem is approximation resistant if it is $NP$-hard to beat the approximation factor of a random assignment. For instance, MAX 3-LIN is approximation resistant since a random ...