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Recently I have come across a paper which talks of "(1+ε)-inapproximability" and of "logarithmic approximation".

While I have a basic knowledge of computational complexity (I more or less know what it means to be in P, NP, NP-hard and so on), I tried to study some basic texts on approximability but I'm not sure I really get what this inapproximability is. Could you enlighten me on what the above expressions mean?

If you want to know the paper I was referring to, it was DasGupta & Desai. «On the complexity of Newmanʼs community finding approach for biological and social networks». Journal of Computer and System Sciences 79, n. 1 (2013): 50–67. The found inapproximability is a property of the modularity measure (the so-called Q) related to the Newman-Girvan algorithm for finding community structure in networks, a measure whose maximization has been proved NP-complete by Brandes, Delling, Gaertler, Gorke, Hoefer, Nikoloski, and Wagner in 2008. Thank you in advance.

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  • $\begingroup$ What is "ELI10"? $\endgroup$ – David Richerby Aug 26 '16 at 16:43
  • $\begingroup$ @DavidRicherby: ELI10 = "explain like I'm 10". There is a large subreddit which is dedicated to ELI5-questions. reddit.com/r/explainlikeimfive The expression ELI5 is more common than ELI10. $\endgroup$ – Ching Chong Aug 26 '16 at 16:58
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    $\begingroup$ @ChingChong OK, so it's redundant, since any decent explanation should be comprehensible to somebody with the mathematical sophistication to ask the question in the first place. And we're not Reddit. $\endgroup$ – David Richerby Aug 26 '16 at 17:17
  • $\begingroup$ You might find Erik Demaine's MIT 6.890 lectures on inapproximability (intro, examples, gap problems) to be useful. (Disclaimer: I was in the audience.) $\endgroup$ – Jeffrey Bosboom Aug 26 '16 at 21:19
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Optimization problems come in two flavors: minimization and maximization. For definiteness, in this answer we consider minimization problems; for maximization problems the situation is completely analogous.

Generally speaking, when we say that a minimization problem $\Pi$ is $c$-hard to approximate, we mean the following:

If there is a polynomial time approximation algorithm for $\Pi$ whose approximation ratio is better than $c$ then [something bad happens].

Usually the "something bad" is $\mathsf{P} = \mathsf{NP}$, but sometimes we can only prove that, say $\mathsf{NP} \subseteq \mathsf{DTime}(n^{\log \log n})$ (that is, SAT can be solved in time $O(n^{ \log \log n})$). Since we believe that the "something bad" is wrong, it follows that probably no polynomial time approximation algorithm for $\Pi$ has an approximation ratio better than $c$.

Why this form of result? If $\mathsf{P} = \mathsf{NP}$, then all NP optimization problems (that is, all optimization problems whose decision version is in NP) can be solved in polynomial time. Therefore in order to prove that a certain problem cannot be approximated well we need to make some complexity assumption such as $\mathsf{P} \neq \mathsf{NP}$.

The best one can hope for is to prove that is an NP-hard to approximate $\Pi$ better than $c$. What this means is that for each $\epsilon > 0$ there is a polynomial reduction $A$ from SAT to $\Pi$ with the following properties:

  1. If the input formula $\varphi$ is satisfiable than the optimal value of $A(\varphi)$ is at most $x$, for some value $x$ which the reduction explicitly calculates (in polynomial time).

  2. If $\varphi$ is not satisfiable than the optimal value of $A(\varphi)$ is more than $(c+\epsilon)x$.

In particular, any polynomial time $(c+\epsilon)$-approximation algorithm would be able to distinguish the two cases, and thus to solve SAT in polynomial time.

Technically, what we have described is known as a promise problem: decide whether an instance of $\Pi$ has value at most $x$ or more than $(c+\epsilon)x$; or stated differently, find the optimal value of an instance of $\Pi$ (or at least distinguish the two cases) under the promise that the optimal value is either at most $x$ or more than $(c+\epsilon)x$.

In some cases we are able to prove that a certain problem $\Pi$ is $c$-hard to approximate, and also show that a polynomial time $c$-approximation algorithm exists. In this case we say that the inapproximability threshold of $\Pi$ is $c$. For example, Håstad shows that the threshold of inapproximability for 3SAT is $7/8$.

Our definition above of $c$-inapproximability states that in polynomial time it is impossible to approximate a certain problem better than $c$, but allows an approximation ratio of $c$. Sometimes we also want to disallow an approximation ratio of $c$. The terminology doesn't reflect this fine point, and formal statements of theorems have to be consulted.

Finally, $(1+\epsilon)$-inapproximability means that (under some complexity assumptions) it is impossible (in polynomial time) to approximate the problem to within $1+\epsilon$ for some $\epsilon > 0$ (in terms of our notation above, the problem is $1$-hard to approximate). We also say that the problem is APX-hard.

Logarithmic inapproximability means that it is impossible to approximate the problem to within $O(\log n)$, where $n$ is some natural parameter inherent in the problem, usually the number of vertices in the input graph.

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  • $\begingroup$ Thank you very much for your lengthy and accurate response. Do you have any advice about reasonably clear textx, books or papers, which explain the above notions? $\endgroup$ – Cesar Moncalvo Aug 26 '16 at 18:30
  • $\begingroup$ You can try Wikipedia and the references therein, such as Luca Trevisan's survey. $\endgroup$ – Yuval Filmus Aug 26 '16 at 18:33
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There are some NP-complete problems where you can get good approximations to an optimal solution, and some where you can't. Take colouring of a planar graph: Deciding whether a planar graph can be coloured in 3 colours is NP complete. On the other hand, every planar graph can be coloured in 4 colours. But there is no graph that would take say 3.01 colours. So if you try to colour a 3 colourable graph, you either solve the problem, or your solution is at least 33.3333% worse than the optimal solution. No (1 + eps) approximation if eps < 1/3.

Other problems allow arbitrarily good approximations that get harder to find as you try to improve the approximation.

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  • $\begingroup$ So, basically, you're saying that there's no sense in which approximating the chromatic number of a planar graph makes much sense. But what does this have to do with the question? $\endgroup$ – David Richerby Aug 26 '16 at 17:21
  • $\begingroup$ @DavidRicherby: Seriously? "but I'm not sure I really get what this inapproximability is. Could you enlighten me on what the above expressions mean" $\endgroup$ – gnasher729 Aug 26 '16 at 19:41

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