Can all packing problems be rephrased as set packing problems?

Can all covering problems be rephrased as set covering problems?

In other words, I was wondering if set packing/covering problems are the most general forms of packing/covering problems?

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    $\begingroup$ Packing/covering problems are larger classes of problems, because the "standard" set packing/covering problems only deal with finite sets (see their definitions). Furthermore you can formulate packing/covering problems using uncomputable stuff; for example: "How many circles of ray $K(x)$ are needed to cover a $x \times x$ square; where $K(x)$ is the Kolmogorov complexity of the binary representation of $x$?" $\endgroup$ – Vor Oct 29 '12 at 0:01
  • $\begingroup$ I edited out the fourth, less related question. If you are still interested, please ask it separately. $\endgroup$ – Raphael Feb 23 '15 at 11:36

If we restrict ourselves to problems in NP, then yes. There are NP-complete problems of either category, so (many-one poly-time) reductions exist; we can view these as "rephrasing".

Since integer programming is also NP-complete, these same is true for that one.

Problems outside of NP do not reduce to integer programming, but I am sure that any (meaningful) complexity class contains examples of both covering and packing problems.

It's not the "type" of a problem that determines its hardness. There are satisfiability problems all the way from linear complexity to undecidability, to name just one example.

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