Testwiki:Reference desk/Archives/Mathematics/2016 June 25

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June 25

Variation on Qudratic Programming

Is the following problem NP-complete? $L=\{<Q,\lambda >|Qisn\times nmatrix,,\lambda \in ℝ,\mathrm{\exists }x(x^{T}Qx<\lambda ,\mathrm{\exists }i(x_i\in [0,1],\mathrm{\forall }j\ne i{x}_j\in \{0,1\}))\}$ עברית (talk) 07:53, 25 June 2016 (UTC)

There is always a solution: set all x's equal to 0. Loraof (talk) 15:02, 25 June 2016 (UTC) That's assuming lambda is positive. Loraof (talk) 15:04, 25 June 2016 (UTC)

After thinking about it for a while, it seems NP-hard to me, but I can't find a proof of that. Loraof (talk) 19:49, 25 June 2016 (UTC)

I don't assume that $\lambda$ is positive. (nor I assume Q's entries to be positive) 132.67.104.216 (talk) 15:24, 26 June 2016 (UTC)

I'm not an expert but AFAIK questions of computability and complexity are usually asked about countable sets. The inputs and witnesses are based on integers. I think you have wholly different questions when you put real numbers in the mix. A classical Turing machine can't operate on real numbers.

That said, I think the problem will not materially change if you replace all occurrences of real numbers with rational numbers, and then I think they can be handled safely. -- Meni Rosenfeld (talk) 10:49, 27 June 2016 (UTC)