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June 25

Variation on Qudratic Programming

Is the following problem NP-complete? $L=\{<Q,\lambda >|Q\ is\ n\times n\ matrix,\ ,\lambda \in \mathbb {R} ,\ \exists x(x^{T}Qx<\lambda ,\ \exists i(x_{i}\in [0,1],\forall j\neq i\ x_{j}\in \{0,1\}))\}$ עברית (talk) 07:53, 25 June 2016 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]