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3

The problem is NP-hard, at least for a particular simplified configuration. Assume that each $m_l$ is effectively infinite - we can scan a particular libraries' books all on the day we get access. Let all $d_l = 1$ - each library takes one day to get access to, meaning we get access to exactly $n$ libraries. Now if $P_b = 1$ the thing that maximizes our ...


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Computer Science is far too diverse to have a single notion of program. For a theoretician, it might be an input tape to a universal Turing Machine, or any lambda expression in lambda calculus. For someone working on the front end of a compiler, it could be defined by a formal grammar while for someone working on the back end, or to the HW architect, it ...


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This is not IMO a 'computer science' matter. A 'program' is necessarily expressed in a particular programming language, whether that's a programming language with an implementation on a computer, or a 'mere' logical formalism. The particular language will tell you what a program is. Which means that it's a grubby implementation issue, not a pristine ...


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Given an instance of the second problem, we can easily reduce it to an instance of the (decision version of) the first problem: simply take $W = k$. There is a subset of sum at least $k$ and at most $W$ iff there is a subset of sum exactly $k$. In the other direction, the idea is to add to the collection a small set of items satisfying the following two ...


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Search for Bellare and Goldwasser "The Complexity of Decision vs. Search" SIAM J. of Computing 23:1 (Feb 1994), pp. 97-119, or Bellare's class note on the matter.


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It is easy to see that if both $L$ and it's complement $\bar{L}$ are recognizable by Turing machine, they can be decided (copy the input to a new tape, then run the machines for $L$ and $\bar{L}$ in parallel, the first one to stop tells you what the answer is; by assumption at least one of them stops in finite time). So the possibilities are: $L$ and $\bar{...


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An answer to this precise question is given by Bellare and Goldwasser, "The Complexity of Decision Versus Search", SIAM Journal on Computing, 23:1 (February 1994), DOI /10.1137/S0097539792228289; a more expository version of the above is Bellare's class note on this. The short answer is that if the decision problem is NP-complete, the search problem is "NP-...


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A clause in 3-CNF can be converted to k-CNF by adding extra padding: (l1 V l2 V l3) can be converted to (l1 V l2 V l3 V y) ∧ (l1 V l2 V l3 V y') Keep adding this extra padding until each clause contains k literals. Same way, a k-CNF clause can be broken until each clause contains 3 literals. (l1 V l2 V l3 V l4) can be broken into (l1 V l2 V y) ∧ (l3 V ...


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The direction of reduction that you are asking for is a bit strange. Typically, we reduce from $A_{TM}$, in order to show undecidability. Perhaps you meant to ask about the other direction? At any rate, in answer to your question: your attempt was actually quite close, it just needs a bit of modification. Here's how you can proceed: Given $M_1,M_2$ as ...


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Using the proof of the Cook–Levin theorem, for every input $x$ you can construct in polynomial time a SAT instance $\phi(r,z)$ which encodes "$M$ accepts when run on input $x$ and randomness $r$". Here $r$ is a vector of $m = \mathit{poly}(n)$ bits, representing the random bits of $M$, and $z$ is an auxiliary vector, with the following property: in any ...


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I don't know of a reference, but there's a standard reduction from Hamiltonian paths/cycles to SAT which is fairly well-known. The reduction from SAT to 3SAT is then done by introducing additional variables to break up clauses, e.g.: $$v_1 \vee \neg v_2 \vee v_3 \vee \neg v_4 \Rightarrow \left( v_1 \vee \neg v_2 \vee X\right) \wedge \left( v_3 \vee \neg v_4 ...


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The decision problem that corresponds to $L$ deals with the behaviour of Turing machines, not with strings of $1$'s and $0$'s. More precisely, the decision problem is Given a Turing machine $M$, is it the case that $L(M) = \{ 1^n0^n \mid > n \geq 0 \}$ A decider for this decision problem would take as input the description $\langle M \rangle$ of an ...


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You're using the terminology of recursion theory. Let me rephrase the argument in terms of computer programs. $K$ is the set of all programs $x$ which halt when run on $x$ as input. $K_1$ is the set of all programs $x$ which halt on some input. Given a program $x$, consider the following program $f(x)$: For $n=1,2,3,\ldots$: run program $x$ on inputs $1,\...


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I know an example of a problem that is missing two of the four features you ask for - it is not NP-complete, and it is not a problem on graphs. Buchfuhrer and Umans (2011) show that the minimum equivalent expression problem in Boolean logic is complete for $\Sigma^P_2$ under polynomial-time Turing reductions. Given a Boolean $(\wedge;\vee;\neg)$-formula $F$...


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As in the comment, consider the collection of problems $N$-SAT (Is $\phi$, a logical formula in $N$-CNF, satisfiable?). Or $N$-coloring of graphs, for $N \ge 3$ (Can the graph be colored with $N$ colors?). Many NP-complete problems have some parameter (Is there a clique of size $k$ in the graph? Has the digraph a feedback vertex set of size $k$?).


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For every integer k, take the travelling salesman problem with n > 1 cities where n is a power of k. (Picked the problem that way because all the instances are distinct, so we can say with good conscience that these are distinct problems).


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