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Questions about problems that entail selecting the best element from some set of available alternatives, and methods to solve them.

Full details are in Robert L. Karg and James L. Thompson, A Heuristic Approach to Solving Traveling Salesman Problems (Management Science, 10(2):225–248, 1964). The PDF is available from JStor and In …
answered Mar 18 '15 by David Richerby
This general form of problem is a facility location problem. The specific version you're asking about is a covering problem – you have a set of points (the houses) and you're trying to cover them with …
answered Nov 15 '14 by David Richerby
narek gives the polynomial-time algorithm. The more general problem is the facility location problem. This includes several problems but the basic idea is that you want to know where to build your res …
answered Jul 9 '18 by David Richerby
This is a knapsack problem. In the most basic version, you have a single resource (e.g., space in your knapsack) and you're trying to choose which items to put in it so you can carry the greatest valu …
answered Apr 8 '17 by David Richerby
It appears that you have a periodic scheduling system on a single processor where task $i$ must run for $C_i$ seconds out of every $T_i$. If this is the case, then the requirement that $\tfrac{C_i}{T …
answered Mar 23 '15 by David Richerby
This is just the assignment problem: you need to find a maximum-weight matching between the users and frequencies. The Hungarian algorithm solves this in time $O(n^3)$.
answered Nov 9 '15 by David Richerby
There's no possibility of an efficient algorithm here, unless you have more information about $T$. $T$ might only return true for one input and, in the worst case, you'd have to try all $2^n$ possibil …
answered May 22 '15 by David Richerby
Represent the events as a bipartite graphs: the vertices are the users and subjects and the edges are events (so there's an edge from user $x$ to subject $y$ if there's an event involving that user an …
answered Aug 25 '14 by David Richerby
You need to formally define the computational problem. It appears that the input is "a programming language" and the output is the length of the shortest quine in that language. But how is the program …
answered May 26 '14 by David Richerby
problem is what is known as an NP optimization problem. It asks us to minimize a certain function $f$ and the decision problem, "Given $S$ and $k$, is $f(S)\leq k$?" is in NP. Now, if this decision … problem is NP-complete, the optimization problem is NP-hard: there's a trivial reduction from the decision version to the optimization problem. Going the other way around, if you had an efficient algorithm for the decision problem, you could find out the optimum by binary search. …
answered Dec 9 '13 by David Richerby
You seem to have chosen a strange metric, in that your wheelchair user apparently prefers travelling 1000km over concrete to even 1cm over gravel. However, in general, the way to proceed is to combin …
answered Dec 16 '16 by David Richerby
You can write $O(f)$ for any function $f$ and it makes perfect sense. As per the definition, $g(n)=O(f(n))$ if there is some constant $c$ such that $g(n)\leq c\,f(n)$ for all large enough $n$. Nothing …
answered May 9 by David Richerby
I am reading on how [...] to solve unweighted bipartite graph matching problem. [...] The goal of the problem seems to be to find a maximum matching in a complete bipartite graph No, the goal of …
answered Jun 1 '16 by David Richerby
The problem of finding a longest simple cycle in a digraph is NP-hard, since the problem of finding a longest simple cycle in an undirected graph is a special case: you can consider an undirected grap …
answered Dec 21 '15 by David Richerby
An optimization problem is an example of a function problem: i.e., one where the task is to map some input to some output. The class of function problems solvable in polynomial time is FP. See, for … example, the Complexity Zoo. (Note that there is a class OptP but that's not the polynomial-time optimization problems. Perhaps confusingly, it's the optimization analogue of NP: it's the class of …
answered Mar 8 '17 by David Richerby

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