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Quickest known integer relation algorithm in the case of signs

Let $x_1,\cdots,x_n,k$ be integers such that $|k|\le\sum_{i=1}^n|x_i|$. What is the quickest known algorithm to determine $w_i\in\{-1,0,1\}$ such that $k=\sum_{i=1}^nw_ix_i$ where they exist? What is ...
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How to model the following Constraint

I would like to model the following: $B \le \alpha \implies \sum_i W(i) \ge \beta$, where $B$ a continuous variable, $W(i)$ binary variables, $\alpha$ a real constant number, $\beta$ an integer ...
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How to identify constraints that make problem not solvable in polynomial time?

I am reading this paper, available for free viewing, which contains an example of job shop scheduling, shown below. The details of the variable definitions, etc., can be found in the paper, but it's ...
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Using binary variable or integer variable in Constraint Programming (CP)?

I am trying to solve an assignment problem by Constraint Programming (CP). Let us consider a task assignment problem that have $|M|$ tasks and $|N|$ machines, each task has different processing time ...