Combinatorial redundancy detection

The problem of detecting and removing redundant constraints is fundamental in optimization. We focus on the case of linear programs (LPs) in dictionary form, given by n equality constraints in $$n+d$$ n + d variables, where the variables are constrained to be nonnegative. A variable $$x_r$$ x r is called redundant, if after removing $$x_r \ge 0$$ x r ≥ 0 the LP still has the same feasible region. The time needed to solve such an LP is denoted by $$\textit{LP}(n,d)$$ LP ( n , d ) . It is easy to see that solving $$n+d$$ n + d LPs of the above size is sufficient to detect all redundancies. The currently fastest practical method is the one by Clarkson: it solves $$n+d$$ n + d linear programs, but each of them has at most s variables, where s is the number of nonredundant constraints. In the first part we show that knowing all of the finitely many dictionaries of the LP is sufficient for the purpose of redundancy detection. A dictionary is a matrix that can be thought of as an enriched encoding of a vertex in the LP. Moreover—and this is the combinatorial aspect—it is enough to know only the signs of the entries, the actual values do not matter. Concretely we show that for any variable $$x_r$$ x r one can find a dictionary, such that its sign pattern is either a redundancy or nonredundancy certificate for $$x_r$$ x r . In the second part we show that considering only the sign patterns of the dictionary, there is an output sensitive algorithm of running time $$\mathcal {O}(d \cdot (n+d) \cdot s^{d-1} \cdot \textit{LP}(s,d) + d \cdot s^{d} \cdot \textit{LP}(n,d))$$ O ( d · ( n + d ) · s d - 1 · LP ( s , d ) + d · s d · LP ( n , d ) ) to detect all redundancies. In the case where all constraints are in general position, the running time is $$\mathcal {O}(s \cdot \textit{LP}(n,d) + (n+d) \cdot \textit{LP}(s,d))$$ O ( s · LP ( n , d ) + ( n + d ) · LP ( s , d ) ) , which is essentially the running time of the Clarkson method. Our algorithm extends naturally to a more general setting of arrangements of oriented topological hyperplane arrangements.