A row-invariant parameterized algorithm for integer programming
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A long line of research on fixed parameter tractability of integer programming culminated with showing that integer programs with n variables and a constraint matrix with tree-depth d and largest entry D are solvable in in time g(d,D)poly(n) for some function g, i.e., fixed parameter tractable when parameterized by tree-depth d and D. However, the tree-depth of a constraint matrix depends on the positions of its non-zero entries and thus does not reflect its geometric nature, in particular, is not invariant under row operations. We consider a parameterization of the constraint matrix by a matroid parameter called branch-depth, which is invariant under row operations. Our main result asserts that integer programs whose matrix has branch-depth d and largest entry D are solvable in time f(d,D)poly(n). Since every constraint matrix with small tree-depth has small branch-depth, our result extends the result above. The parameterization by branch-depth cannot be replaced by the more permissive notion of branch-width.
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