Fast Algorithms for Separable Linear Programs

Sally Dong, Gramoz Goranci, Lawrence Li, Sushant Sachdeva, Guanghao Ye

Abstract: In numerical linear algebra, considerable effort has been devoted to obtaining faster algorithms for linear systems whose underlying matrices exhibit structural properties. A prominent success story is the method of generalized nested dissection [Lipton-Rose-Tarjan’79] for separable matrices. On the other hand, the majority of recent developments in the design of efficient linear program (LP) solvers have not leveraged the ideas underlying these faster linear system solvers nor exploited the separable structure of the constraint matrix.In this work, we consider LPs of the form minAx=b,l≤x≤u CTx, where the graphical support of the constraint matrix A ∈ ℝn×m is nα-separable. We present an Õ((m + m1/2+2α) log(1/ɛ))-time algorithm for solving these LPs to e relative accuracy.Our new solver has two important implications: for the k-multicommodity flow problem on planar graphs, we obtain an Õ(k5/2 m3/2 log(1/ɛ))-time algorithm; and when the support of A is nα-separable with α ≤ 1/4, our runtime of Õ(m log(1/ɛ)) is nearly optimal. The latter significantly improves upon the natural approach of combining interior point methods and nested dissection, whose time complexity is lower bounded by where ω ≈ 2.373 is the matrix multiplication exponent. Lastly, our solver can be applied to low-treewidth LPs to recover the results of [DLY21,GS22] while using significantly simpler data structure machinery.