Parameterized Algorithms for Matching Integer Programs with Additional Rows and Columns
Alexandra Lassota, Koen Ligthart
TL;DR
This work studies integer linear programs whose constraint matrices are near the generalized matching structure (columns with $\ell_1$-norm at most $2$) through backdoor parameters. It establishes an FPT algorithm parameterized by the number $p$ of variables to delete to reach generalized matching, and a randomized XP algorithm for mixed backdoors with parameters $p+h$ under unary encoding, while proving W[1]-hardness when parameterized by the constraint backdoor $h$ alone. The key techniques combine a master reduction to perfect $b$-matching, lattice-convexity and SBO jump M-convexity to encode non-backdoor variables polyhedrally, and proximity-based reductions to (constrained) exact matching problems solvable by specialized algorithms. These results illuminate the boundary between tractable and intractable ILPs within a distance-to-triviality framework and provide new algorithmic tools for matching-structured ILPs, along with rich directions for future exploration (e.g., Graver-basis methods, stronger proximity bounds, and exact matching approaches).
Abstract
We study integer linear programs (ILP) of the form $\min\{c^\top x\ \vert\ Ax=b,l\le x\le u,x\in\mathbb Z^n\}$ and analyze their parameterized complexity with respect to their distance to the generalized matching problem, following the well-established approach of capturing the hardness of a problem by the distance to triviality. The generalized matching problem is an ILP where each column of the constraint matrix has a $1$-norm of at most $2$. It captures several well-known polynomial time solvable problems such as matching and flow problems. We parameterize by the size of variable and constraint backdoors, which measure the least number of columns or rows that must be deleted to obtain a generalized matching ILP. We present the following results: (i) a fixed-parameter tractable (FPT) algorithm for ILPs parameterized by the size $p$ of a minimum variable backdoor to generalized matching; (ii) a randomized slice-wise polynomial (XP) time algorithm for ILPs parameterized by the size $p+h$ of a mixed variable plus constraint backdoor to generalized matching as long as $c$ and $A$ are encoded in unary; (iii) we complement (ii) by proving that solving ILPs is W[1]-hard when parameterized by the size of a minimum constraint backdoor $h$ even when all coefficients are bounded. To obtain (i), we prove a variant of lattice-convexity of the degree sequences of weighted $b$-matchings, which we study in the light of SBO jump M-convex functions. This allows us to model the matching part as a polyhedral constraint on the integer backdoor variables. The resulting ILP is solved using an FPT integer programming algorithm. For (ii), the randomized XP time algorithm is obtained by pseudo-polynomially reducing the problem to the exact matching problem. To prevent an exponential blowup in terms of the encoding length of $b$, we bound the proximity of the ILP through a subdeterminant based circuit bound.