🤖 AI Summary
This paper investigates the parameterized complexity of Integer Linear Programming (ILP) with lower and upper bounds, parameterized by the “distance to generalized matching”—the minimum number of modifications required to transform the constraint matrix into one where each column has ℓ₁-norm at most two (encompassing polynomial-time solvable matching and flow problems). The authors introduce two novel structural parameters: a *variable backdoor* (minimum column deletions to achieve the generalized matching structure) and a *constraint backdoor* (minimum row deletions). They present the first fixed-parameter tractable (FPT) algorithm parameterized by variable backdoor size (p). In contrast, they prove W[1]-hardness parameterized by constraint backdoor size (h) and devise a randomized XP algorithm for unary-encoded instances. Key technical innovations include a variant of lattice convexity, Graver basis-enhanced local search, and a pseudo-polynomial reduction to Exact Matching—enabling both tight complexity classification and significant algorithmic advances.
📝 Abstract
We study integer linear programs (ILP) of the form $min{c^ op x vert Ax=b,lle xle u,xinmathbb Z^n}$ 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 $1$-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 $h$ of a minimum constraint backdoor to generalized matching as long as $c$ and $A$ are encoded in unary; (iii) we complement (ii) by proving that solving an ILP is W[1]-hard when parameterized by $h$ even when $c,A,l,u$ have coefficients of constant size. 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 in FPT time using an 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 Graver complexity of the constraint matrix and employ a Graver augmentation local search framework.