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.

翻译：我们研究形式为 $\min\{c^\top x\ \vert\ Ax=b,l\le x\le u,x\in\mathbb Z^n\}$ 的整数线性规划（ILP），并遵循“通过距离平凡性捕捉问题难度”这一成熟方法，分析其相对于广义匹配问题距离的参数化复杂度。广义匹配问题是一种ILP，其约束矩阵的每一列的1-范数至多为2。它涵盖了几个著名的多项式时间可解问题，如匹配和流问题。我们以变量后门和约束后门的大小作为参数，它们分别度量了为获得广义匹配ILP所需删除的最小列数或行数。我们提出了以下结果：(i) 针对以最小变量后门大小 $p$ 为参数的ILP，给出了一个固定参数可解（FPT）算法；(ii) 针对以混合变量加约束后门大小 $p+h$ 为参数的ILP，只要 $c$ 和 $A$ 以一元编码，给出了一个随机切片多项式（XP）时间算法；(iii) 我们通过证明即使所有系数有界，以最小约束后门大小 $h$ 为参数求解ILP也是W[1]-难的，从而对(ii)进行了补充。为得到(i)，我们证明了加权 $b$-匹配的度序列的一种格凸性变体，我们结合SBO跳跃M凸函数对其进行了研究。这使我们能够将匹配部分建模为整数后门变量上的多面体约束。所得的ILP使用一个FPT整数规划算法求解。对于(ii)，随机XP时间算法是通过将问题伪多项式归约到精确匹配问题而得到的。为了防止 $b$ 的编码长度出现指数级爆炸，我们通过基于子行列式的回路界来界定ILP的邻近性。