Bounded Integer Programming in Almost $N^{2d}$ Time
Lin Chen, Yuchen Mao, Guochuan Zhang
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作者提出确定性算法，将标准形式有界整数规划的运行时间从 N^{O(d^2)} 改进到约 N^{2d+ε}（固定 d 与 ε）。
cs.DS
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We study integer programs in standard form $\min\{\mathbf{c}^{\mathsf T}\mathbf{x}:A\mathbf{x}=\mathbf{b},\mathbf{l}\leq\mathbf{x}\leq\mathbf{u},\ \mathbf{x}\in\mathbb{Z}^n\}$, where $A\in\mathbb{Z}^{d\times n}$, all entries of $\mathbf{b},\mathbf{c},\mathbf{l},\mathbf{u}$ are integers, and every variable bound is finite. Let $N$ be an upper bound on the absolute values of the entries of $A$, and $c_{\max}$ one on the absolute values of the costs. For every fixed $d$ and every fixed $\varepsilon>0$ we give a deterministic algorithm running in time \[ T_{\textrm LP}+O_d(n)+O_{d,\varepsilon}\bigl(N^{2d+\varepsilon}\log^2(c_{\max}+2)\bigr), \] where $T_{\textrm LP}$ is the time to solve one linear programming relaxation exactly. Prior to our work, the best known running time for bounded variables was $N^{O(d^2)}$ [Eisenbrand \& Weismantel'20], and it has been a fundamental open problem whether the exponent $O(d^2)$ can be improved to one linear in $d$. We answer this question affirmatively, with the exponent $2d+\varepsilon$.
We study the fair allocation of indivisible items to agents with budget constraints and additive valuations. Each agent compares its own bundle with the subsets of another bundle that fit its budget. An allocation is envy-free up to one item (EF1) if, for every agent and every nonempty subset under comparison, removing an item of maximum value to that agent eliminates its envy. We require EF1 both between agents and towards the charity, which contains all unallocated items. EF1 allocations exist, but computing them in polynomial time has remained open even for homogeneous valuations. We resolve this problem with a polynomial-time algorithm for EF1 allocations under heterogeneous valuations. We also prove that detecting an EF1 violation is NP-complete. Equivalently, checking whether a given allocation is EF1 is coNP-complete. Finally, we give polynomial-time algorithms for envy-freeness up to any item (EFX) both when each agent's valuation is proportional to item size and when each agent has a fixed number of distinct positive item values. As an application, for homogeneous valuations and any $\varepsilon>0$, we obtain an EF1 allocation with at least a $(1/2-\varepsilon)$ fraction of the maximum social welfare in time polynomial in the input length and $1/\varepsilon$.