发表机构
Peking University（北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种Tardos型强多项式精确算法，用于求解整数等式矩阵的连续可分离凸二次规划，通过压缩与邻近性转移实现高效求解，并扩展至分段二次函数及多商品流应用。
AI 中文摘要
我们给出了一种用于连续可分离凸二次规划的精确算法，该规划具有整数等式矩阵和非负变量。有理算术运算和比较的次数在维数和约束矩阵的编码长度上是多项式的，与右侧、线性成本以及所有非负二次权重无关。中间有理编码在整个输入上是多项式的。这为每一类具有多项式有界条目编码的整数矩阵类（包括条目属于$\{0,\pm1\}$的任意矩阵）产生了一种强多项式算法。局部求解器在整数盒上压缩二次目标，并且仅求解压缩后的连续问题。在具有有理界的盒上的双侧邻近性将其解转移到附近的原始最优解；整数最优解仅用于证明。一个线性规划提供误差界和对偶势，而紧坐标揭示和固定可行锚点确保终止并控制编码长度。一个显式提升将该结果扩展到有符号整数仿射特征的凸分段二次函数，操作次数与断点和目标系数无关。应用包括具有共享容量和聚合拥塞的分段二次成本的连续多商品流。
英文摘要
We give an exact algorithm for continuous separable convex quadratic programming with an integer equality matrix and nonnegative variables. The number of rational arithmetic operations and comparisons is polynomial in the dimensions and the encoding length of the constraint matrix, independently of the right-hand side, linear costs, and all nonnegative quadratic weights. Intermediate rational encodings are polynomial in the complete input. This yields a strongly polynomial algorithm for every integer matrix class with polynomially bounded entry encodings, including arbitrary matrices with entries in $\{0,\pm1\}$. The local solver compresses a quadratic objective on an integer box and solves only the compressed continuous problem. Two-sided proximity on boxes with rational bounds transfers its solution to a nearby original optimum; integer optima are used only in the proof. A single linear program supplies an error bound and a dual potential, while tight-coordinate revelation and fixed feasible anchors ensure termination and control encoding lengths. An explicit lifting extends the result to convex piecewise quadratic functions of signed integer affine features, with operation counts independent of breakpoints and objective coefficients. Applications include continuous multicommodity flow with shared capacities and piecewise quadratic costs of aggregate congestion.
Comments17 pages, no figures
