We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $$k\in {\mathbb {N}}$$ that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and we show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both.

中文翻译：

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增量最大化的一般界限

我们提出了一个理论框架来捕获基数约束最大化问题的增量解决方案。我们框架的定义特征是解决方案的基数/支持受一个值 $$k\in {\mathbb {N}}$$ 的限制，该值随着时间的推移而增长，我们允许将解决方案扩展到一个元素一个时间。我们研究了这种增量解决方案的最佳可能竞争比率，即 k 步之后的增量解决方案与基数 k 的最佳解决方案之间在所有 k 中的最差比率。我们考虑一大类问题，其中包含许多重要的基数约束最大化问题，如最大匹配、背包和包装/覆盖问题。我们为此类问题提供了通用的 2.618 竞争增量算法，并且我们证明没有算法的竞争比率一般低于 2.18。在论文的第二部分，我们重点介绍了在每一步都尽可能增加目标值的固有增量贪婪算法。已知该算法对于子模块目标函数的竞争率为 1.58，但对于上述增量问题类别，它具有无限的竞争率。我们为目标函数定义了一个宽松的子模条件，捕获诸如最大（加权）d 维匹配、最大（加权）（b-）匹配和最大流问题的变体等问题。我们展示了贪心算法在满足这种宽松的子模块性条件的问题类别上的竞争比率的一般界限。我们的界限概括了 1 的（紧）界限。58 略微超出子模函数，并为最大（加权）（b-）匹配产生 2.313 的严格界限。随着相关参数趋于无穷大，我们的界限对于更一般的函数类也很严格。请注意，我们对竞争比率的上限转化为潜在基数约束问题的近似比率，而我们对贪婪算法的限制则适用于两者。