In the area of query complexity of Boolean functions, the most widely studied cost measure of an algorithm is the worst-case number of queries made by it on an input. Motivated by the most natural cost measure studied in online algorithms, the competitive ratio, we consider a different cost measure for query algorithms for Boolean functions that captures the ratio of the cost of the algorithm and the cost of an optimal algorithm that knows the input in advance. The cost of an algorithm is its largest cost over all inputs. Grossman, Komargodski and Naor [ITCS'20] introduced this measure for Boolean functions, and dubbed it instance complexity. Grossman et al. showed, among other results, that monotone Boolean functions with instance complexity 1 are precisely those that depend on one or two variables. We complement the above-mentioned result of Grossman et al. by completely characterizing the instance complexity of symmetric Boolean functions. As a corollary we conclude that the only symmetric Boolean functions with instance complexity 1 are the Parity function and its complement. We also study the instance complexity of some graph properties like Connectivity and k-clique containment. In all the Boolean functions we study above, and those studied by Grossman et al., the instance complexity turns out to be the ratio of query complexity to minimum certificate complexity. It is a natural question to ask if this is the correct bound for all Boolean functions. We show a negative answer in a very strong sense, by analyzing the instance complexity of the Greater-Than and Odd-Max-Bit functions. We show that the above-mentioned ratio is linear in the input size for both of these functions, while we exhibit algorithms for which the instance complexity is a constant.

翻译：在布尔函数的查询复杂度领域中，算法最广泛研究的代价度量是其对输入进行查询的最坏情况次数。受在线算法中最自然代价度量——竞争比——的启发，我们考虑布尔函数查询算法的另一种代价度量，该度量刻画算法代价与预知输入的优化算法代价之比。算法的代价是其所有输入上的最大代价。Grossman、Komargodski与Naor [ITCS'20] 针对布尔函数引入此度量，并将其命名为实例复杂度。Grossman等人证明了实例复杂度为1的单调布尔函数恰好是仅依赖一个或两个变量的函数。我们通过完全刻画对称布尔函数的实例复杂度，对上述结果进行了补充。作为推论，我们得出结论：实例复杂度为1的对称布尔函数仅有奇偶函数及其补函数。我们还研究了连通性和k-团包含等图性质的实例复杂度。在我们研究的所有布尔函数以及Grossman等人研究的函数中，实例复杂度均表现为查询复杂度与最小证书复杂度之比。一个自然的问题是这是否适用于所有布尔函数。我们通过分析大于函数和奇数最大比特函数的实例复杂度，从极强意义上给出了否定答案。我们表明上述比率对这两个函数均与输入规模呈线性关系，同时我们展示的算法可实现常数级别的实例复杂度。