Suppose you are given a function $f\colon [n] \to [n]$ via (black-box) query access to the function. You are looking to find something local, like a collision (a pair $x \neq y$ s.t. $f(x)=f(y)$). The question is whether knowing the "shape" of the function helps you or not (by shape we mean that some permutation of the function is known). Formally, we investigate the unlabeled instance optimality of substructure detection problems in graphs and functions. A problem is $g(n)$-instance optimal if it admits an algorithm $A$ satisfying that for any possible input, the (randomized) query complexity of $A$ is at most $g(n)$ times larger than the query complexity of any algorithm $A'$ which solves the same problem while holding an unlabeled copy of the input (i.e., any $A'$ that "knows the structure of the input"). Our results point to a trichotomy of unlabeled instance optimality among substructure detection problems in graphs and functions: 1. A few very simple properties have an $O(1)$-instance optimal algorithm. 2. Most properties of graphs and functions, with examples such as containing a fixed point or a $3$-collision in functions, or a triangle in graphs, are $n^{Ω(1)}$-far from instance optimality. 3. The problems of collision detection in functions and finding a claw in a graph serve as a middle ground between the two regimes. We show that these two properties are $Ω(\log n)$-far from instance optimality, and conjecture that this bound is tight. We provide evidence towards this conjecture, by proving that finding a claw in a graph is $O(\log(n))$-instance optimal among all input graphs for which the query complexity of an algorithm holding an unlabeled certificate is $O\left(\sqrt{\frac{n}{\log n}}\right)$.

A tournament is an orientation of a complete graph. A vertex that can reach every other vertex within two steps is called a king. We study the complexity of finding k kings in a tournament graph. We show that the randomized query complexity of finding k ≤ 3 kings is O(n), and for the deterministic case it takes the same amount of queries (up to a constant) as finding a single king (the best known deterministic algorithm makes O(n^{3/2}) queries). On the other hand, we show that finding k ≥ 4 kings requires Ω(n²) queries, even in the randomized case. We consider the RAM model for k ≥ 4. We show an algorithm that finds k kings in time O(kn²), which is optimal for constant values of k. Alternatively, one can also find k ≥ 4 kings in time n^{ω} (the time for matrix multiplication). We provide evidence that this is optimal for large k by suggesting a fine-grained reduction from a variant of the triangle detection problem.

In this paper, we show that every $(2^{n-1}+1)$-vertex induced subgraph of the $n$-dimensional cube graph has maximum degree at least $\sqrt{n}$. This result is best possible, and improves a logarithmic lower bound shown by Chung, Füredi, Graham and Seymour in 1988. As a direct consequence, we prove that the sensitivity and degree of a boolean function are polynomially related, solving an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.