arXiv subjects

Conrad Schecker

Publications and source records attributed to Conrad Schecker.

3 recordsLinked to original sources

arXiv2026

On the Complexity of the Minimum-($k,\rho$)-Shortcut Problem

We consider the Minimum-$(k,\rho)$-$\mathrm{Shortcut}$ problem ($\min(k,\rho)\text{-}\mathrm{Shortcut}$), where the goal is to find the smallest set of shortcut edges such that every vertex in a given graph can reach its $\rho$ closest vertices using paths of at most $k$ edges. This is a fundamental graph optimization problem used to accelerate parallel shortest path algorithms. It is well-known that the problem is trivially solvable for the cases $k=1$ and $k\geq\rho$. While recent work by Leonhardt, Meyer, and Penschuck (ESA 2024) showed that in undirected graphs $\min(k,\rho)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq 3$ if $\rho=\Theta(n^\epsilon)$, the boundary where the problem transitions from polynomial-time solvable to NP-hard remained open. In this paper, we narrow this gap significantly. We present a simpler and more direct reduction from the Hitting Set problem which establishes that $\min(k,\rho)\text{-}\mathrm{Shortcut}$ is NP-hard for $k\geq2$ and $\rho\geq k+2$ in both directed and undirected graphs. Complementing this, we use the symmetry of the undirected case to show that $\rho=k+1$ is solvable in polynomial time, a regime where the directed version remains a candidate for NP-hardness. Therefore, we obtain an almost complete characterization of the complexity of $\min(k,\rho)\text{-}\mathrm{Shortcut}$, with the sole remaining open case being $\rho = k+1$ in the directed setting.

cs.CC↗

arXiv2024

Designing Exploration Contracts

We study a natural application of contract design in the context of sequential exploration problems. In our principal-agent setting, a search task is delegated to an agent. The agent performs a sequential exploration of $n$ boxes, suffers the exploration cost for each inspected box, and selects the content (called the prize) of one inspected box as outcome. Agent and principal obtain an individual value based on the selected prize. To influence the search, the principal a-priori designs a contract with a non-negative payment to the agent for each potential prize. The goal of the principal is to maximize her expected reward, i.e., value minus payment. Interestingly, this natural contract scenario shares close relations with the Pandora's Box problem. We show how to compute optimal contracts for the principal in several scenarios. A popular and important subclass is that of linear contracts, and we show how to compute optimal linear contracts in polynomial time. For general contracts, we obtain optimal contracts under the standard assumption that the agent suffers cost but obtains value only from the transfers by the principal. More generally, for general contracts with non-zero agent values for outcomes we show how to compute an optimal contract in two cases: (1) when each box has only one prize with non-zero value for principal and agent, (2) for i.i.d. boxes with a single prize with positive value for the principal.

cs.GT↗

arXiv2022

Delegated Online Search

In a delegation problem, a principal P with commitment power tries to pick one out of $n$ options. Each option is drawn independently from a known distribution. Instead of inspecting the options herself, P delegates the information acquisition to a rational and self-interested agent A. After inspection, A proposes one of the options, and P can accept or reject. Delegation is a classic setting in economic information design with many prominent applications, but the computational problems are only poorly understood. In this paper, we study a natural online variant of delegation, in which the agent searches through the options in an online fashion. For each option, he has to irrevocably decide if he wants to propose the current option or discard it, before seeing information on the next option(s). How can we design algorithms for P that approximate the utility of her best option in hindsight? We show that in general P can obtain a $Θ(1/n)$-approximation and extend this result to ratios of $Θ(k/n)$ in case (1) A has a lookahead of $k$ rounds, or (2) A can propose up to $k$ different options. We provide fine-grained bounds independent of $n$ based on two parameters. If the ratio of maximum and minimum utility for A is bounded by a factor $α$, we obtain an $Ω(\log \log α/ \log α)$-approximation algorithm, and we show that this is best possible. Additionally, if P cannot distinguish options with the same value for herself, we show that ratios polynomial in $1/α$ cannot be avoided. If the utilities of P and A for each option are related by a factor $β$, we obtain an $Ω(1/ \log β)$-approximation, where $O(\log \log β/ \log β)$ is best possible.

cs.GT↗