Motivated by digital marketplaces such as Uber and Airbnb, we study how a two-sided platform designs a menu of qualities, consumer prices, and provider wages. Consumers and providers self-select according to their private values and costs, so the platform must design its menu to attract consumers while inducing enough supply to meet their demand. We study this coordination problem over a continuous quality space, requiring supply to cover demand at every quality threshold. We formulate menu design as an optimization over trading distributions, which specify how much trade occurs at or above each quality. Through convexification of the technological cost function, we characterize the profit- and welfare-optimal trading distributions. This characterization shows how production technology alone can reduce menu size without sacrificing optimal profit or welfare, regardless of the type distributions, and determines when optimal menus can be finite. We then show that profit maximization can be highly efficient: under monotone hazard rate conditions on both sides, a profit-optimal menu achieves at least $1/(e-1)$ of optimal social welfare. We next consider the practical setting in which the menu size is restricted to a prescribed number of offers. We establish how the restricted menu approximates the unrestricted optimum in profit. We develop an algorithm that computes a menu within this size limit in polynomial time. The computed menu simultaneously approximates the unrestricted optimal profit and social welfare with additive error.
The Directed Feedback Vertex Set problem (DFVS) asks whether a digraph can be made acyclic by deleting at most $k$ vertices. Whether DFVS admits a polynomial kernel parameterized by $k$ is a major open problem in kernelization, even for planar digraphs. We resolve the planar case by giving a deterministic kernel with $O(k^{66}\log^2 k)$ vertices and arcs. Our algorithm proceeds in three stages. First, we apply structural reduction rules to the input digraph, bounding the number of directed faces and some special vertices. Second, we pass to the planar dual, where vertex deletion corresponds to adding groups of reverse arcs to make each weakly connected component strongly connected. The structural bounds in the first stage yield a small retained vertex set in the dual. We then compress the dual instance by identifying vertices with the same distance records from this retained vertex set. The main technical contribution is a directed-cut argument showing that this identification preserves feasibility. Finally, we transform the polynomial-size dual instance back into an instance of Planar Directed Feedback Vertex Set via a $3$-CNF encoding and a planar graph construction.
In \(d\)-Hitting Set, the input consists of a finite universe \(U\), a family \(\mathcal S\) of subsets of \(U\) with size at most \(d\), and an integer \(k\). The task is to decide whether at most \(k\) elements of \(U\) can intersect every set in \(\mathcal S\). Previously known kernels contain \(O(k^{d-1})\) elements and \(O(k^d)\) sets. Whether the number of elements can be reduced to \(O(k^{d-1-\varepsilon})\) for some constant \(\varepsilon>0\) is a longstanding open problem in kernelization. In this paper, we answer this question affirmatively by giving a near-linear randomized element-kernel and a subquadratic deterministic element-kernel through a re-encoding of the instance. For every fixed \(d\geq3\), we give a one-sided randomized kernel with \(O_d(k\log^3k)\) elements and a deterministic kernel with \(O_d(k^{7/4}\log^{3/2}k)\) elements. Our kernels may still contain \(k^{O(d)}\) sets and the parameter \(k\) may grow polynomially.