We re-visit the complexity of polynomial time pre-processing (kernelization) for the d-Hitting Set problem. This is one of the most classic problems in Parameterized Complexity by itself, and, furthermore, it encompasses several other of the most well-studied problems in this field, such as Vertex Cover, Feedback Vertex Set in Tournaments (FVST) and Cluster Vertex Deletion (CVD). In fact, d-Hitting Set encompasses any deletion problem to a hereditary property that can be characterized by a finite set of forbidden induced subgraphs. With respect to bit size, the kernelization complexity of d-Hitting Set is essentially settled: there exists a kernel with 𝒪(k^d) bits (𝒪(k^d) sets and 𝒪(k^{d-1}) elements) and this it tight by the result of Dell and van Melkebeek [STOC 2010, JACM 2014]. Still, the question of whether there exists a kernel for d-Hitting Set with fewer elements has remained one of the most major open problems in Kernelization. In this paper, we first show that if we allow the kernelization to be lossy with a qualitatively better loss than the best possible approximation ratio of polynomial time approximation algorithms, then one can obtain kernels where the number of elements is linear for every fixed d. Further, based on this, we present our main result: we show that there exist approximate Turing kernelizations for d-Hitting Set that even beat the established bit-size lower bounds for exact kernelizations - in fact, we use a constant number of oracle calls, each with "near linear" (𝒪(k^{1+ε})) bit size, that is, almost the best one could hope for. Lastly, for two special cases of implicit 3-Hitting set, namely, FVST and CVD, we obtain the "best of both worlds" type of results - (1+ε)-approximate kernelizations with a linear number of vertices. In terms of size, this substantially improves the exact kernels of Fomin et al. [SODA 2018, TALG 2019], with simpler arguments.

Optimal transport has received much attention during the past few years to deal with domain adaptation tasks. The goal is to transfer knowledge from a source domain to a target domain by finding a transportation of minimal cost moving the source distribution to the target one. In this paper, we address the challenging task of privacy preserving domain adaptation by optimal transport. Using the Johnson-Lindenstrauss transform together with some noise, we present the first differentially private optimal transport model and show how it can be directly applied on both unsupervised and semi-supervised domain adaptation scenarios. Our theoretically grounded method allows the optimization of the transportation plan and the Wasserstein distance between the two distributions while protecting the data of both domains.We perform an extensive series of experiments on various benchmarks (VisDA, Office-Home and Office-Caltech datasets) that demonstrates the efficiency of our method compared to non-private strategies.

We consider four well-studied NP-complete packing/covering problems on graphs: F eedback V ertex S et in T ournaments (FVST), C luster V ertex D eletion (CVD), T riangle P acking in T ournaments (TPT) and I nduced P 3 -P acking . For these four problems, kernels with O ( k 2 ) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-S et P acking ) or a hitting set of size at most k for a family of sets of size at most 3 (3-H itting S et ). In this article, we give the first kernels for FVST, CVD, TPT, and I nduced P 3 -P acking with a subquadratic number of vertices. Specifically, we obtain the following results. • FVST admits a kernel with O ( k 3/2 ) vertices. • CVD admits a kernel with O ( k 5/3 ) vertices. • TPT admits a kernel with O ( k 3/2 ) vertices. • I nduced P 3 -P acking admits a kernel with O ( k 5/3 ) vertices. Our results resolve an open problem from WorKer 2010 on the existence of kernels with O( k 2−ϵ ) vertices for FVST and CVD. All of our results are based on novel uses of old and new “expansion lemmas” and a weak form of crown decomposition where (i) almost all of the head is used by the solution (as opposed to all ), (ii) almost none of the crown is used by the solution (as opposed to none ), and (iii) if H is removed from G , then there is almost no interaction between the head and the rest (as opposed to no interaction at all).

We consider four well-studied NP-complete packing/covering problems on graphs: Feedback Vertex Set in Tournaments (FVST), Cluster Vertex Deletion (CVD), Triangle Packing in Tournaments (TPT) and Induced P3-Packing. For these four problems kernels with O(k2) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-Set Packing) or a hitting set of size at most k for a family of sets of size at most 3 (3-Hitting Set). In this paper, we give the first kernels for FVST, CVD, TPT and Induced P3-Packing with a subquadratic number of vertices. Specifically, we obtain the following results. • FVST admits a kernel with vertices. • CVD admits a kernel with vertices. • TPT admits a kernel with vertices. • Induced P3-Packing admits a kernel with vertices. Our results resolve an open problem from WorKer 2010 on the existence of kernels with

It was conjectured by Jaeger et al. in 1992 that for any prime number $p$, there is a constant $c$ such that for any $n$, the union (with repetition) of the vectors of any family of $c$ linear bases of $\mathbb{Z}_p^n$ forms an additive basis of $\mathbb{Z}_p^n$ (i.e., any element of $\mathbb{Z}_p^n$ can be expressed as the sum of a subset of these vectors). In this note, we prove this conjecture when each vector contains at most two nonzero entries. As an application, we prove several results on flows in highly edge-connected graphs, extending known results. For instance, assume that $p\geqslant 3$ is a prime number and $\vec{G}$ is a directed, highly edge-connected graph in which each arc is given a list of two distinct values in $\mathbb{Z}_p$. Then $\vec{G}$ has a $\mathbb{Z}_p$-flow in which each arc is assigned a value of its own list.