The Complexity of Cluster Vertex Splitting and Company

Clustering a graph when the clusters can overlap can be seen from three different angles: We may look for cliques that cover the edges of the graph, we may look to add or delete few edges to uncover the cluster structure, or we may split vertices to separate the clusters from each other. Splitting a vertex $v$ means to remove it and to add two new copies of $v$ and to make each previous neighbor of $v$ adjacent with at least one of the copies. In this work, we study the underlying computational problems regarding the three angles to overlapping clusterings, in particular when the overlap is small. We show that the above-mentioned covering problem, which also has been independently studied in different contexts,is NP-complete. Based on a previous so-called critical-clique lemma, we leverage our hardness result to show that Cluster Editing with Vertex Splitting is also NP-complete, resolving an open question by Abu-Khzam et al. [ISCO 2018]. We notice, however, that the proof of the critical-clique lemma is flawed and we give a counterexample. Our hardness result also holds under a version of the critical-clique lemma to which we currently do not have a counterexample. On the positive side, we show that Cluster Vertex Splitting admits a vertex-linear problem kernel with respect to the number of splits.Comment: 30 pages, 9 figure

Enhanced cytotoxicity and decreased CD8 dependence of human cancer-specific cytotoxic T lymphocytes after vaccination with low peptide dose.

In mice, vaccination with high peptide doses generates higher frequencies of specific CD8+ T cells, but with lower avidity compared to vaccination with lower peptide doses. To investigate the impact of peptide dose on CD8+ T cell responses in humans, melanoma patients were vaccinated with 0.1 or 0.5 mg Melan-A/MART-1 peptide, mixed with CpG 7909 and Incomplete Freund's adjuvant. Neither the kinetics nor the amplitude of the Melan-A-specific CD8+ T cell responses differed between the two vaccination groups. Also, CD8+ T cell differentiation and cytokine production ex vivo were similar in the two groups. Interestingly, after low peptide dose vaccination, Melan-A-specific CD8+ T cells showed enhanced degranulation upon peptide stimulation, as assessed by CD107a upregulation and perforin release ex vivo. In accordance, CD8+ T cell clones derived from low peptide dose-vaccinated patients showed significantly increased degranulation and stronger cytotoxicity. In parallel, Melan-A-specific CD8+ T cells and clones from low peptide dose-vaccinated patients expressed lower CD8 levels, despite similar or even stronger binding to tetramers. Furthermore, CD8+ T cell clones from low peptide dose-vaccinated patients bound CD8 binding-deficient tetramers more efficiently, suggesting that they may express higher affinity TCRs. We conclude that low peptide dose vaccination generated CD8+ T cell responses with stronger cytotoxicity and lower CD8 dependence