On the Complexity of Local Distributed Graph Problems

This paper is centered on the complexity of graph problems in the well-studied LOCAL model of distributed computing, introduced by Linial [FOCS '87]. It is widely known that for many of the classic distributed graph problems (including maximal independent set (MIS) and $(\Delta+1)$-vertex coloring), the randomized complexity is at most polylogarithmic in the size $n$ of the network, while the best deterministic complexity is typically $2^{O(\sqrt{\log n})}$. Understanding and narrowing down this exponential gap is considered to be one of the central long-standing open questions in the area of distributed graph algorithms. We investigate the problem by introducing a complexity-theoretic framework that allows us to shed some light on the role of randomness in the LOCAL model. We define the SLOCAL model as a sequential version of the LOCAL model. Our framework allows us to prove completeness results with respect to the class of problems which can be solved efficiently in the SLOCAL model, implying that if any of the complete problems can be solved deterministically in $\log^{O(1)} n$ rounds in the LOCAL model, we can deterministically solve all efficient SLOCAL-problems (including MIS and $(\Delta+1)$-coloring) in $\log^{O(1)} n$ rounds in the LOCAL model. We show that a rather rudimentary looking graph coloring problem is complete in the above sense: Color the nodes of a graph with colors red and blue such that each node of sufficiently large polylogarithmic degree has at least one neighbor of each color. The problem admits a trivial zero-round randomized solution. The result can be viewed as showing that the only obstacle to getting efficient determinstic algorithms in the LOCAL model is an efficient algorithm to approximately round fractional values into integer values

Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces

MUC1 immunotherapy against a metastatic mammary adenocarcinoma model: Importance of IFN-gamma

Abstract Immunotherapy using mucin 1 (MUC1) linked to oxidised mannan (MFP) was investigated in an aggressive MUC1+ metastatic tumour, DA3-MUC1 because, unlike many MUC1+ tumour models, DA3-MUC1 is not spontaneously rejected in mice making it an alternative model for immunotherapy studies. Further, DA3-MUC1 cells are resistant to lysis by anti-MUC1 cytotoxic T cells (CTLs). The inability of DA3-MUC1 tumours to be rejected in naïve mice as well as vaccination to MUC1 was attributed to a deficiency of expression of MHC class I molecules on the tumour cell surface. In vitro and in vivo analysis of subcutaneous tumours and lung metastases demonstrated that DA3-MUC1 tumour cells have a low expression (&lt; 6%) of MHC class I which can be upregulated (&gt; 90%) following culturing with IFN-γ. Results from flow cytometry analysis and immunoperoxidase staining indicated that the in vitro up-regulation of MHC class I could be maintained for up to seven days in vivo, without affecting the expression levels of MUC1 antigen. Interestingly, MUC1-specific CTL that lyse DA3-MUC1 targets in vitro were induced in MFP immunised mice but failed to protect mice from a DA3-MUC1 tumour challenge. These results highlight the importance of MHC class I molecules in the induction of anti-tumour immunity and the MFP immune response.</jats:p

Combinatorial Bounds for Conflict-free Coloring on Open Neighborhoods

In an undirected graph $G$, a conflict-free coloring with respect to open neighborhoods (denoted by CFON coloring) is an assignment of colors to the vertices such that every vertex has a uniquely colored vertex in its open neighborhood. The minimum number of colors required for a CFON coloring of $G$ is the CFON chromatic number of $G$, denoted by $\chi_{ON}(G)$. The decision problem that asks whether $\chi_{ON}(G) \leq k$ is NP-complete. We obtain the following results: * Bodlaender, Kolay and Pieterse [WADS 2019] showed the upper bound $\chi_{ON}(G)\leq {\sf fvs}(G)+3$, where ${\sf fvs}(G)$ denotes the size of a minimum feedback vertex set of $G$. We show the improved bound of $\chi_{ON}(G)\leq {\sf fvs}(G)+2$, which is tight, thereby answering an open question in the above paper. * We study the relation between $\chi_{ON}(G)$ and the pathwidth of the graph $G$, denoted ${\sf pw}(G)$. The above paper from WADS 2019 showed the upper bound $\chi_{ON}(G) \leq 2{\sf tw}(G)+1$ where ${\sf tw}(G)$ stands for the treewidth of $G$. This implies an upper bound of $\chi_{ON}(G) \leq 2{\sf pw}(G)+1$. We show an improved bound of $\chi_{ON}(G) \leq \lfloor \frac{5}{3}({\sf pw}(G)+1) \rfloor$. * We prove new bounds for $\chi_{ON}(G)$ with respect to the structural parameters neighborhood diversity and distance to cluster, improving existing results. * We also study the partial coloring variant of the CFON coloring problem, which allows vertices to be left uncolored. Let $\chi^*_{ON}(G)$ denote the minimum number of colors required to color $G$ as per this variant. Abel et. al. [SIDMA 2018] showed that $\chi^*_{ON}(G) \leq 8$ when $G$ is planar. They asked if fewer colors would suffice for planar graphs. We answer this question by showing that $\chi^*_{ON}(G) \leq 5$ for all planar $G$. All our bounds are a result of constructive algorithmic procedures.Comment: 30 page

Mutations That Confer Drug-Resistance, Oncogenicity and Intrinsic Activity on the ERK MAP Kinases-Current State of the Art

10.3390/cells9010129Cells9