Spotting Trees with Few Leaves

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph in $O^*(1.657^k2^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^*(\min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for $k$-Path and Hamiltonicity in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8

Receptor Tyrosine Kinases in Osteosarcoma: 2019 Update

The primary conclusions of our 2014 contribution to this series were as follows: Multiple receptor tyrosine kinases (RTKs) likely contribute to aggressive phenotypes in osteosarcoma and, therefore, inhibition of multiple RTKs is likely necessary for successful clinical outcomes. Inhibition of multiple RTKs may also be useful to overcome resistance to inhibitors of individual RTKs as well as resistance to conventional chemotherapies. Different combinations of RTKs are likely important in individual patients. AXL, EPHB2, FGFR2, IGF1R, and RET were identified as promising therapeutic targets by our in vitro phosphoproteomic/siRNA screen of 42 RTKs in the highly metastatic LM7 and 143B human osteosarcoma cell lines. This chapter is intended to provide an update on these topics as well as the large number of osteosarcoma clinical studies of inhibitors of multiple tyrosine kinases (multi-TKIs) that were recently published

Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization

Recently we investigated in "The operator $\Psi$ for the Chromatic Number of a Graph" hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. In particular, we introduced two hierarchies of lower bounds, the `$\psi$'-hierarchy converging to the fractional chromatic number, and the `$\Psi$'-hierarchy converging to the chromatic number of a graph. In both hierarchies the first order bounds are related to the Lov\' asz theta number, while the second order bounds would already be too costly to compute for large graphs. As an alternative, relaxations of the second order bounds are proposed. We present here our experimental results with these relaxed bounds for Hamming graphs, Kneser graphs and DIMACS benchmark graphs. Symmetry reduction plays a crucial role as it permits to compute the bounds using more compact semidefinite programs. In particular, for Hamming and Kneser graphs, we use the explicit block-diagonalization of the Terwilliger algebra given by Schrijver. Our numerical results indicate that the new bounds can be much stronger than the Lov\' asz theta number. For some of the DIMACS instances we improve the best known lower bounds significantly

The operator Ψ\Psi for the Chromatic Number of a Graph

We investigate hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. We introduce an operator $\Psi$ mapping any graph parameter $\beta(G)$, nested between the stability number $\alpha(G)$ and \chi\left( {\ol G} \right), to a new graph parameter $\Psi_\beta(G)$, nested between \alpha (\ol G) and $\chi(G)$; $\Psi_\beta(G)$ is polynomial time computable if $\beta(G)$ is. As an application, there is no polynomial time computable graph parameter nested between the fractional chromatic number $\chi^*(\cdot)$ and $\chi(\cdot)$ unless P=NP. Moreover, based on Motzkin-Straus formulation for $\alpha(G)$, we give (quadratically constrained) quadratic and copositive programming formulations for $\chi(G)$. Under some mild assumption, $n/\beta(G)\le \Psi_\beta(G)$ but, while $n/\beta(G)$ remains below $\chi^*(G)$, $\Psi_\beta(G)$ can reach $\chi(G)$ (e.g., for $\beta(\cdot)=\alpha(\cdot)$). We also define new polynomial time computable lower bounds for $\chi(G)$, improving the classic Lov\'{a}sz theta number (and its strengthenings obtained by adding nonnegativity and triangle inequalities); experimental results on Hamming graphs, Kneser graphs and DIMACS benchmark graphs will be given in the follow-up paper

Computing semidefinite programming lower bounds for the (fractional) chromatic number via block-diagonalization

Recently we investigated in "The operator $\Psi$ for the Chromatic Number of a Graph" hierarchies of semidefinite approximations for the chromatic number $\chi(G)$ of a graph $G$. In particular, we introduced two hierarchies of lower bounds, the `$\psi$'-hierarchy converging to the fractional chromatic number, and the `$\Psi$'-hierarchy converging to the chromatic number of a graph. In both hierarchies the first order bounds are related to the Lov\' asz theta number, while the second order bounds would already be too costly to compute for large graphs. As an alternative, relaxations of the second order bounds are proposed. We present here our experimental results with these relaxed bounds for Hamming graphs, Kneser graphs and DIMACS benchmark graphs. Symmetry reduction plays a crucial role as it permits to compute the bounds using more compact semidefinite programs. In particular, for Hamming and Kneser graphs, we use the explicit block-diagonalization of the Terwilliger algebra given by Schrijver. Our numerical results indicate that the new bounds can be much stronger than the Lov\' asz theta number. For some of the DIMACS instances we improve the best known lower bounds significantly