Dimension Reduction via Colour Refinement

Colour refinement is a basic algorithmic routine for graph isomorphism testing, appearing as a subroutine in almost all practical isomorphism solvers. It partitions the vertices of a graph into "colour classes" in such a way that all vertices in the same colour class have the same number of neighbours in every colour class. Tinhofer (Disc. App. Math., 1991), Ramana, Scheinerman, and Ullman (Disc. Math., 1994) and Godsil (Lin. Alg. and its App., 1997) established a tight correspondence between colour refinement and fractional isomorphisms of graphs, which are solutions to the LP relaxation of a natural ILP formulation of graph isomorphism. We introduce a version of colour refinement for matrices and extend existing quasilinear algorithms for computing the colour classes. Then we generalise the correspondence between colour refinement and fractional automorphisms and develop a theory of fractional automorphisms and isomorphisms of matrices. We apply our results to reduce the dimensions of systems of linear equations and linear programs. Specifically, we show that any given LP L can efficiently be transformed into a (potentially) smaller LP L' whose number of variables and constraints is the number of colour classes of the colour refinement algorithm, applied to a matrix associated with the LP. The transformation is such that we can easily (by a linear mapping) map both feasible and optimal solutions back and forth between the two LPs. We demonstrate empirically that colour refinement can indeed greatly reduce the cost of solving linear programs

CCI-779 (Temsirolimus) exhibits increased anti-tumor activity in low EGFR expressing HNSCC cell lines and is effective in cells with acquired resistance to cisplatin or cetuximab

Background: The mammalian target of rapamycin (mTOR) signaling pathway plays a pivotal role in numerous cellular processes involving growth, proliferation and survival. The purpose of this study was to investigate the anti-tumoral effect of the mTOR inhibitor (mTORi) CCI-779 in HNSCC cell lines and its potency in cisplatin- and cetuximab-resistant cells. Methods: A panel of 10 HNSCC cell lines with differences in TP53 mutational status and basal cisplatin sensitivity and two isogenic models of acquired resistance to cisplatin and cetuximab, respectively were studied. Cell survival after treatment with CCI-779, cisplatin and cetuximab alone or in combination was determined by MTT assays. Potential predictive biomarkers for tumor cell sensitivity to CCI-779 were evaluated. Results: We observed considerable heterogeneity in sensitivity of HNSCC cell lines to CCI-779 monotherapy. Sensitivity was observed in TP53 mutated as well as wild-type cell lines. Total and p-EGFR expression levels but not the basal activity of the mTOR and MAPK signaling pathways were correlated with sensitivity to CCI-779. Resistant cells with increased EGFR activation could be sensitized by the combination of CCI-779 with cetuximab. Interestingly, cell lines with acquired resistance to cisplatin displayed a higher sensitivity to CCI-779 whereas cetuximab-resistant cells were less sensitive to the drug, but could be sensitized to CCI-779 by EGFR blockade. Conclusions: Activity of CCI-779 in HNSCC cells harboring TP53 mutations and displaying a phenotype of cisplatin resistance suggests its clinical potential even in patients with dismal outcome after current standard treatment. Cetuximab/mTORi combinations might be useful for treatment of tumors with high expression of EGFR/p-EGFR and/or acquired cetuximab resistance. This combinatorial treatment modality needs further evaluation in future translational and clinical studies

Limitations of Algebraic Approaches to Graph Isomorphism Testing

We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gr\"obner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gr\"obner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic different from 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations. We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus

b-coloring is NP-hard on co-bipartite graphs and polytime solvable on tree-cographs

A b-coloring of a graph is a proper coloring such that every color class contains a vertex that is adjacent to all other color classes. The b-chromatic number of a graph G, denoted by \chi_b(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is called b-continuous if it admits a b-coloring with t colors, for every t = \chi(G),\ldots,\chi_b(G), and b-monotonic if \chi_b(H_1) \geq \chi_b(H_2) for every induced subgraph H_1 of G, and every induced subgraph H_2 of H_1. We investigate the b-chromatic number of graphs with stability number two. These are exactly the complements of triangle-free graphs, thus including all complements of bipartite graphs. The main results of this work are the following: - We characterize the b-colorings of a graph with stability number two in terms of matchings with no augmenting paths of length one or three. We derive that graphs with stability number two are b-continuous and b-monotonic. - We prove that it is NP-complete to decide whether the b-chromatic number of co-bipartite graph is at most a given threshold. - We describe a polynomial time dynamic programming algorithm to compute the b-chromatic number of co-trees. - Extending several previous results, we show that there is a polynomial time dynamic programming algorithm for computing the b-chromatic number of tree-cographs. Moreover, we show that tree-cographs are b-continuous and b-monotonic

Spheroid Culture of Head and Neck Cancer Cells Reveals an Important Role of EGFR Signalling in Anchorage Independent Survival

In solid tumours millions of cells are shed into the blood circulation each day. Only a subset of these circulating tumour cells (CTCs) survive, many of them presumable because of their potential to form multi-cellular clusters also named spheroids. Tumour cells within these spheroids are protected from anoikis, which allows them to metastasize to distant organs or re-seed at the primary site. We used spheroid cultures of head and neck squamous cell carcinoma (HNSCC) cell lines as a model for such CTC clusters for determining the role of the epidermal growth factor receptor (EGFR) in cluster formation ability and cell survival after detachment from the extra-cellular matrix. The HNSCC cell lines FaDu, SCC-9 and UT-SCC-9 (UT-SCC-9P) as well as its cetuximab (CTX)-resistant sub-clone (UT-SCC-9R) were forced to grow in an anchorage- independent manner by coating culture dishes with the anti-adhesive polymer poly-2-hydroxyethylmethacrylate (poly-HEMA). The extent of apoptosis, clonogenic survival and EGFR signalling under such culture conditions was evaluated. The potential of spheroid formation in suspension culture was found to be positively correlated with the proliferation rate of HNSCC cell lines as well as their basal EGFR expression levels. CTX and gefitinib blocked, whereas the addition of EGFR ligands promoted anchorage-independent cell survival and spheroid formation. Increased spheroid formation and growth were associated with persistent activation of EGFR and its downstream signalling component (MAPK/ERK). Importantly, HNSCC cells derived from spheroid cultures retained their clonogenic potential in the absence of cell-matrix contact. Addition of CTX under these conditions strongly inhibited colony formation in CTX- sensitive cell lines but not their resistant subclones. Altogether, EGFR activation was identified as crucial factor for anchorage-independent survival of HNSCC cells. Targeting EGFR in CTC cluster formation might represent an attractive anti-metastatic treatment approach in HNSCC

On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio possible for a one-pass deterministic online algorithm is $1/2$, which is achieved by any greedy algorithm. D\"urr et al. recently presented a $2$-pass algorithm called Category-Advice that achieves approximation ratio $3/5$. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the $k$-pass Category-Advice algorithm for all $k \ge 1$, and show that the approximation ratio converges to the inverse of the golden ratio $2/(1+\sqrt{5}) \approx 0.618$ as $k$ goes to infinity. The convergence is extremely fast --- the $5$-pass Category-Advice algorithm is already within $0.01\%$ of the inverse of the golden ratio. We then consider a natural greedy algorithm in the online stochastic IID model---MinDegree. This algorithm is an online version of a well-known and extensively studied offline algorithm MinGreedy. We show that MinDegree cannot achieve an approximation ratio better than $1-1/e$, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek, we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model

A comparative study of machine learning methods for time-to-event survival data for radiomics risk modelling

Radiomics applies machine learning algorithms to quantitative imaging data to characterise the tumour phenotype and predict clinical outcome. For the development of radiomics risk models, a variety of different algorithms is available and it is not clear which one gives optimal results. Therefore, we assessed the performance of 11 machine learning algorithms combined with 12 feature selection methods by the concordance index (C-Index), to predict loco- regional tumour control (LRC) and overall survival for patients with head and neck squamous cell carcinoma. The considered algorithms are able to deal with continuous time-to-event survival data. Feature selection and model building were performed on a multicentre cohort (213 patients) and validated using an independent cohort (80 patients). We found several combinations of machine learning algorithms and feature selection methods which achieve similar results, e.g., MSR-RF: C-Index = 0.71 and BT-COX: C-Index = 0.70 in combination with Spearman feature selection. Using the best performing models, patients were stratified into groups of low and high risk of recurrence. Significant differences in LRC were obtained between both groups on the validation cohort. Based on the presented analysis, we identified a subset of algorithms which should be considered in future radiomics studies to develop stable and clinically relevant predictive models for time-to-event endpoints

Negative enrichment by immunomagnetic nanobeads for unbiased characterization of circulating tumor cells from peripheral blood of cancer patients

BACKGROUND: A limitation of positive selection strategies to enrich for circulating tumor cells (CTCs) is that there might be CTCs with insufficient expression of the surface target marker which may be missed by the procedure. We optimized a method for enrichment, subsequent detection and characterization of CTCs based on depletion of the leukocyte fraction. METHODS: The 2-step protocol was developed for processing 20 mL blood and based on red blood cell lysis followed by leukocyte depletion. The remaining material was stained with the epithelial markers EpCAM and cytokeratin (CK) 7/8 or for the melanoma marker HMW-MAA/MCSP. CTCs were detected by flow cytometry. CTCs enriched from blood of patients with carcinoma were defined as EpCAM+CK+CD45-. CTCs enriched from blood of patients with melanoma were defined as MCSP+CD45-. One-hundred-sixteen consecutive blood samples from 70 patients with metastatic carcinomas (n = 48) or metastatic melanoma (n = 22) were analyzed. RESULTS: CTCs were detected in 47 of 84 blood samples (56%) drawn from carcinoma patients, and in 17 of 32 samples (53%) from melanoma patients. CD45-EpCAM-CK+ was detected in pleural effusion specimens, as well as in peripheral blood samples of patients with NSCLC. EpCAM-CK+ cells have been successfully cultured and passaged longer than six months suggesting their neoplastic origin. This was confirmed by CGH. By defining CTCs in carcinoma patients as CD45-CK+ and/or EpCAM+, the detection rate increased to 73% (61/84). CONCLUSION: Enriching CTCs using CD45 depletion allowed for detection of epithelial cancer cells not displaying the classical phenotype. This potentially leads to a more accurate estimation of the number of CTCs. If detection of CTCs without a classical epithelial phenotype has clinical relevance need to be determined