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

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

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

Radiosurgery and fractionated stereotactic body radiotherapy for patients with lung oligometastases

Background: Patients with oligometastatic disease can potentially be cured by using an ablative therapy for all active lesions. Stereotactic body radiotherapy (SBRT) is a non-invasive treatment option that lately proved to be as effective and safe as surgery in treating lung metastases (LM). However, it is not clear which patients benefit most and what are the most suitable fractionation regimens. The aim of this study was to analyze treatment outcomes after single fraction radiosurgery (SFRS) and fractionated SBRT (fSBRT) in patients with lung oligometastases and identify prognostic clinical features for better survival outcomes. Methods: Fifty-two patients with 94 LM treated with SFRS or fSBRT between 2010 and 2016 were analyzed. The characteristics of primary tumor, LM, treatment, toxicity profiles and outcomes were assessed. Kaplan-Meier and Cox regression analyses were used for estimation of local control (LC), overall survival (OS) and progression-free survival. Results: Ninety-four LM in 52 patients were treated using SFRS/fSBRT with a median of 2 lesions per patient (range: 1-5). The median planning target volume (PTV)-encompassing dose for SFRS was 24 Gy (range: 17-26) compared to 45 Gy (range: 20-60) in 2-12 fractions with fSBRT. The median follow-up time was 21 months (range: 3-68). LC rates at 1 and 2 years for SFSR vs. fSBRT were 89 and 83% vs. 75 and 59%, respectively (p = 0.026). LM treated with SFSR were significantly smaller (p = 0.001). The 1 and 2-year OS rates for all patients were 84 and 71%, respectively. In univariate analysis treatment with SFRS, an interval of ≥12 months between diagnosis of LM and treatment, non-colorectal cancer histology and BED 70% and time to first metastasis ≥12 months. There was no grade 3 acute or late toxicity. Conclusions: Longer time to first metastasis, good KPS and N0 predicted better OS. Good LC and low toxicity rates were achieved after short SBRT schedules

Establishment and validation of cyberknife irradiation in a syngeneic glioblastoma mouse model

CyberKnife stereotactic radiosurgery (CK-SRS) precisely delivers radiation to intracranial tumors. However, the underlying radiobiological mechanisms at high single doses are not yet fully understood. Here, we established and evaluated the early radiobiological effects of CK-SRS treatment at a single dose of 20 Gy after 15 days of tumor growth in a syngeneic glioblastoma-mouse model. Exact positioning was ensured using a custom-made, non-invasive, and trackable frame. One superimposed target volume for the CK-SRS planning was created from the fused tumor volumes obtained from MRIs prior to irradiation. Dose calculation and delivery were planned using a single-reference CT scan. Six days after irradiation, tumor volumes were measured using MRI scans, and radiobiological effects were assessed using immunofluorescence staining. We found that CK-SRS treatment reduced tumor volume by approximately 75%, impaired cell proliferation, diminished tumor vasculature, and increased immune response. The accuracy of the delivered dose was demonstrated by staining of DNA double-strand breaks in accordance with the planned dose distribution. Overall, we confirmed that our proposed setup enables the precise irradiation of intracranial tumors in mice using only one reference CT and superimposed MRI volumes. Thus, our proposed mouse model for reproducible CK-SRS can be used to investigate radiobiological effects and develop novel therapeutic approaches

Immune-related pan-cancer gene expression signatures of patient survival revealed by NanoString-based analyses

The immune system plays a central role in the onset and progression of cancer. A better understanding of transcriptional changes in immune cell-related genes associated with cancer progression, and their significance in disease prognosis, is therefore needed. NanoString-based targeted gene expression profiling has advantages for deployment in a clinical setting over RNA-seq technologies. We analysed NanoString PanCancer Immune Profiling panel gene expression data encompassing 770 genes, and overall survival data, from multiple previous studies covering 10 different cancer types, including solid and blood malignancies, across 515 patients. This analysis revealed an immune gene signature comprising 39 genes that were upregulated in those patients with shorter overall survival; of these 39 genes, three (MAGEC2, SSX1 and ULBP2) were common to both solid and blood malignancies. Most of the genes identified have previously been reported as relevant in one or more cancer types. Using Cibersort, we investigated immune cell levels within individual cancer types and across groups of cancers, as well as in shorter and longer overall survival groups. Patients with shorter survival had a higher proportion of M2 macrophages and γδ T cells. Patients with longer overall survival had a higher proportion of CD8+ T cells, CD4+ T memory cells, NK cells and, unexpectedly, T regulatory cells. Using a transcriptomics platform with certain advantages for deployment in a clinical setting, our multi-cancer meta-analysis of immune gene expression and overall survival data has identified a specific transcriptional profile associated with poor overall survival

Definition and validation of a radiomics signature for loco-regional tumour control in patients with locally advanced head and neck squamous cell carcinoma

Purpose: To develop and validate a CT-based radiomics signature for the prognosis of loco-regional tumour control (LRC) in patients with locally advanced head and neck squamous cell carcinoma (HNSCC) treated by primary radiochemotherapy (RCTx) based on retrospective data from 6 partner sites of the German Cancer Consortium - Radiation Oncology Group (DKTK-ROG). Material and methods: Pre-treatment CT images of 318 patients with locally advanced HNSCC were col-lected. Four-hundred forty-six features were extracted from each primary tumour volume and then fil-tered through stability analysis and clustering. First, a baseline signature was developed from demographic and tumour-associated clinical parameters. This signature was then supplemented by CT imaging features. A final signature was derived using repeated 3-fold cross-validation on the discovery cohort. Performance in external validation was assessed by the concordance index (C-Index). Furthermore, calibration and patient stratification in groups with low and high risk for loco-regional recurrence were analysed. Results: For the clinical baseline signature, only the primary tumour volume was selected. The final sig-nature combined the tumour volume with two independent radiomics features. It achieved moderatel

Broad targeting of resistance to apoptosis in cancer

Apoptosis or programmed cell death is natural way of removing aged cells from the body. Most of the anti-cancer therapies trigger apoptosis induction and related cell death networks to eliminate malignant cells. However, in cancer, de-regulated apoptotic signaling, particularly the activation of an anti-apoptotic systems, allows cancer cells to escape this program leading to uncontrolled proliferation resulting in tumor survival, therapeutic resistance and recurrence of cancer. This resistance is a complicated phenomenon that emanates from the interactions of various molecules and signaling pathways. In this comprehensive review we discuss the various factors contributing to apoptosis resistance in cancers. The key resistance targets that are discussed include (1) Bcl-2 and Mcl-1 proteins; (2) autophagy processes; (3) necrosis and necroptosis; (4) heat shock protein signaling; (5) the proteasome pathway; (6) epigenetic mechanisms; and (7) aberrant nuclear export signaling. The shortcomings of current therapeutic modalities are highlighted and a broad spectrum strategy using approaches including (a) gossypol; (b) epigallocatechin-3-gallate; (c) UMI-77 (d) triptolide and (e) selinexor that can be used to overcome cell death resistance is presented. This review provides a roadmap for the design of successful anti-cancer strategies that overcome resistance to apoptosis for better therapeutic outcome in patients with cancer