Multiply Balanced k − Partitioning

Abstract. The problem of partitioning an edge-capacitated graph on n vertices into k balanced parts has been amply researched. Motivated by applications such as load balancing in distributed systems and market segmentation in social networks, we propose a new variant of the problem, called Multiply Balanced k Partitioning, where the vertex-partition must be balanced under d vertex-weight functions simultaneously. We design bicriteria approximation algorithms for this problem, i.e., they partition the vertices into up to k parts that are nearly balanced simultaneously for all weight functions, and their approximation factor for the capacity of cut edges matches the bounds known for a single weight function times d. For the case where d = 2, for vertex weights that are integers bounded by a polynomial in n and any fixed ɛ> 0, we obtain a (2 + ɛ, O ( √ log n log k))-bicriteria approximation, namely, we partition the graph into parts whose weight is at most 2+ɛ times that of a perfectly balanced part (simultaneously for both weight functions), and whose cut capacity is O ( √ log n log k) · OPT. For unbounded (exponential) vertex weights, we achieve approximation (3, O(log n)). Our algorithm generalizes to d weight functions as follows: For vertex weights that are integers bounded by a polynomial in n and any fixed ɛ> 0, we obtain a (2d + ɛ, O(d √ log n log k))-bicriteria approximation. For unbounded (exponential) vertex weights, we achieve approximation (2d + 1, O(d log n)).

53BP1 as a potential predictor of response in PARP inhibitor-treated homologous recombination-deficient ovarian cancer

Contains fulltext : 202591.pdf (publisher's version ) (Closed access)OBJECTIVE: Poly(ADP-ribose) polymerase (PARP) inhibitors have shown substantial activity in homologous recombination- (HR-) deficient ovarian cancer and are undergoing testing in other HR-deficient tumors. For reasons that are incompletely understood, not all patients with HR-deficient cancers respond to these agents. Preclinical studies have demonstrated that changes in alternative DNA repair pathways affect PARP inhibitor (PARPi) sensitivity in ovarian cancer models. This has not previously been assessed in the clinical setting. METHODS: Clonogenic and plasmid-based HR repair assays were performed to compare BRCA1-mutant COV362 ovarian cancer cells with or without 53BP1 gene deletion. Archival biopsies from ovarian cancer patients in the phase I, open-label clinical trial of PARPi ABT-767 were stained for PARP1, RAD51, 53BP1 and multiple components of the nonhomologous end-joining (NHEJ) DNA repair pathway. Modified histochemistry- (H-) scores were determined for each repair protein in each sample. HRD score was determined from tumor DNA. RESULTS: 53BP1 deletion increased HR in BRCA1-mutant COV362 cells and decreased PARPi sensitivity in vitro. In 36 women with relapsed ovarian cancer, responses to the PARPi ABT-767 were observed exclusively in cancers with HR deficiency. In this subset, 7 of 18 patients (39%) had objective responses. The actual HRD score did not further correlate with change from baseline tumor volume (r=0.050; p=0.87). However, in the HR-deficient subset, decreased 53BP1 H-score was associated with decreased antitumor efficacy of ABT-767 (r=-0.69, p=0.004). CONCLUSION: Differences in complementary repair pathways, particularly 53BP1, correlate with PARPi response of HR-deficient ovarian cancers