Atypical and painless presentation was more common among women delaying presentation. Barriers included cost, access, time and embarrassment. Education should emphasize atypical symptoms, the high-cure rate and the need for early presentation. Reduced cost and improved access to clinics would enhance early consultation.

We develop a new randomized method, random separation, for solving fixed-cardinality optimization problems on graphs, i.e., problems concerning solutions with exactly a fixed number k of elements (e.g., k vertices V) that optimize solution values (e.g., the number of edges covered by V). The key idea of the method is to partition the vertex set of a graph randomly into two disjoint sets to separate a solution from the rest of the graph into connected components, and then select appropriate components to form a solution. We can use universal sets to derandomize algorithms obtained from this method. This new method is versatile and powerful as it can be used to solve a wide range of fixed-cardinality optimization problems for degree-bounded graphs, graphs of bounded degeneracy (a large family of graphs that contains degree-bounded graphs, planar graphs, graphs of bounded treewidth, and nontrivial minor-closed families of graphs), and even general graphs.

Indecision over symptom meaning comprised the main component of Appraisal and Total Delay, suggesting that educational strategies targeting atypical symptoms should reduce avoidable delays following self-discovered breast symptoms.

We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are no more powerful than programs arising from a constant number of rounds of the Sherali-Adams hierarchy.In particular, any polynomial-sized linear program for Max Cut has an integrality gap of

As new proposals aim to sequence ever larger collection of humans, it is critical to have a quantitative framework to evaluate the statistical power of these projects. We developed a new algorithm, UnseenEst, and applied it to the exomes of 60,706 individuals to estimate the frequency distribution of all protein-coding variants, including rare variants that have not been observed yet in the current cohorts. Our results quantified the number of new variants that we expect to identify as sequencing cohorts reach hundreds of thousands of individuals. With 500K individuals, we find that we expect to capture 7.5% of all possible loss-of-function variants and 12% of all possible missense variants. We also estimate that 2,900 genes have loss-of-function frequency of <0.00001 in healthy humans, consistent with very strong intolerance to gene inactivation.

This study showed high patient satisfaction and good cosmetic outcome after oncoplastic breast-conserving surgery, even in small breast sized Asian women. The application of oncoplastic technique allows large volume excision, and satisfaction remains high with breast volume excision less than 20% regardless of tumor location or distance of the tumor from the nipple. More complicated oncoplastic techniques, e.g., breast replacement, might be required if breast volume excision exceeds 20%.

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We show optimal (up to constant factor) NP-hardness for Max-k-CSP over any domain, whenever k is larger than the domain size. This follows from our main result concerning predicates over abelian groups. We show that a predicate is approximation resistant if it contains a subgroup that is balanced pairwise independent. This gives an unconditional analogue of Austrin-Mossel hardness result, bypassing the Unique-Games Conjecture for predicates with an abelian subgroup structure.Our main ingredient is a new gap-amplification technique inspired by XOR-lemmas. Using this technique, we also improve the NP-hardness of approximating Independent-Set on bounded-degree graphs, Almost-Coloring, Two-Prover-OneRound-Game, and various other problems.