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Characterizing intratumor heterogeneity (ITH) is crucial to understanding cancer development, but it is hampered by limits of available data sources. Bulk DNA sequencing is the most common technology to assess ITH, but mixes many genetically distinct cells in each sample, which must then be computationally deconvolved. Single-cell sequencing (SCS) is a promising alternative, but its limitations --e.g., high noise, difficulty scaling to large populations, technical artifacts, and large data sets --have so far made it impractical for studying cohorts of sufficient size to identify statistically robust features of tumor evolution. We have developed strategies for deconvolution and tumor phylogenetics combining limited amounts of bulk and single-cell data to gain some advantages of single-cell resolution with much lower cost, with specific focus on deconvolving genomic copy number data. We developed a mixed membership model for clonal deconvolution via non-negative matrix factorization (NMF) balancing deconvolution quality with similarity to single-cell samples via an associated efficient coordinate descent algorithm. We then improve on that algorithm by integrating deconvolution with clonal phylogeny inference, using a mixed integer linear programming (MILP) model to incorporate a minimum evolution phylogenetic tree cost in the problem objective. We demonstrate the effectiveness of these methods on semi-simulated data of known ground truth, showing improved deconvolution accuracy relative to bulk data alone.

We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a novel pairwise IB-representable class of CDCs, CDCs admitting junction trees, and provide a combinatorial procedure to build MIP formulations for those constraints. Generalized special ordered sets (SOS k) can be modeled by CDCs admitting junction trees and we also obtain MIP formulations of SOS k. Furthermore, we provide a novel ideal extended formulation of any combinatorial disjunctive constraints with fewer auxiliary binary variables with an application in planar obstacle avoidance.

In this paper, we show that the biclique partition number (bp) of a co-chordal (complementary graph of chordal) graph G = (V, E) is less than the number of maximal cliques (mc) of its complementary graph: a chordal graph G c = (V, E c ). We first provide a general framework of the "divided and conquer" heuristic of finding minimum biclique partition on co-chordal graphs based on clique trees. Then, an O[|V |(|V | + |E c |)]-time heuristic is proposed by applying lexicographic breadth-first search. Either heuristic gives us a biclique partition of G with a size of mc(G c ) − 1. Eventually, we prove that our heuristic can solve the minimum biclique partition problem on G exactly if its complement G c is chordal and clique vertex irreducible. We also show that mc(G c ) − 2 ≤ bp(G) ≤ mc(G c ) − 1 if G is a split graph.