We consider the Arrow–Debreu exchange market model under the assumption that the agents’ demands satisfy the weak gross substitutes (WGS) property. We present a simple auction algorithm that obtains an approximate market equilibrium for WGS demands assuming the availability of a price update oracle. We exhibit specific implementations of such an oracle for WGS demands with bounded price elasticities and for Gale demand systems.

A multi-assembly problem asks to reconstruct multiple genomic sequences from mixed reads sequenced from all of them. Standard formulations of such problems model a solution as a path cover in a directed acyclic graph, namely a set of paths that together cover all vertices of the graph. Since multi-assembly problems admit multiple solutions in practice, we consider an approach commonly used in standard genome assembly: output only partial solutions (contigs, or safe paths), that appear in all path cover solutions. We study constrained path covers, a restriction on the path cover solution that incorporate practical constraints arising in multi-assembly problems. We give efficient algorithms finding all maximal safe paths for constrained path covers. We compute the safe paths of splicing graphs constructed from transcript annotations of different species. Our algorithms run in less than 15 seconds per species and report RNA contigs that are over 99% precise and are up to 8 times longer than unitigs. Moreover, RNA contigs cover over 70% of the transcripts and their coding sequences in most cases. With their increased length to unitigs, high precision, and fast construction time, maximal safe paths can provide a better base set of sequences for transcript assembly programs.

We consider the problem of approximating maximum Nash social welfare (NSW) while allocating a set of indivisible items to n agents. The NSW is a popular objective that provides a balanced tradeoff between the often conflicting requirements of fairness and efficiency, defined as the weighted geometric mean of the agents’ valuations. For the symmetric additive case of the problem, where agents have the same weight with additive valuations, the first constant-factor approximation algorithm was obtained in 2015. Subsequent work has obtained constant-factor approximation algorithms for the symmetric case under mild generalizations of additive, and O(n)-approximation algorithms for subadditive valuations and for the asymmetric case. In this paper, we make significant progress towards both symmetric and asymmetric NSW problems. We present the first constant-factor approximation algorithm for the symmetric case under Rado valuations. Rado valuations form a general class of valuation functions that arise from maximum cost independent matching problems, including as special cases assignment (OXS) valuations and weighted matroid rank functions. Furthermore, our approach also gives the first constant-factor approximation algorithm for the asymmetric case under Rado valuations, provided that the maximum ratio between the weights is bounded by a constant.

Dense subgraph detection is a fundamental problem in network analysis for which few worst-case guarantees are known, motivating its study through the lens of fixed-parameter tractability. But for what parameter? Recent work has proposed parameterizing graphs by their degree of triadic closure, with a $c$-closed graph defined as one in which every vertex pair with at least $c$ common neighbors are themselves connected by an edge. The special case of enumerating all maximal cliques (and hence computing a maximum clique) of a $c$-closed graph is known to be fixed-parameter tractable with respect to $c$ (Fox et al., SICOMP 2020). In network analysis, sufficiently dense subgraphs are typically as notable and meaningful as cliques. We investigate the fixed-parameter tractability (with respect to $c$) of optimization and enumeration in $c$-closed graphs, for several notions of dense subgraphs. We focus on graph families that are the complements of the most well-studied notions of sparse graphs, including graphs with bounded degree, bounded treewidth, or bounded degeneracy, and provide fixed-parameter tractable enumeration and optimization algorithms for these families. To go beyond the special case of maximal cliques, we use a new combinatorial bound (generalizing the Moon-Moser theorem); new techniques for exploiting the $c$-closed condition; and more sophisticated enumeration algorithms.

We consider the exchange market models with divisible goods where the demands of the agents satisfy the weak gross substitutes (WGS) property. This is a well-studied property, in particular, it gives a sufficient condition for the convergence of the classical tatonnement dynamics. In this paper, we present a simple auction algorithm that obtains an approximate market equilibrium for WGS demands. Such auction algorithms have been previously known for restricted classes of WGS demands only. As an application of our result, we obtain an efficient algorithm to find an approximate spending-restricted market equilibrium for WGS demands, a model that has been recently introduced as a continuous relaxation of the Nash Social Welfare problem.

The class of even-hole-free graphs is very similar to the class of perfect graphs, and was indeed a cornerstone in the tools leading to the proof of the Strong Perfect Graph Theorem. However, the complexity of computing a maximum independent set (MIS) is a long-standing open question in even-hole-free graphs. From the hardness point of view, MIS is W[1]-hard in the class of graphs without induced 4-cycle (when parameterized by the solution size). Halfway of these, we show in this paper that MIS is FPT when parameterized by the solution size in the class of even-hole-free graphs. The main idea is to apply twice the well-known technique of augmenting graphs to extend some initial independent set.

Motivation: Discovering the evolution of a tumor may help identify driver mutations and provide a more comprehensive view on the history of the tumor. Recent studies have tackled this problem using multiple samples sequenced from a tumor, and due to clinical implications, this has attracted great interest. However, such samples usually mix several distinct tumor subclones, which confounds the discovery of the tumor phylogeny. Results: We study a natural problem formulation requiring to decompose the tumor samples into several subclones with the objective of forming a minimum perfect phylogeny. We propose an Integer Linear Programming formulation for it, and implement it into a method called MIPUP. We tested the ability of MIPUP and of four popular tools LICHeE, AncesTree, CITUP, Treeomics to reconstruct the tumor phylogeny. On simulated data, MIPUP shows up to a 34% improvement under the ancestor‐descendant relations metric. On four real datasets, MIPUP's reconstructions proved to be generally more faithful than those of LICHeE. Availability and implementation: MIPUP is available at https://github.com/zhero9/MIPUP as open source. Supplementary information: Supplementary data are available at Bioinformatics online.

Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, 2018) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdurović et al. proved that both problems are NP-hard, gave a related characterization of transitively orientable graphs, and proposed a polynomial-time heuristic algorithm for the MCRS problem based on coloring cocomparability graphs. We give new, more transparent formulations of the two problems, showing that the problems are equivalent to two optimization problems on branchings in a derived directed acyclic graph. Building on these formulations, we obtain new results on the two problems, including (1) a strengthening of the heuristic by Hujdurović et al. via a new min-max result in digraphs generalizing Dilworth’s theorem, which may be of independent interest; (2) APX-hardness results for both problems; (3) approximation algorithms; and (4) exponential-time algorithms solving the two problems to optimality faster than the naïve brute-force approach. Our work relates to several well-studied notions in combinatorial optimization: chain partitions in partially ordered sets, laminar hypergraphs, and (classical and weighted) colorings of graphs.