On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201

On Generalizations of Network Design Problems with Degree Bounds

Iterative rounding and relaxation have arguably become the method of choice in dealing with unconstrained and constrained network design problems. In this paper we extend the scope of the iterative relaxation method in two directions: (1) by handling more complex degree constraints in the minimum spanning tree problem (namely, laminar crossing spanning tree), and (2) by incorporating `degree bounds' in other combinatorial optimization problems such as matroid intersection and lattice polyhedra. We give new or improved approximation algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure

Matroid and Knapsack Center Problems

In the classic $k$-center problem, we are given a metric graph, and the objective is to open $k$ nodes as centers such that the maximum distance from any vertex to its closest center is minimized. In this paper, we consider two important generalizations of $k$-center, the matroid center problem and the knapsack center problem. Both problems are motivated by recent content distribution network applications. Our contributions can be summarized as follows: 1. We consider the matroid center problem in which the centers are required to form an independent set of a given matroid. We show this problem is NP-hard even on a line. We present a 3-approximation algorithm for the problem on general metrics. We also consider the outlier version of the problem where a given number of vertices can be excluded as the outliers from the solution. We present a 7-approximation for the outlier version. 2. We consider the (multi-)knapsack center problem in which the centers are required to satisfy one (or more) knapsack constraint(s). It is known that the knapsack center problem with a single knapsack constraint admits a 3-approximation. However, when there are at least two knapsack constraints, we show this problem is not approximable at all. To complement the hardness result, we present a polynomial time algorithm that gives a 3-approximate solution such that one knapsack constraint is satisfied and the others may be violated by at most a factor of $1+\epsilon$. We also obtain a 3-approximation for the outlier version that may violate the knapsack constraint by $1+\epsilon$.Comment: A preliminary version of this paper is accepted to IPCO 201

Spotting Trees with Few Leaves

We show two results related to the Hamiltonicity and $k$-Path algorithms in undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10]. First, we demonstrate that the technique used can be generalized to finding some $k$-vertex tree with $l$ leaves in an $n$-vertex undirected graph in $O^*(1.657^k2^{l/2})$ time. It can be applied as a subroutine to solve the $k$-Internal Spanning Tree ($k$-IST) problem in $O^*(\min(3.455^k, 1.946^n))$ time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time we break the natural barrier of $O^*(2^n)$. Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for $k$-Path and Hamiltonicity in any graph of maximum degree $\Delta=4,\ldots,12$ or with vector chromatic number at most 8

On the Complexity of the Asymmetric VPN Problem

We give the first constant factor approximation algorithm for the asymmetric Virtual Private Network (VPN) problem with arbitrary concave costs. We even show the stronger result, that there is always a tree solution of cost at most 2 OPT and that a tree solution of (expected) cost at most 49.84 OPT can be determined in polynomial time. Furthermore, we answer an outstanding open question about the complexity status of the so called balanced VPN problem by proving its NP-hardness

Multiple small RNAs identified in Mycobacterium bovis BCG are also expressed in Mycobacterium tuberculosis and Mycobacterium smegmatis

Tuberculosis (TB) is a major global health problem, infecting millions of people each year. The causative agent of TB, Mycobacterium tuberculosis, is one of the world’s most ancient and successful pathogens. However, until recently, no work on small regulatory RNAs had been performed in this organism. Regulatory RNAs are found in all three domains of life, and have already been shown to regulate virulence in well-known pathogens, such as Staphylococcus aureus and Vibrio cholera. Here we report the discovery of 34 novel small RNAs (sRNAs) in the TB-complex M. bovis BCG, using a combination of experimental and computational approaches. Putative homologues of many of these sRNAs were also identified in M. tuberculosis and/or M. smegmatis. Those sRNAs that are also expressed in the non-pathogenic M. smegmatis could be functioning to regulate conserved cellular functions. In contrast, those sRNAs identified specifically in M. tuberculosis could be functioning in mediation of virulence, thus rendering them potential targets for novel antimycobacterials. Various features and regulatory aspects of some of these sRNAs are discussed

Combinatorial Auctions without Money

Algorithmic Mechanism Design attempts to marry computation and incentives, mainly by leveraging monetary transfers between designer and selfish agents involved. This is principally because in absence of money, very little can be done to enforce truthfulness. However, in certain applications, money is unavailable, morally unacceptable or might simply be at odds with the objective of the mechanism. For example, in Combinatorial Auctions (CAs), the paradigmatic problem of the area, we aim at solutions of maximum social welfare, but still charge the society to ensure truthfulness. We focus on the design of incentive-compatible CAs without money in the general setting of k-minded bidders. We trade monetary transfers with the observation that the mechanism can detect certain lies of the bidders: i.e., we study truthful CAs with verification and without money. In this setting, we characterize the class of truthful mechanisms and give a host of upper and lower bounds on the approximation ratio obtained by either deterministic or randomized truthful mechanisms. Our results provide an almost complete picture of truthfully approximating CAs in this general setting with multi-dimensional bidders

Inhibiting mycobacterial tryptophan synthase by targeting the inter-subunit interface

Drug discovery efforts against the pathogen Mycobacterium tuberculosis (Mtb) have been advanced through phenotypic screens of extensive compound libraries. Such a screen revealed sulfolane 1 and indoline-5-sulfonamides 2 and 3 as potent inhibitors of mycobacterial growth. Optimization in the sulfolane series led to compound 4, which has proven activity in an in vivo murine model of Mtb infection. Here we identify the target and mode of inhibition of these compounds based on whole genome sequencing of spontaneous resistant mutants, which identified mutations locating to the essential α- and β-subunits of tryptophan synthase. Over-expression studies confirmed tryptophan synthase as the biological target. Biochemical techniques probed the mechanism of inhibition, revealing the mutant enzyme complex incurs a fitness cost but does not prevent inhibitor binding. Mapping of the resistance conferring mutations onto a low-resolution crystal structure of Mtb tryptophan synthase showed they locate to the interface between the α- and β-subunits. The discovery of anti-tubercular agents inhibiting tryptophan synthase highlights the therapeutic potential of this enzyme and draws attention to the prospect of other amino acid biosynthetic pathways as future Mtb drug targets

The Effect of Iron Limitation on the Transcriptome and Proteome of Pseudomonas fluorescens Pf-5

One of the most important micronutrients for bacterial growth is iron, whose bioavailability in soil is limited. Consequently, rhizospheric bacteria such as Pseudomonas fluorescens employ a range of mechanisms to acquire or compete for iron. We investigated the transcriptomic and proteomic effects of iron limitation on P. fluorescens Pf-5 by employing microarray and iTRAQ techniques, respectively. Analysis of this data revealed that genes encoding functions related to iron homeostasis, including pyoverdine and enantio-pyochelin biosynthesis, a number of TonB-dependent receptor systems, as well as some inner-membrane transporters, were significantly up-regulated in response to iron limitation. Transcription of a ribosomal protein L36-encoding gene was also highly up-regulated during iron limitation. Certain genes or proteins involved in biosynthesis of secondary metabolites such as 2,4-diacetylphloroglucinol (DAPG), orfamide A and pyrrolnitrin, as well as a chitinase, were over-expressed under iron-limited conditions. In contrast, we observed that expression of genes involved in hydrogen cyanide production and flagellar biosynthesis were down-regulated in an iron-depleted culture medium. Phenotypic tests revealed that Pf-5 had reduced swarming motility on semi-solid agar in response to iron limitation. Comparison of the transcriptomic data with the proteomic data suggested that iron acquisition is regulated at both the transcriptional and post-transcriptional levels

Transcriptional regulation of the ilv-leu operon of Bacillus subtilis.

We used primer extension and mutational analysis to identify a promoter upstream of ilvB, the first gene in the ilv-leu operon of Bacillus subtilis. Between the promoter and ilvB, there is a 482-bp leader region which contains a sequence that resembles a factor-independent transcription terminator. In in vitro transcription experiments, 90% of transcripts initiated at the ilvB promoter ended at a site near this terminator. Primer extension analysis of RNA synthesized in vivo showed that the steady-state level of mRNA upstream of the terminator was twofold higher from cells limited for leucine than it was from cells grown with excess leucine. mRNA downstream of the terminator was 14-fold higher in cells limited for leucine than in cells grown with excess leucine. Measurement of mRNA degradation rates showed that the half-life of ilv-leu mRNA was the same when the cells were grown with or without leucine. These data demonstrate that the ilv-leu operon is regulated by transcription attenuation