Approximating the minimum directed tree cover

Given a directed graph $G$ with non negative cost on the arcs, a directed tree cover of $G$ is a rooted directed tree such that either head or tail (or both of them) of every arc in $G$ is touched by $T$. The minimum directed tree cover problem (DTCP) is to find a directed tree cover of minimum cost. The problem is known to be $NP$-hard. In this paper, we show that the weighted Set Cover Problem (SCP) is a special case of DTCP. Hence, one can expect at best to approximate DTCP with the same ratio as for SCP. We show that this expectation can be satisfied in some way by designing a purely combinatorial approximation algorithm for the DTCP and proving that the approximation ratio of the algorithm is $\max\{2, \ln(D^+)\}$ with $D^+$ is the maximum outgoing degree of the nodes in $G$.Comment: 13 page

Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.Comment: 25 pages, 12 figures, Latex. To appear in "Discrete and Computational Geometry". Previous version (extended abstract) appears in SODA 2004, pp. 430-43

The complexity of the Pk partition problem and related problems in bipartite graphs

International audienceIn this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems

Tight-binding parameters for charge transfer along DNA

We systematically examine all the tight-binding parameters pertinent to charge transfer along DNA. The $\pi$ molecular structure of the four DNA bases (adenine, thymine, cytosine, and guanine) is investigated by using the linear combination of atomic orbitals method with a recently introduced parametrization. The HOMO and LUMO wavefunctions and energies of DNA bases are discussed and then used for calculating the corresponding wavefunctions of the two B-DNA base-pairs (adenine-thymine and guanine-cytosine). The obtained HOMO and LUMO energies of the bases are in good agreement with available experimental values. Our results are then used for estimating the complete set of charge transfer parameters between neighboring bases and also between successive base-pairs, considering all possible combinations between them, for both electrons and holes. The calculated microscopic quantities can be used in mesoscopic theoretical models of electron or hole transfer along the DNA double helix, as they provide the necessary parameters for a tight-binding phenomenological description based on the $\pi$ molecular overlap. We find that usually the hopping parameters for holes are higher in magnitude compared to the ones for electrons, which probably indicates that hole transport along DNA is more favorable than electron transport. Our findings are also compared with existing calculations from first principles.Comment: 15 pages, 3 figures, 7 table

Approximation Algorithms for Min-Max Path Cover Problems with Service Handling Time

20th International Symposium on Algorithms and Computation (ISAAC 2009), Honolulu, Hawaii, USA, December 16-18, 2009This paper presents improved approximation algorithms and inapproximability results for min-max path cover problems with service handling time, which have wide applications in practice when the latest service completion time for customers is critical. We study three variants of this problem, where paths must start (i) from a given depot, (ii) from any depot of a given set, and (iii) from any vertex of the given graph, respectively. For these three variants, we are able to achieve approximation ratios of 3, (4 + ε), and (5 + ε), respectively, for any ε> 0. We have further shown that approximation ratios less than 4/3, 3/2, and 3/2 are impossible for them, respectively, unless NP = P.Department of Logistics and Maritime StudiesRefereed conference pape

A liquid chromatography/mass spectrometry method for screening disulfide tethering fragments

We report the refinement of a high-throughput, liquid chromatography/mass spectrometry (LC/MS)-based screening method for the identification of covalent small-molecule binders to proteins. Using a custom library of 1600 disulfide-capped fragments targeting surface cysteine residues, we optimize sample preparation, chromatography, and ionization conditions to maximize the reliability and flexibility of the approach. Data collection at a rate of 84 s per sample balances speed with reliability for sustained screening over multiple, diverse projects run over a 24-month period. The method is applicable to protein targets of various classes and a range of molecular masses. Data are processed in a custom pipeline that calculates a percent bound value for each compound and identifies false positives by calculating significance of detected masses (signal significance). An example pipeline is available through Biovia's ScienceCloud Protocol Exchange. Data collection and analysis methods for the screening of covalent adducts of intact proteins are now fast enough to screen the largest covalent compound libraries in 1 to 2 days

A 3/2-Approximation Algorithm for the Multiple TSP with a Fixed Number of Depots

2009-2010 > Academic research: refereed > Refereed conference pape