Scheduling Independent Multiprocessor Tasks

We study the problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a fixed number of processors. We propose a linear time algorithm that finds a schedule of minimum makespan in the preemptive model, and a linear time approximation algorithm that finds a schedule of makespan within a factor of (1+\eps) of optimal in the non-preemptive model. We extend our results by obtaining a polynomial time approximation scheme for the parallel processors variant of the multiprocessor task model.     

Scheduling Independent Multiprocessor Tasks

Abstract. We study the problem of scheduling a set of n independent multiprocessor tasks with prespecified processor allocations on a fixed number of processors. We propose a linear time algorithm that finds a schedule of minimum makespan in the preemptive model, and a linear time approximation algorithm that finds a schedule of makespan within a factor of (1+\eps) of optimal in the non-preemptive model. We extend our results by obtaining a polynomial time approximation scheme for the parallel processors variant of the multiprocessor task model.     

Paths and trails in edge-colored graphs

This paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s$s$ and t$t$ in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s−t$s-t$ path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s$s$ and t$t$ for a particular class of graphs and characterize edge-colored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k$k$ pairwise vertex/edge disjoint properly edge-colored s−t$s-t$ paths/trails in a c$c$-edge-colored graph G^c$G^c$ is NP-complete even for k=2$k=2$ and c=Ω(n²)$c=\Omega \left(n^2\right)$, where n$n$ denotes the number of vertices in G^c$G^c$. Moreover, we prove that these problems remain NP-complete for c$c$-edge-colored graphs containing no properly edge-colored cycles and c=Ω(n)$c=\Omega \left(n\right)$. We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs.     

Detection of anti-Ro(SSA) antibodies in autoimmune diseases: comparison of five methods

Counterimmunoelectrophoresis (CIE), RNA precipitation, ELISA and immunoblotting against cytoplasmic HeLa cell extract (IB-HeLa) and erythrocyte extract (IB-RBC) were applied to detect anti-Ro(SSA) antibodies in 93 sera selected from patients with various autoimmune diseases [47 were anti-Ro(SSA) positive by CIE]. The RNA precipitation assay, which demonstrated the highest sensitivity was selected as the reference method. CIE was found to be reliable with a specificity of 100% and a sensitivity of 89%. ELISA showed a comparable specificity (95%) but somewhat lower sensitivity (72%). Antibodies to 52 or 60 kDa Ro(SSA) proteins by IB-HeLa demonstrated a high specificity (95 and 97% respectively) but a low overall sensitivity (36 and 17% respectively). Anti-Ro(SSA) antibodies to 52, 54 and 60 kDa erythrocyte proteins by IB-RBC, had a variable overall specificity (95, 97 and 57%) and sensitivity (51, 13 and 34%). The anti-52 kDa antibodies detected by IB-HeLa correlated to those found by IB-RBC (P < 0.001) and occurred predominantly in primary Sj?gren's syndrome (P < 0.001, sensitivity: 71 and 77%) as well as in sera with anti-Ro(SSA) and anti-La(SSB) antibodies (P < 0.001). These findings confirm that RNA precipitation assay has the highest sensitivity and specificity for anti-Ro(SSA) antibody detection. However, until a more sensitive ELISA is available, CIE because of its reliability appears to be the method of choice. Finally IB-RBC was found to be more sensitive than IB-HeLa for the detection of anti-Ro52 kDa antibodies.     

Parallel algorithms for the hamiltonian cycle and hamiltonian path problems in semicomplete bipartite digraphs

We give anO(log⁴ n)-timeO(n ²)-processor CRCW PRAM algorithm to find a hamiltonian cycle in a strong semicomplete bipartite digraph,B, provided that a factor ofB (i.e., a collection of vertex disjoint cycles covering the vertex set ofB) is computed in a preprocessing step. The factor is found (if it exists) using a bipartite matching algorithm, hence placing the whole algorithm in the class Random-NC. We show that any parallel algorithm which can check the existence of a hamiltonian cycle in a strong semicomplete bipartite digraph in timeO(r(n)) usingp(n) processors can be used to check the existence of a perfect matching in a bipartite graph in timeO(r(n)+n ² /p(n)) usingp(n) processors. Hence, our problem belongs to the class NC if and only if perfect matching in bipartite graphs belongs to NC. We also consider the problem of finding a hamiltonian path in a semicomplete bipartite digraph.