Geometry of Scheduling on Multiple Machines

We consider the following general scheduling problem: there are m identical machines and n jobs all released at time 0. Each job j has a processing time pj, and an arbitrary non-decreasing function fj that specifies the cost incurred for j, for each possible completion time. The goal is to find a preemptive migratory schedule of minimum cost. This models several natural objectives such as weighted norm of completion time, weighted tardiness and much more. We give the first O(1) approximation algorithm for this problem, improving upon the O(loglognP) bound due to Moseley (2019). To do this, we first view the job-cover inequalities of Moseley geometrically, to reduce the problem to that of covering demands on a line by rectangular and triangular capacity profiles. Due to the non-uniform capacities of triangles, directly using quasi-uniform sampling loses a O(loglogP) factor, so a second idea is to adapt it to our setting to only lose an O(1) factor. Our ideas for covering points with non-uniform capacity profiles (which have not been studied before) may be of independent int

Non-uniform geometric set cover and scheduling on multiple machines

We consider the following general scheduling problem studied recently by Moseley [27]. There are n jobs, all released at time 0, where job j has size pj and an associated arbitrary non-decreasing cost function fj of its completion time.

Improved approximations for min sum vertex cover and generalized min sum set cover

We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges E with arbitrary covering requirements {ke ∈ Z+ : e ∈ E}, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge e is considered covered by the first time when ke many of its vertices appear in the ordering. We give a 4.642 approximation algorithm for GMSSC, coming close to the best possible bound of 4, already for the classical special case (with all ke = 1) of min sum set cover (MSSC) studied by Feige, Lovász and Tetali [11], and improving upon the previous best known bound of 12.4 due to Im, Sviridenko and van der Zwaan [20]. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. This also gives an LP-based 4 approximation for MSSC. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility. Another well-known special case is the min sum vertex cover (MSVC) problem, in which the input hypergraph is a graph (i.e., |e| = 2) and ke = 1, for every edge e ∈ E. We give a 16/9 ' 1.778 approximation for MSVC, and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best 1.999946 approximation of Barenholz, Feige and Peleg [6]. (The claimed 1.79 approximation result of Iwata, Tetali and Tripathi [21] for the MSVC turned out have an unfortunate, seemingly unfixable, mistake in it.) Finally, we revisit MSSC and consider the lp norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm simultaneously achieves approximation guarantees of (p + 1)1+1/p, for all p ≥ 1, giving another proof of the result of Golovin, Gupta, Kumar and Tangwongsan [13], and showing its tightness up to NP-hardness. For p = 1, this gives yet another proof of the 4 approximation for MSSC

Large expert-curated database for benchmarking document similarity detection in biomedical literature search

Document recommendation systems for locating relevant literature have mostly relied on methods developed a decade ago. This is largely due to the lack of a large offline gold-standard benchmark of relevant documents that cover a variety of research fields such that newly developed literature search techniques can be compared, improved and translated into practice. To overcome this bottleneck, we have established the RElevant LIterature SearcH consortium consisting of more than 1500 scientists from 84 countries, who have collectively annotated the relevance of over 180 000 PubMed-listed articles with regard to their respective seed (input) article/s. The majority of annotations were contributed by highly experienced, original authors of the seed articles. The collected data cover 76% of all unique PubMed Medical Subject Headings descriptors. No systematic biases were observed across different experience levels, research fields or time spent on annotations. More importantly, annotations of the same document pairs contributed by different scientists were highly concordant. We further show that the three representative baseline methods used to generate recommended articles for evaluation (Okapi Best Matching 25, Term Frequency-Inverse Document Frequency and PubMed Related Articles) had similar overall performances. Additionally, we found that these methods each tend to produce distinct collections of recommended articles, suggesting that a hybrid method may be required to completely capture all relevant articles. The established database server located at https://relishdb.ict.griffith.edu.au is freely available for the downloading of annotation data and the blind testing of new methods. We expect that this benchmark will be useful for stimulating the development of new powerful techniques for title and title/abstract-based search engines for relevant articles in biomedical research.Peer reviewe

Constant factor approximation algorithm for weighted flow-time on a single machine in pseudopolynomial time

In the weighted flow-time problem on a single machine, we are given a set of n jobs, where each job has a processing requirement pj, release date rj, and weight wj. The goal is to find a preemptive schedule which minimizes the sum of weighted flow-time of jobs, where the flow-time of a job is the difference between its completion time and its released date. We give the first pseudopolynomial time constant approximation algorithm for this problem. The algorithm also extends directly to the problem of minimizing the ℓp norm of weighted flow-times. The running time of our algorithm is polynomial in n, the number of jobs, and P, which is the ratio of the largest to the smallest processing requirement of a job. Our algorithm relies on a novel reduction of this problem to a generalization of the multicut problem on trees, which we call the Demand MultiCut problem. Even though we do not give a constant factor approximation algorithm for the Demand MultiCut problem on trees, we show that the specific instances of Demand MultiCut obtained by reduction from weighted flow-time problem instances have more structure in them, and we are able to employ techniques based on dynamic programming. Our dynamic programming algorithm relies on showing that there are near optimal solutions which have nice smoothness properties, and we exploit these properties to reduce the size of the dynamic programming table