On a generalization of iterated and randomized rounding

We give a general method for rounding linear programs that combines the commonly used iterated rounding and randomized rounding techniques. In particular, we show that whenever iterated rounding can be applied to a problem with some slack, there is a randomized procedure that returns an integral solution that satisfies the guarantees of iterated rounding and also has concentration properties. We use this to give new results for several classic problems such as rounding column-sparse LPs, makespan minimization on unrelated machines, degree-bounded spanning trees and multi-budgeted matchings

Potential-Function Proofs for First-Order Methods

This note discusses proofs for convergence of first-order methods based on simple potential-function arguments. We cover methods like gradient descent (for both smooth and non-smooth settings), mirror descent, and some accelerated variants

On-Line Balancing of Random Inputs

We consider an online vector balancing game where vectors vt, chosen uniformly at random in {−1,+1}n, arrive over time and a sign xt∈{−1,+1} must be picked immediately upon the arrival of vt. The goal is to minimize the L∞ norm of the signed sum ∑txtvt. We give an online strategy for picking the signs xt that has value O(n1/2) with high probability. Up to constants, this is the best possible even when the vectors are given in advance

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an O((tlogt)1/2) discrepancy bound in the regime when n ≤ m and an O(1) bound when n≫mt. In this paper, we give a tight O(√t) bound for the entire range of n and m, under a mild assumption that t=Ω(log log m)2. The result is based on two steps. First, applying the partial coloring method to the case when n=mlogO(1)m and using the properties of the random set system we show that the overa

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

Faster space-efficient algorithms for Subset Sum, k -Sum, and related problems

We present randomized algorithms that solve subset sum and knapsack instances with n items in O∗ (20.86n) time, where the O∗ (∙ ) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve binary integer programming on n variables with few constraints in a similar running time. We also show that for any constant k ≥ 2, random instances of k-Sum can be solved using O(nk -0.5polylog(n)) time and O(log n) space, without the assumption of random access to random bits.Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(log n) space significantly faster than the trivial O(n2) time algorithm if no value occurs too often in the same list.</p

New Notions and Constructions of Sparsification for Graphs and Hypergraphs

A sparsifier of a graph G (Benczu´r and Karger; Spielman and Teng) is a sparse weighted subgraph ˜ G that approximately retains the same cut structure of G. For general graphs, non-trivial sparsification is possible only by using weighted graphs in which different edges have different weights. Even for graphs that admit unweighted sparsifiers (that is, sparsifiers in which all the edge weights are equal to the same scaling factor), there are no known polynomial time algorithms that find such unweighted sparsifiers. We study a weaker notion of sparsification suggested by Oveis Gharan, in which the number of cut edges in each cut (S, ¯ S) is not approximated within a multiplicative factor (1 + ǫ), but is, instead, approximated up to an additive term bounded by ǫ times d · |S| + vol(S), where d is the average

On the Lovász theta function for independent sets in sparse graphs

We consider the maximum independent set problem on sparse graphs with maximum degree d. We show that the Lovász ϑ-function based semidefinite program (SDP) has an integrality gap of O(d/log3/2 d), improving on the previous best result of O(d/log d). This improvement is based on a new Ramsey-theoretic bound on the independence number of Kr-free graphs for large values of r. We also show that for stronger SDPs, namely, those obtained using polylog(d) levels of the SA+ semidefinite hierarchy, the integrality gap reduces to O(d/log2 d). This matches the best unique-games-based hardness result up to lower-order poly(log log d) factors. Finally, we give an algorithmic version of this SA+-based integrality gap result, albeit using d levels of SA+, via a coloring algorithm of Johansson

Achievable performance of blind policies in heavy traffic

For a GI/GI/1 queue, we show that the average sojourn time under the (blind) Randomized Multilevel Feedback algorithm is no worse than that under the Shortest Remaining Processing Time algorithm times a logarithmic function of the system load. Moreover, it is verified that this bound is tight in heavy traffic, up to a constant multiplicative factor. We obtain this result by combining techniques from two disparate areas: competitive analysis and applied probability

Penerapan Model Pembelajaran Numbered Heads Together dengan Media Flashcard dalam Peningkatan Pembelajaran IPS tentang Jenis Pekerjaan pada Siswa Kelas III SD Negeri 1 Bumirejo Tahun Ajaran 2017/2018

ABSTRAK Lilis Muryani. PENERAPAN MODEL PEMBELAJARAN NUMBERED HEADS TOGETHER DENGAN MEDIA FLASHCARD DALAM PENINGKATAN PEMBELAJARAN IPS TENTANG JENIS PEKERJAAN PADA SISWA KELAS III SD NEGERI 1 BUMIREJO TAHUN AJARAN 2017/2018. Skripsi, Fakultas Keguruan dan Ilmu Pendidikan Universitas Sebelas Maret Surakarta. Mei 2018. Tujuan penelitian ini yaitu: (1) mendeskripsikan langkah penerapan model Numbered Heads Together dengan media Flashcard dalam peningkatan pembelajaran IPS tentang jenis pekerjaan pada siswa kelas III SD Negeri 1 Bumirejo tahun ajaran 2017/2018; (2) meningkatkan pembelajaran IPS tentang jenis pekerjaan pada siswa kelas III SD Negeri 1 Bumirejo tahun ajaran 2017/2018 melalui penerapan model Numbered Heads Together dengan media Flashcard; (3) mendeskripsikan kendala dan solusi penerapan model Numbered Heads Together dengan media Flashcard dalam peningkatan pembelajaran IPS tentang jenis pekerjaan pada siswa kelas III SD Negeri 1 Bumirejo tahun ajaran 2017/2018. Penelitian ini merupakan penelitian tindakan kelas (PTK) kolaboratif. Penelitian dilaksanakan selama tiga siklus. Sumber data dari penelitian ini yaitu guru dan siswa. Alat pengumpul data yang digunakan adalah observasi, pedoman wawancara dan tes hasil belajar siswa. Validitas data menggunakan triangulasi sumber dan triangulasi teknik. Teknik analisis data pada penelitian ini yaitu reduksi data, penyajian data, dan penarikan kesimpulan. Hasil penelitian menunjukkan bahwa: (1) penerapan model pembelajaran numbered heads together dengan media flashcard dilaksanakan dengan langkah-langkah: (a) membagi kelas menjadi beberapa kelompok kecil lalu membagi nomor kepala, (b) memberikan tugas, (c) diskusi kelompok, (d) memanggil nomor siswa untuk presentasi, (e) diskusi secara klasikal, (f) kesimpulan; (2) penerapan model pembelajaran numbered heads together dengan media flashcard dapat meningkatkan pembelajaran IPS tentang jenis pekerjaan di kelas III SD Negeri 1 Bumirejo tahun ajaran 2017/2018, baik dari segi proses maupun hasil pembelajaran.; (3) kendala dan solusi yang dihadapi yaitu: a) guru kurang maksimal dalam membimbing pelaksanaan diskusi kelompok dan menentukan jawaban atau hasil diskusi; b) siswa malu dalam bertanya apabila mengalami kesulitan dan menyampaikan hasil diskusi kelompoknya; c) siswa belum menanggapi hasil diskusi kelompok lain, d) ada beberapa anak yang mengganggu mahasiswa ketika sedang mendokumentasikan kegiatan pembelajaran. Adapun solusi untuk menyelesaikan kendala-kendala tersebut, antara lain: a) mengarahkan guru untuk membimbing pelaksanaan diskusi kelompok dan memberikan jawaban atau hasil diskusinya secara maksimal; b) membangkitkan semangat siswa agar tidak malu bertanya ketika mengalami kesulitan serta tidak malu dalam menyampaikan hasil diskusi kelompoknya; d) mengkondisikan siswa agar mau menganggapi hasil diskusi kelompok lain, e) menegur siswa tersebut agar fokus pada pembelajaran dan tidak mengganggu mahasiswa yang sedang mendokumentasikan kegiatan pembelajaran. Simpulan penelitian ini adalah penerapan model pembelajaran numbered heads together dengan media flashcard dapat meningkatkan pembelajaran IPS tentang jenis pekerjaan di kelas III SD Negeri 1 Bumirejo. Kata Kunci : numbered heads together, media flashcard, pembelajaran, IPS ABSTRACT Lilis Muryani. THE USE OF NUMBERED HEADS TOGETHER LEARNING MODEL USING FLASHCARD MEDIA TO IMPROVE LEARNING OUTCOMES OF SOCIAL SCIENCE ABOUT TYPE OF WORK FOR THE THIRD-GRADE STUDENTS OF SD NEGERI 1 BUMIREJO IN THE ACADEMIC YEAR OF 2017/2018. Thesis, Teacher Training and Education Faculity, Sebelas Maret University of Surakarta, May 2018. The objectives of this research are: (1) to describe the step on the use of numbered heads together learning model using flashcard media to improving learning on social science about type of work for the third-grade students of SD Negeri 1 Bumirejo in the academic year of 2017/2018; (2) to improve learning outcomes of social science about type of work for the third-grade students of SD Negeri 1 Bumirejo in the academic year of 2017/2018; and (3) to describe obstacle and solutions to the use of numbered heads together learning model using flashcard media to improving learning outcomes of social science about type of work for the third-grade for the third-grade of SD Negeri 1 Bumirejo in the academic year of 2017/2018. This research is a collaborative Classroom Action Research (CAR) conducted within three cycles. Subjects of this research were of third-grade student. The data were collected from the teacher and third-grade students. Tecniques of collecting data were observation, interview, and test of student’s learning outcome. Validity of data in this research was analyzed using triangulation of sources and triangulation of technique. Data analysis technique used in research are data reduction, data display, and drawing conclusion. The result of this research show that: (1) the steps on the use of numbered heads together learning model using flashcard media namely: (a) teachers divide the class into small groups. Each group member gets a number, (b) the teacher assigns a set of questions to be answered by each group, (c) each group thinks of answers to teacher questions. And make sure each group member knows the answer correctly, (d) the teacher dials the student number and the number dialed out of his or her group reports or explains the results of their cooperation, (e) teachers develop deeper discussion, so learners can find the answers to the questions as a whole knowledge, (f) conclusion; (2) the use of numbered heads together learning model using flashcard media can improve learning outcomes of social science.; (3) there were obstacles on the use of numbered heads together learning model using flashcard media namely: (a) teachers have not guided the implementation of group discussions, (b) the teacher has not determined the correct answer or discussion result, (c) students are embarrassed to ask when they have difficulties, (d) students are embarrassed when they come to the front of the class and deliver the results of their group discussions, (e) students have not responded to the results of other group discussions, (f) there are some students who annoy students while documenting learning activities. The conclusion of this research is the use of numbered heads together learning model using flashcard media can improve learning on social science about type of work for the third-grade students of SD Negeri 1 Bumirejo in the academic year of 2017/2018. Keywords: numbered heads together, flashcard media, improve learning, social scienc