scholarly journals Anticoncentration versus the Number of Subset Sums

2021 ◽  
Author(s):  
Vishesh Jain ◽  
Ashwin Sah ◽  
Mehtaab Sawhney
Keyword(s):  
Recent Result ◽  
Bin Packing ◽  
Similar Improvement ◽  
Subset Sums

Let w=(w_1,...,w_n) be a vector from R^n. We show that for any n^{-2}<=eps<=1, if the number of zero-one vectors xi such that <xi,w>=tau is at least 2^{-eps*n}*2^n for some tau, then the number of values <xi,w> is at most 2^{O(eps^{1/2}n)} where xi ranges over all zero-one vectors. This exponentially improves the eps dependence in a recent result of Nederlof, Pawlewicz, Swennenhuis, and Węgrzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.

scholarly journals Additive kinematic formulas for flag area measures

2021 ◽  
Author(s):  
Judit Abardia-Evéquoz ◽  
Andreas Bernig
Keyword(s):  
Recent Result ◽  
Flag Manifold ◽  
Unit Vector ◽  
Algebraic Framework

AbstractWe show the existence of additive kinematic formulas for general flag area measures, which generalizes a recent result by Wannerer. Building on previous work by the second named author, we introduce an algebraic framework to compute these formulas explicitly. This is carried out in detail in the case of the incomplete flag manifold consisting of all $$(p+1)$$ ( p + 1 ) -planes containing a unit vector.

Tight Bounds for Clairvoyant Dynamic Bin Packing

2019 ◽  
Vol 6 (3) ◽  
pp. 1-21 ◽  
Author(s):  
Yossi Azar ◽  
Danny Vainstein
Keyword(s):  
Bin Packing

Power of d Choices for Large-Scale Bin Packing

2015 ◽  
Vol 43 (1) ◽  
pp. 321-334 ◽  
Author(s):  
Qiaomin Xie ◽  
Xiaobo Dong ◽  
Yi Lu ◽  
Rayadurgam Srikant
Keyword(s):  
Large Scale ◽  
Bin Packing

scholarly journals An Exponential Separation Between MA and AM Proofs of Proximity

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Tom Gur ◽  
Yang P. Liu ◽  
Ron D. Rothblum
Keyword(s):  
Quantum Information ◽  
Lower Bound ◽  
Recent Result ◽  
Query Complexity ◽  
Random String ◽  
Natural Property ◽  
Public And Private ◽  
Alternate Proof ◽  
Interactive Proofs ◽  
Exponential Separation

AbstractInteractive proofs of proximity allow a sublinear-time verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work, we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof. The first variant, known as -proofs of Proximity (), is fully non-interactive, meaning that the proof is a function of the input only. The second variant, known as -proofs of Proximity (), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both s and s is the total number of bits that the verifier observes—namely, the sum of the proof length and query complexity. Our main result is an exponential separation between the power of s and s. Specifically, we exhibit an explicit and natural property $$\Pi$$ Π that admits an with complexity $$O(\log n)$$ O ( log n ) , whereas any for $$\Pi$$ Π has complexity $$\tilde{\Omega}(n^{1/4})$$ Ω ~ ( n 1 / 4 ) , where n denotes the length of the input in bits. Our lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014). Also, Aaronson (Quantum Information & Computation 2012) has shown a $$\Omega(n^{1/6})$$ Ω ( n 1 / 6 ) lower bound for the same property $$\Pi$$ Π .Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting, we show that s can only be quadratically stronger than s. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity.

scholarly journals Dynamic Windows Scheduling with Reallocation

2021 ◽  
Vol 26 ◽  
pp. 1-19
Author(s):  
Martín Farach-Colton ◽  
Katia Leal ◽  
Miguel A. Mosteiro ◽  
Christopher Thraves Caro
Keyword(s):  
Supply Chain ◽  
Bin Packing ◽  
Time Slot ◽  
Peak Load ◽  
Communication Channels ◽  
Current Load ◽  
Constant Amount ◽  
Unit Fractions ◽  
Consecutive Time ◽  
Multiple Variants

We consider the Windows Scheduling (WS) problem, which is a restricted version of Unit-Fractions Bin Packing, and it is also called Inventory Replenishment in the context of Supply Chain. In brief, WS problem is to schedule the use of communication channels to clients. Each client c i is characterized by an active cycle and a window w i . During the period of time that any given client c i is active, there must be at least one transmission from c i scheduled in any w i consecutive time slots, but at most one transmission can be carried out in each channel per time slot. The goal is to minimize the number of channels used. We extend previous online models, where decisions are permanent, assuming that clients may be reallocated at some cost. We assume that such cost is a constant amount paid per reallocation. That is, we aim to minimize also the number of reallocations. We present three online reallocation algorithms for Windows Scheduling. We evaluate experimentally multiple variants of these protocols showing that, in practice, all three achieve constant amortized reallocations with close to optimal channel usage. Our simulations also expose interesting tradeoffs between reallocations and channel usage. We introduce a new objective function for WS with reallocations that can be also applied to models where reallocations are not possible. We analyze this metric for one of the algorithms that, to the best of our knowledge, is the first online WS protocol with theoretical guarantees that applies to scenarios where clients may leave and the analysis is against current load rather than peak load. Using previous results, we also observe bounds on channel usage for one of the algorithms.

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