In two papers, Bürgisser and Ikenmeyer (STOC 2011, STOC 2013) used an adaption of the geometric complexity theory (GCT) approach by Mulmuley and Sohoni (Siam J Comput 2001, 2008) to prove lower bounds on the border rank of the matrix multiplication tensor. A key ingredient was information about certain Kronecker coefficients. While tensors are an interesting test bed for GCT ideas, the far-away goal is the separation of algebraic complexity classes. The role of the Kronecker coefficients in that setting is taken by the so-called plethysm coefficients: These are the multiplicities in the coordinate rings of spaces of polynomials. Even though several hardness results for Kronecker coefficients are known, there are almost no results about the complexity of computing the plethysm coefficients or even deciding their positivity. In this paper, we show that deciding positivity of plethysm coefficients is -hard and that computing plethysm coefficients is #-hard. In fact, both problems remain hard even if the inner parameter of the plethysm coefficient is fixed. In this way, we obtain an inner versus outer contrast: If the outer parameter of the plethysm coefficient is fixed, then the plethysm coefficient can be computed in polynomial time. Moreover, we derive new lower and upper bounds and in special cases even combinatorial descriptions for plethysm coefficients, which we consider to be of independent interest. Our technique uses discrete tomography in a more refined way than the recent work on Kronecker coefficients by Ikenmeyer, Mulmuley, and Walter (Comput Compl 2017). This makes our work the first to apply techniques from discrete tomography to the study of plethysm coefficients. Quite surprisingly, that interpretation also leads to new equalities between certain plethysm coefficients and Kronecker coefficients.
This paper is concerned with the 1|| pjUj problem, the problem of minimizing the total processing time of tardy jobs on a single machine. This is not only a fundamental scheduling problem, but also a very important problem from a theoretical point of view as it generalizes the Subset Sum problem and is closely related to the 0/1-Knapsack problem. The problem is well-known to be NP-hard, but only in a weak sense, meaning it admits pseudo-polynomial time algorithms. The fastest known pseudo-polynomial time algorithm for the problem is the famous Lawler and Moore algorithm which runs in O(P • n) time, where P is the total processing time of all n jobs in the input. This algorithm has been developed in the late 60s, and has yet to be improved to date.In this paper we develop two new algorithms for 1|| pjUj, each improving on Lawler and Moore's algorithm in a different scenario:Our first algorithm runs in Õ(P 7/4 ) time 1 , and outperforms Lawler and Moore's algorithm in instances where n = ω(P 3/4 ).Our second algorithm runs in Õ(min{P • D # , P + D}) time, where D # is the number of different due dates in the instance, and D is the sum of all different due dates. This algorithm improves on Lawler and Moore's algorithm when n = ω(D # ) or n = ω(D/P ). Further, it extends the known Õ(P ) algorithm for the single due date special case of 1|| pjUj in a natural way.Both algorithms rely on basic primitive operations between sets of integers and vectors of integers for the speedup in their running times. The second algorithm relies on fast polynomial multiplication as its main engine, while for the first algorithm we define a new "skewed" version of (max, min)convolution which is interesting in its own right. ACM Subject Classification Theory of computation → Design and analysis of algorithmsKeywords and phrases Weighted number of tardy jobs, sumsets, convolutions1 Throughout the paper we use Õ(•) to suppress logarithmic factors.
The McCarthy-era witch hunts marked the culmination of an anticommunist crusade launched after the First World War. With Bolshevism triumphant in Russia and public discontent shaking the United States, conservatives at every level of government and business created a network dedicated to sweeping away the “spider web” of radicalism they saw threatening the nation. This book shines a light on right-wing activities during the interwar period. Conservatives, eager to dispel communism's appeal to the working class, railed against a supposed Soviet-directed conspiracy composed of socialists, trade unions, peace and civil liberties groups, feminists, liberals, aliens, and Jews. The rhetoric and power of anticommunism made for devastating weapons in a systematic war for control of the country against progressive causes. But, as the book shows, the term “spider web” far more accurately described the anticommunist movement than it did the makeup and operations of international communism. The book details how anticommunist myths and propaganda influenced mainstream politics in America, and how its ongoing efforts paved the way for the McCarthyite Fifties—and augured the conservative backlash that would one day transform American politics.
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