Graph Properties in Node-Query Setting: Effect of Breaking Symmetry

The query complexity of graph properties is well-studied when queries are on the edges. We investigate the same when queries are on the nodes. In this setting a graph G = (V,E) on n vertices and a property P are given. A black-box access to an unknown subset S of V is provided via queries of the form "Does i belong to S?". We are interested in the minimum number of queries needed in the worst case in order to determine whether G[S] - the subgraph of G induced on S - satisfies P. Our primary motivation to study this model comes from the fact that it allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on the hereditary graph properties - properties that are closed under deletion of vertices as well as edges. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on G, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., n. We show that in the absence of any symmetry on G it can fall as low as O(n^{1/(d + 1)}) where d denotes the minimum possible degree of a minimal forbidden sub-graph for P. In particular, every hereditary property benefits at least quadratically. The main question left open is: Can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with finitely many forbidden subgraphs by exhibiting a bound of Omega(n^{1/k}) for a constant k depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing Omega(log(n)*log(log(n))) bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdos and Rado

Any monotone property of 3-uniform hypergraphs is weakly evasive

© 2014 Elsevier B.V. For a Boolean function f, let D(f) denote its deterministic decision tree complexity, i.e., minimum number of (adaptive) queries required in worst case in order to determine f. In a classic paper, Rivest and Vuillemin [11] show that any non-constant monotone property P:{0,1}(n2)→{0,1} of n-vertex graphs has D(P)=Ω(n2).We extend their result to 3-uniform hypergraphs. In particular, we show that any non-constant monotone property P:{0,1}(n3)→{0,1} of n-vertex 3-uniform hypergraphs has D(P)=Ω(n3).Our proof combines the combinatorial approach of Rivest and Vuillemin with the topological approach of Kahn, Saks, and Sturtevant [6]. Interestingly, our proof makes use of Vinogradov's Theorem (weak Goldbach Conjecture), inspired by its recent use by Babai et al. [1] in the context of the topological approach. Our work leaves the generalization to k-uniform hypergraphs as an intriguing open question

Disturbing the Dream of Integration: Critical Whiteness and the History of Penn State’s College of Education, 1954-1963

In this study I drawn upon Critical Whiteness frameworks and a deconstructionist historiographical method to explore tensions between espoused and enacted ‘integrationist’ values within the Pennsylvania State University’s College of Education in the decade following Brown v. Board (1954-1963). This site-specific historical approach is a response to the fact that the vast majority of higher education scholarship exploring the history of the Civil Rights era focuses on Southern institutions and their overt struggles over desegregation and racial integration. This focus is warranted given the dramatic and often violent nature of this period of Southern history, but it may serve to obscure more subtle patterns of re-segregation, sidelining, and marginalization of Black concerns on Northern campuses. By examining the curriculum and doctoral dissertations from Penn State’s College of Education through a Critical Whiteness frame, this study contributes to recent scholarship of Northern colleges that seeks to disrupt the overly simplistic master narrative of peaceful campus racial integration, and calls for Northern colleges to recognize, grapple with, and atone for their own histories

On the Sensitivity Complexity of k-Uniform Hypergraph Properties

In this paper we investigate the sensitivity complexity of hypergraph properties. We present a k-uniform hypergraph property with sensitivity complexity O(n^{ceil(k/3)}) for any k >= 3, where n is the number of vertices. Moreover, we can do better when k = 1 (mod 3) by presenting a k-uniform hypergraph property with sensitivity O(n^{ceil(k/3)-1/2}). This result disproves a conjecture of Babai, which conjectures that the sensitivity complexity of k-uniform hypergraph properties is at least Omega(n^{k/2}). We also investigate the sensitivity complexity of other weakly symmetric functions and show that for many classes of transitive-invariant Boolean functions the minimum achievable sensitivity complexity can be O(N^{1/3}), where N is the number of variables. Finally, we give a lower bound for sensitivity of k-uniform hypergraph properties, which implies the sensitivity conjecture of k-uniform hypergraph properties for any constant k

Taking Out the Adversary: The Assault on Progressive Public Interest Lawyers

This Essay concerns laws and doctrines, some very recent, that undermine the capacity of progressive public-interest lawyers to bring cases. It asks a simple-sounding question: how just is the adversary system if one side is not adequately represented in it? And it defends a simple-sounding answer: It is not just at all. As we shall see, however, neither the question nor the answer is quite as simple as it sounds

A Theory of Racialized Judicial Decision-Making

In this Article, I introduce a theory of racialized judicial decision-making as a framework to explain how judicial decision-making as a system contributes to creating and maintaining the racial hierarchy in the United States. Judicial decision-making, I argue, is itself a racialized systemic process in which judges transpose racially-bounded cognitive schemas as they make decisions. In the process, they assign legal burdens differentially across ethnoracial groups, to the disproportionate detriment of ethnoracial minorities. After presenting this argument, I turn to three mechanisms at play in racialized judicial decision-making: (1) whiteness as capital that increases epistemic advantages in the judicial process, (2) color-evasive approaches as effective tools to justify racially disparate outcomes, and (3) the elevation of racial discrimination into a status of exceptionalism that justifies heightened standards in proving racial anti-discrimination claims. I argue that the racialized judicial decision-making process reproducing the social racial hierarchy is institutionalized via the legitimacy courts wield. I conclude with a discussion on the agency and autonomy inherent in the judicial decision-making process, emphasizing judicial decision-making is not simply a reflection of ideology—personal or otherwise—individual biases, or cultural tides, and can as a system be leveraged to further racial equity in a democratic society

USA v. Jamar Hunter

USDC for the Eastern District of Pennsylvani