We study the natural problem of estimating the expansion of subsets of vertices on one side of a bipartite graph. More precisely, given a bipartite graph G(U, V,E) and a parameter β, the goal is to find a subset V ′ ⊆ V containing β fraction of the vertices of V which minimizes the size of N(V ′), the neighborhood of V ′. This problem, which we call Bipartite Expansion, is a special case of sub...

We present a simple new construction of a pseudorandom bit generator, based on the constant depth generators of [N]. It stretches a short string of truly random bits into a long string that looks random to any algorithm from a complexity class C (eg P, NC, PSPACE, ...) using an arbitrary function that is hard for C. This construction reveals an equivalence between the problem of proving lower b...

We consider the problem of amplifying uniform average-case hardness of languages in NP, where hardness is with respect to BPP algorithms. We introduce the notion of monotone errorcorrecting codes, and show that hardness amplification for NP is essentially equivalent to constructing efficiently locally encodable and locally list-decodable monotone codes. The previous hardness amplification resul...

The class TFNP is the search analog of NP with the additional guarantee that any instance has a solution. TFNP has attracted extensive attention due to its natural syntactic subclasses that capture the computational complexity of important search problems from algorithmic game theory, combinatorial optimization and computational topology. Thus, one of the main research objectives in the context...

Pebble games were originally formulated to study time-space tradeoffs in computation, modeled by games played on directed acyclic graphs (DAGs). Close connections between pebbling and cryptography have been known for decades. A series of recent research starting with (Alwen and Serbinenko, STOC 2015) has deepened our understanding of the notion of memory-hardness in cryptography— a useful prope...

We study the isomorphic implication problem for Boolean constraints. We show that this is a natural analog of the subgraph isomorphism problem. We prove that, depending on the set of constraints, this problem is in P, NP-complete, or NP-hard, coNP-hard, and in P || . We show how to extend the NP-hardness and coNP-hardness to P || -hardness for some cases, and conjecture that this can be done in...

In this lecture, we first examine several classes of NP-hardness and polynomial time algorithms which arise from differences in how integers are encoded in problem input. We then look at the 3-partition problem, which is very useful for proving the strongest notion of NP-hardness. Finally, we use reduction from 3-partition to prove NP-hardness for a handful of problems, including a set of 4 pac...

I. The Knoop test of hardness of materials has been used to test the hardness of human nail. 2. The range of hardness of nails was found to be small in healthy persons and large in malnourished children. 3. Nails from malnourished children were in general much harder than those from healthy persons. The nails of a small group of three children became softer while they were recovering from malnu...

A simplified model of the hardness kernel, hð~r ; ~r 0Þydð~r K~r 0Þ, has been evaluated for the calculation of the global hardness in a series of Lewis acids and bases using the MP2 method. It is found that this simple model yields the appropriate ordering of the molecules according to their hardness values when compared to experimental available results. q 2005 Elsevier B.V. All rights reserved.