We study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and... more We study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a new way to transform a PCP with imperfect completeness to a PCP with perfect completeness, when the initial gap is a constant. In particular, we show that PCPc,s[r, q] ⊆ PCP1,1−Ω(1)[r + O(1), q + O(r)], for c − s = Ω(1). This implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NTIME[O(n)] with a O(log n) additive loss in the query complexity q. We show our result by constructing a “robust circuit” using threshold gates. These results are a gap amplification procedure for PCPs (when completeness is imperfect), analogous to questions studied in parallel repetition [23] and pseudorandomness [14]. We also investigate the time complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness,...
The problem of solving linear systems is one of the most fundamental problems in computer science... more The problem of solving linear systems is one of the most fundamental problems in computer science, where given a satisfiable linear system (A, b), for A ∈ Rn×n and b ∈ R, we wish to find a vector x ∈ R such that Ax = b. The current best algorithms for solving dense linear systems reduce the problem to matrix multiplication, and run in time O(n). We consider the problem of finding ε-approximate solutions to linear systems with respect to the L2-norm, that is, given a satisfiable linear system (A ∈ Rn×n, b ∈ R), find an x ∈ R such that ||Ax − b||2 ≤ ε||b||2. Our main result is a fine-grained reduction from computing the rank of a matrix to finding ε-approximate solutions to linear systems. In particular, if the best known Õ(n) time algorithm for computing the rank of n×O(n) matrices is optimal (which we conjecture is true), then finding an ε-approximate solution to a dense linear system also requires Ω̃(n) time, even for ε as large as (1 − 1/poly(n)). We also prove (under some modifie...
Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible ... more Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass (ITCS 2016), yet their broader combinatorial and spectral properties remain poorly understood. We develop a combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders (Dinur and Kaufman FOCS 2017), which offer a broad generalization of the well-studied Johnson and Grassmann graphs. Our characterization, which shows that the spectra of HD-walks lie tightly concentrated in a few combinatorially structured strips, leads to novel structural theorems such as a tight l2-characterization of edge-expansion, as well as to a new understanding of local-to-global graph algorithms on HDX. Towards the latter, we introduce a novel spectral complexity measure called Stripped Threshold Rank, and show how it can replace the (much larger) threshold rank as a parameter controlling the...
Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean funct... more Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and degree over R, are all polynomially related to one another. The sensitivity conjecture states that there is also a polynomial relationship between sensitivity and block sensitivity, thus supplying the "missing link". Since its introduction in 1991, the sensitivity conjecture has remained a challenging open question in the study of Boolean functions. One natural approach is to prove it for special classes of functions. For instance, the conjecture is known to be true for monotone functions, symmetric functions, and functions describing graph properties. In this paper, we consider the conjecture for Boolean functions computa...
We study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and... more We study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a new way to transform a PCP with imperfect completeness to a PCP with perfect completeness, when the initial gap is a constant. In particular, we show that PCPc,s[r, q] ⊆ PCP1,1−Ω(1)[r + O(1), q + O(r)], for c − s = Ω(1). This implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NTIME[O(n)] with a O(log n) additive loss in the query complexity q. We show our result by constructing a “robust circuit” using threshold gates. These results are a gap amplification procedure for PCPs (when completeness is imperfect), analogous to questions studied in parallel repetition [23] and pseudorandomness [14]. We also investigate the time complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness,...
The problem of solving linear systems is one of the most fundamental problems in computer science... more The problem of solving linear systems is one of the most fundamental problems in computer science, where given a satisfiable linear system (A, b), for A ∈ Rn×n and b ∈ R, we wish to find a vector x ∈ R such that Ax = b. The current best algorithms for solving dense linear systems reduce the problem to matrix multiplication, and run in time O(n). We consider the problem of finding ε-approximate solutions to linear systems with respect to the L2-norm, that is, given a satisfiable linear system (A ∈ Rn×n, b ∈ R), find an x ∈ R such that ||Ax − b||2 ≤ ε||b||2. Our main result is a fine-grained reduction from computing the rank of a matrix to finding ε-approximate solutions to linear systems. In particular, if the best known Õ(n) time algorithm for computing the rank of n×O(n) matrices is optimal (which we conjecture is true), then finding an ε-approximate solution to a dense linear system also requires Ω̃(n) time, even for ε as large as (1 − 1/poly(n)). We also prove (under some modifie...
Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible ... more Higher order random walks (HD-walks) on high dimensional expanders (HDX) have seen an incredible amount of study and application since their introduction by Kaufman and Mass (ITCS 2016), yet their broader combinatorial and spectral properties remain poorly understood. We develop a combinatorial characterization of the spectral structure of HD-walks on two-sided local-spectral expanders (Dinur and Kaufman FOCS 2017), which offer a broad generalization of the well-studied Johnson and Grassmann graphs. Our characterization, which shows that the spectra of HD-walks lie tightly concentrated in a few combinatorially structured strips, leads to novel structural theorems such as a tight l2-characterization of edge-expansion, as well as to a new understanding of local-to-global graph algorithms on HDX. Towards the latter, we introduce a novel spectral complexity measure called Stripped Threshold Rank, and show how it can replace the (much larger) threshold rank as a parameter controlling the...
Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean funct... more Various combinatorial/algebraic parameters are used to quantify the complexity of a Boolean function. Among them, sensitivity is one of the simplest and block sensitivity is one of the most useful. Nisan (1989) and Nisan and Szegedy (1991) showed that block sensitivity and several other parameters, such as certificate complexity, decision tree depth, and degree over R, are all polynomially related to one another. The sensitivity conjecture states that there is also a polynomial relationship between sensitivity and block sensitivity, thus supplying the "missing link". Since its introduction in 1991, the sensitivity conjecture has remained a challenging open question in the study of Boolean functions. One natural approach is to prove it for special classes of functions. For instance, the conjecture is known to be true for monotone functions, symmetric functions, and functions describing graph properties. In this paper, we consider the conjecture for Boolean functions computa...
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