Eigenstripping, Spectral Decay, and Edge-Expansion on Posets

Fast mixing of random walks on hypergraphs (simplicial complexes) has recently led to myriad breakthroughs throughout theoretical computer science. Many important applications, however, (e.g. to LTCs, 2-2 games) rely on a more general class of underlying structures called posets, and crucially take advantage of non-simplicial structure. These works make it clear that the global expansion properties of posets depend strongly on their underlying architecture (e.g. simplicial, cubical, linear algebraic), but the overall phenomenon remains poorly understood. In this work, we quantify the advantage of different poset architectures in both a spectral and combinatorial sense, highlighting how regularity controls the spectral decay and edge-expansion of corresponding random walks. We show that the spectra of walks on expanding posets (Dikstein, Dinur, Filmus, Harsha APPROX-RANDOM 2018) concentrate in strips around a small number of approximate eigenvalues controlled by the regularity of the underlying poset. This gives a simple condition to identify poset architectures (e.g. the Grassmann) that exhibit strong (even exponential) decay of eigenvalues, versus architectures like hypergraphs whose eigenvalues decay linearly - a crucial distinction in applications to hardness of approximation and agreement testing such as the recent proof of the 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). We show these results lead to a tight characterization of edge-expansion on expanding posets in the ??-regime (generalizing recent work of Bafna, Hopkins, Kaufman, and Lovett (SODA 2022)), and pay special attention to the case of the Grassmann where we show our results are tight for a natural set of sparsifications of the Grassmann graphs. We note for clarity that our results do not recover the characterization of expansion used in the proof of the 2-2 Games Conjecture which relies on ?_? rather than ??-structure

Algorithms for detecting dependencies and rigid subsystems for CAD

Geometric constraint systems underly popular Computer Aided Design soft- ware. Automated approaches for detecting dependencies in a design are critical for developing robust solvers and providing informative user feedback, and we provide algorithms for two types of dependencies. First, we give a pebble game algorithm for detecting generic dependencies. Then, we focus on identifying the "special positions" of a design in which generically independent constraints become dependent. We present combinatorial algorithms for identifying subgraphs associated to factors of a particular polynomial, whose vanishing indicates a special position and resulting dependency. Further factoring in the Grassmann- Cayley algebra may allow a geometric interpretation giving conditions (e.g., "these two lines being parallel cause a dependency") determining the special position.Comment: 37 pages, 14 figures (v2 is an expanded version of an AGD'14 abstract based on v1

Streaming Hardness of Unique Games

We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by p, the alphabet size of the Unique Game, gives a trivial p-approximation that can be computed in O(log n) space. Meanwhile, with high probability, a sample of O~(n) constraints suffices to estimate the optimal value to (1+epsilon) accuracy. We prove that any single-pass streaming algorithm that achieves a (p-epsilon)-approximation requires Omega_epsilon(sqrt n) space. Our proof is via a reduction from lower bounds for a communication problem that is a p-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to downstream hardness results for streaming approximability for other CSP-like problems

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

A Log-Sobolev Inequality for the Multislice, with Applications

Let kappa in N_+^l satisfy kappa_1 + *s + kappa_l = n, and let U_kappa denote the multislice of all strings u in [l]^n having exactly kappa_i coordinates equal to i, for all i in [l]. Consider the Markov chain on U_kappa where a step is a random transposition of two coordinates of u. We show that the log-Sobolev constant rho_kappa for the chain satisfies rho_kappa^{-1} <= n * sum_{i=1}^l 1/2 log_2(4n/kappa_i), which is sharp up to constants whenever l is constant. From this, we derive some consequences for small-set expansion and isoperimetry in the multislice, including a KKL Theorem, a Kruskal - Katona Theorem for the multislice, a Friedgut Junta Theorem, and a Nisan - Szegedy Theorem

Boolean functions on high-dimensional expanders

We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only $|X(k-1)|=O(n)$ points in contrast to $\binom{n}{k}$ points in the $(k)$-slice (which consists of all $n$-bit strings with exactly $k$ ones).Comment: 48 pages, Extended version of the prior submission, with more details of expanding posets (eposets

UG-Hardness to NP-Hardness by Losing Half

The 2-to-2 Games Theorem of [Subhash Khot et al., 2017; Dinur et al., 2018; Dinur et al., 2018; Dinur et al., 2018] implies that it is NP-hard to distinguish between Unique Games instances with assignment satisfying at least (1/2-epsilon) fraction of the constraints vs. no assignment satisfying more than epsilon fraction of the constraints, for every constant epsilon>0. We show that the reduction can be transformed in a non-trivial way to give a stronger guarantee in the completeness case: For at least (1/2-epsilon) fraction of the vertices on one side, all the constraints associated with them in the Unique Games instance can be satisfied. We use this guarantee to convert the known UG-hardness results to NP-hardness. We show: 1) Tight inapproximability of approximating independent sets in degree d graphs within a factor of Omega(d/(log^2 d)), where d is a constant. 2) NP-hardness of approximate the Maximum Acyclic Subgraph problem within a factor of 2/3+epsilon, improving the previous ratio of 14/15+epsilon by Austrin et al. [Austrin et al., 2015]. 3) For any predicate P^{-1}(1) subseteq [q]^k supporting a balanced pairwise independent distribution, given a P-CSP instance with value at least 1/2-epsilon, it is NP-hard to satisfy more than (|P^{-1}(1)|/(q^k))+epsilon fraction of constraints

NP-Hardness of Almost Coloring Almost 3-Colorable Graphs

A graph G = (V,E) is said to be (k,?) almost colorable if there is a subset of vertices V\u27 ? V of size at least (1-?)|V| such that the induced subgraph of G on V\u27 is k-colorable. We prove that for all k, there exists ? > 0 such for all ? > 0, given a graph G it is NP-hard (under randomized reductions) to distinguish between: 1) Yes case: G is (3,?) almost colorable. 2) No case: G is not (k,?) almost colorable. This improves upon an earlier result of Khot et al. [Irit Dinur et al., 2018], who showed a weaker result wherein in the "yes case" the graph is (4,?) almost colorable