arXiv subjects

Yuval Filmus

Publications and source records attributed to Yuval Filmus.

At least 55 records · Page 3Linked to original sources

arXiv2021

Approximate polymorphisms

For a function $g\colon\{0,1\}^m\to\{0,1\}$, a function $f\colon \{0,1\}^n\to\{0,1\}$ is called a $g$-polymorphism if their actions commute: $f(g(\mathsf{row}_1(Z)),\ldots,g(\mathsf{row}_n(Z))) = g(f(\mathsf{col}_1(Z)),\ldots,f(\mathsf{col}_m(Z)))$ for all $Z\in\{0,1\}^{n\times m}$. The function $f$ is called an approximate polymorphism if this equality holds with probability close to $1$, when $Z$ is sampled uniformly. We study the structure of exact polymorphisms as well as approximate polymorphisms. Our results include: - We prove that an approximate polymorphism $f$ must be close to an exact polymorphism; - We give a characterization of exact polymorphisms, showing that besides trivial cases, only the functions $g = \mathsf{AND}, \mathsf{XOR}, \mathsf{OR}, \mathsf{NXOR}$ admit non-trivial exact polymorphisms. We also study the approximate polymorphism problem in the list-decoding regime (i.e., when the probability equality holds is not close to $1$, but is bounded away from some value). We show that if $f(x \land y) = f(x) \land f(y)$ with probability larger than $s_\land \approx 0.815$ then $f$ correlates with some low-degree character, and $s_\land$ is the optimal threshold for this property. Our result generalize the classical linearity testing result of Blum, Luby and Rubinfeld, that in this language showed that the approximate polymorphisms of $g = \mathsf{XOR}$ are close to XOR's, as well as a recent result of Filmus, Lifshitz, Minzer and Mossel, showing that the approximate polymorphisms of AND can only be close to AND functions.

cs.DM↗

arXiv2021

Revisiting the Complexity Analysis of Conflict-Based Search: New Computational Techniques and Improved Bounds

The problem of Multi-Agent Path Finding (MAPF) calls for finding a set of conflict-free paths for a fleet of agents operating in a given environment. Arguably, the state-of-the-art approach to computing optimal solutions is Conflict-Based Search (CBS). In this work we revisit the complexity analysis of CBS to provide tighter bounds on the algorithm's run-time in the worst-case. Our analysis paves the way to better pinpoint the parameters that govern (in the worst case) the algorithm's computational complexity. Our analysis is based on two complementary approaches: In the first approach we bound the run-time using the size of a Multi-valued Decision Diagram (MDD) -- a layered graph which compactly contains all possible single-agent paths between two given vertices for a specific path length. In the second approach we express the running time by a novel recurrence relation which bounds the algorithm's complexity. We use generating functions-based analysis in order to tightly bound the recurrence. Using these technique we provide several new upper-bounds on CBS's complexity. The results allow us to improve the existing bound on the running time of CBS for many cases. For example, on a set of common benchmarks we improve the upper-bound by a factor of at least $2^{10^{7}}$.

cs.MA↗

arXiv2020

Agreement tests on graphs and hypergraphs

Agreement tests are a generalization of low degree tests that capture a local-to-global phenomenon, which forms the combinatorial backbone of most PCP constructions. In an agreement test, a function is given by an ensemble of local restrictions. The agreement test checks that the restrictions agree when they overlap, and the main question is whether average agreement of the local pieces implies that there exists a global function that agrees with most local restrictions. There are very few structures that support agreement tests, essentially either coming from algebraic low degree tests or from direct product tests (and recently also from high-dimensional expanders). In this work, we prove a new agreement theorem which extends direct product tests to higher dimensions, analogous to how low degree tests extend linearity testing. As a corollary of our main theorem, it follows that an ensemble of small graphs on overlapping sets of vertices can be glued together to one global graph assuming they agree with each other on average. We prove the agreement theorem by (re)proving the agreement theorem for dimension 1, and then generalizing it to higher dimensions (with the dimension 1 case being the direct product test, and dimension 2 being the graph case). A key technical step in our proof is the reverse union bound, which allows us to treat dependent events as if they are disjoint, and may be of independent interest. An added benefit of the reverse union bound is that it can be used to show that the "majority decoded" function also serves as a global function that explains the local consistency of the agreement theorem, a fact that was not known even in the direct product setting (dimension 1) prior to our work.

cs.CC↗

arXiv2020

Hypercontractivity on the symmetric group

The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of \emph{global functions} on $S_n$, which are functions whose $2$-norm remains small when restricting $O(1)$ coordinates of the input, and assert that low-degree, global functions have small $q$-norms, for $q>2$. As applications, we show: 1. An analog of the level-$d$ inequality on the hypercube, asserting that the mass of a global function on low-degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group $A_n$. 2. Isoperimetric inequalities on the transposition Cayley graph of $S_n$ for global functions, that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube. 3. Hypercontractive inequalities on the multi-slice, and stability versions of the Kruskal--Katona Theorem in some regimes of parameters.

cs.DM↗

arXiv2020

Complexity Measures on the Symmetric Group and Beyond

We extend the definitions of complexity measures of functions to domains such as the symmetric group. The complexity measures we consider include degree, approximate degree, decision tree complexity, sensitivity, block sensitivity, and a few others. We show that these complexity measures are polynomially related for the symmetric group and for many other domains. To show that all measures but sensitivity are polynomially related, we generalize classical arguments of Nisan and others. To add sensitivity to the mix, we reduce to Huang's sensitivity theorem using "pseudo-characters", which witness the degree of a function. Using similar ideas, we extend the characterization of Boolean degree 1 functions on the symmetric group due to Ellis, Friedgut and Pilpel to the perfect matching scheme. As another application of our ideas, we simplify the characterization of maximum-size $t$-intersecting families in the symmetric group and the perfect matching scheme.

cs.CC↗

arXiv2020

Boolean degree 1 functions on some classical association schemes

We investigate Boolean degree 1 functions for several classical association schemes, including Johnson graphs, Grassmann graphs, graphs from polar spaces, and bilinear forms graphs, as well as some other domains such as multislices (Young subgroups of the symmetric group). In some settings, Boolean degree 1 functions are also known as \textit{completely regular strength 0 codes of covering radius 1}, \textit{Cameron--Liebler line classes}, and \textit{tight sets}. We classify all Boolean degree $1$ functions on the multislice. On the Grassmann scheme $J_q(n, k)$ we show that all Boolean degree $1$ functions are trivial for $n \geq 5$, $k, n-k \geq 2$ and $q \in \{ 2, 3, 4, 5 \}$, and that for general $q$, the problem can be reduced to classifying all Boolean degree $1$ functions on $J_q(n, 2)$. We also consider polar spaces and the bilinear forms graphs, giving evidence that all Boolean degree $1$ functions are trivial for appropriate choices of the parameters.

math.CO↗

arXiv2020

Explicit SoS lower bounds from high-dimensional expanders

We construct an explicit family of 3XOR instances which is hard for $O(\sqrt{\log n})$ levels of the Sum-of-Squares hierarchy. In contrast to earlier constructions, which involve a random component, our systems can be constructed explicitly in deterministic polynomial time. Our construction is based on the high-dimensional expanders devised by Lubotzky, Samuels and Vishne, known as LSV complexes or Ramanujan complexes, and our analysis is based on two notions of expansion for these complexes: cosystolic expansion, and a local isoperimetric inequality due to Gromov. Our construction offers an interesting contrast to the recent work of Alev, Jeronimo and the last author~(FOCS 2019). They showed that 3XOR instances in which the variables correspond to vertices in a high-dimensional expander are easy to solve. In contrast, in our instances the variables correspond to the edges of the complex.

cs.CC↗

arXiv2020

Tight Approximation for Unconstrained XOS Maximization

A set function is called XOS if it can be represented by the maximum of additive functions. When such a representation is fixed, the number of additive functions required to define the XOS function is called the width. In this paper, we study the problem of maximizing XOS functions in the value oracle model. The problem is trivial for the XOS functions of width $1$ because they are just additive, but it is already nontrivial even when the width is restricted to $2$. We show two types of tight bounds on the polynomial-time approximability for this problem. First, in general, the approximation bound is between $O(n)$ and $Ω(n / \log n)$, and exactly $Θ(n / \log n)$ if randomization is allowed, where $n$ is the ground set size. Second, when the width of the input XOS functions is bounded by a constant $k \geq 2$, the approximation bound is between $k - 1$ and $k - 1 - ε$ for any $ε> 0$. In particular, we give a linear-time algorithm to find an exact maximizer of a given XOS function of width $2$, while we show that any exact algorithm requires an exponential number of value oracle calls even when the width is restricted to $3$.

cs.DS↗

arXiv2020

A Sauer-Shelah-Perles Lemma for Lattices

We study lattice-theoretical extensions of the celebrated Sauer-Shelah-Perles Lemma. We conjecture that a general Sauer-Shelah-Perlem Lemma holds for a lattice $L$ if and only if $L$ is relatively complemented, and prove partial results towards this conjecture.

math.CO↗

arXiv2019

Asymptotic performance of the Grimmett-McDiarmid heuristic

Grimmett and McDiarmid suggested a simple heuristic for finding stable sets in random graphs. They showed that the heuristic finds a stable set of size $\sim\log_2 n$ (with high probability) on a $G(n, 1/2)$ random graph. We determine the asymptotic distribution of the size of the stable set found by the algorithm.

cs.DS↗

arXiv2019

High dimensional Hoffman bound and applications in extremal combinatorics

One powerful method for upper-bounding the largest independent set in a graph is the Hoffman bound, which gives an upper bound on the largest independent set of a graph in terms of its eigenvalues. It is easily seen that the Hoffman bound is sharp on the tensor power of a graph whenever it is sharp for the original graph. In this paper, we introduce the related problem of upper-bounding independent sets in tensor powers of hypergraphs. We show that many of the prominent open problems in extremal combinatorics, such as the Turán problem for (hyper-)graphs, can be encoded as special cases of this problem. We also give a new generalization of the Hoffman bound for hypergraphs which is sharp for the tensor power of a hypergraph whenever it is sharp for the original hypergraph. As an application of our Hoffman bound, we make progress on the problem of Frankl on families of sets without extended triangles from 1990. We show that if $\frac{1}{2}n\le2k\le\frac{2}{3}n,$ then the extremal family is the star, i.e. the family of all sets that contains a given element. This covers the entire range in which the star is extremal. As another application, we provide spectral proofs for Mantel's theorem on triangle-free graphs and for Frankl-Tokushige theorem on $k$-wise intersecting families.

math.CO↗

arXiv2019

AND Testing and Robust Judgement Aggregation

A function $f\colon\{0,1\}^n\to \{0,1\}$ is called an approximate AND-homomorphism if choosing ${\bf x},{\bf y}\in\{0,1\}^n$ randomly, we have that $f({\bf x}\land {\bf y}) = f({\bf x})\land f({\bf y})$ with probability at least $1-ε$, where $x\land y = (x_1\land y_1,\ldots,x_n\land y_n)$. We prove that if $f\colon \{0,1\}^n \to \{0,1\}$ is an approximate AND-homomorphism, then $f$ is $δ$-close to either a constant function or an AND function, where $δ(ε) \to 0$ as $ε\to0$. This improves on a result of Nehama, who proved a similar statement in which $δ$ depends on $n$. Our theorem implies a strong result on judgement aggregation in computational social choice. In the language of social choice, our result shows that if $f$ is $ε$-close to satisfying judgement aggregation, then it is $δ(ε)$-close to an oligarchy (the name for the AND function in social choice theory). This improves on Nehama's result, in which $δ$ decays polynomially with $n$. Our result follows from a more general one, in which we characterize approximate solutions to the eigenvalue equation $\mathrm T f = λg$, where $\mathrm T$ is the downwards noise operator $\mathrm T f(x) = \mathbb{E}_{\bf y}[f(x \land {\bf y})]$, $f$ is $[0,1]$-valued, and $g$ is $\{0,1\}$-valued. We identify all exact solutions to this equation, and show that any approximate solution in which $\mathrm T f$ and $λg$ are close is close to an exact solution.

cs.DM↗

arXiv2019

On the Bhattacharya-Mesner rank of third order hypermatrices

We introduce the Bhattacharya-Mesner rank of third order hypermatrices as a relaxation to the tensor rank and devise from it some bounds for the tensor rank. We use the Bhattacharya-Mesner rank to extend to third order hypermatrices the connection relating the rank to a notion of linear dependence. We also derive explicit necessary and sufficient conditions for the existence of third order hypermatrix inverse pair. Finally we use inverse pair to extend to third order hypermatrices the formulation and proof of the matrix rank-nullity theorem.

math.CO↗

arXiv2019

Biasing Boolean Functions and Collective Coin-Flipping Protocols over Arbitrary Product Distributions

The seminal result of Kahn, Kalai and Linial shows that a coalition of $O(\frac{n}{\log n})$ players can bias the outcome of any Boolean function $\{0,1\}^n \to \{0,1\}$ with respect to the uniform measure. We extend their result to arbitrary product measures on $\{0,1\}^n$, by combining their argument with a completely different argument that handles very biased coordinates. We view this result as a step towards proving a conjecture of Friedgut, which states that Boolean functions on the continuous cube $[0,1]^n$ (or, equivalently, on $\{1,\dots,n\}^n$) can be biased using coalitions of $o(n)$ players. This is the first step taken in this direction since Friedgut proposed the conjecture in 2004. Russell, Saks and Zuckerman extended the result of Kahn, Kalai and Linial to multi-round protocols, showing that when the number of rounds is $o(\log^* n)$, a coalition of $o(n)$ players can bias the outcome with respect to the uniform measure. We extend this result as well to arbitrary product measures on $\{0,1\}^n$. The argument of Russell et al. relies on the fact that a coalition of $o(n)$ players can boost the expectation of any Boolean function from $ε$ to $1-ε$ with respect to the uniform measure. This fails for general product distributions, as the example of the AND function with respect to $μ_{1-1/n}$ shows. Instead, we use a novel boosting argument alongside a generalization of our first result to arbitrary finite ranges.

cs.DM↗

arXiv2019

Harmonicity and invariance on slices of the Boolean cube

In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree functions. Here we provide an alternative proof for general low-degree functions, with no constraints on the influences. We show that any real-valued function on the slice, whose degree when written as a harmonic multi-linear polynomial is $o(\sqrt{n})$, has approximately the same distribution under the slice and cube measure. Our proof is based on a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, ours decomposition is applied in a non-reversible non-stationary setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof. Finally, we provide independent simple proofs for the facts that 1) one cannot distinguish between the slice and the cube based on functions of $o(n)$ coordinates and 2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is $o(\sqrt{n})$.

math.PR↗

arXiv2018

Online Submodular Maximization: Beating 1/2 Made Simple

The Submodular Welfare Maximization problem (SWM) captures an important subclass of combinatorial auctions and has been studied extensively from both computational and economic perspectives. In particular, it has been studied in a natural online setting in which items arrive one-by-one and should be allocated irrevocably upon arrival. In this setting, it is well known that the greedy algorithm achieves a competitive ratio of 1/2, and recently Kapralov et al. (2013) showed that this ratio is optimal for the problem. Surprisingly, despite this impossibility result, Korula et al. (2015) were able to show that the same algorithm is 0.5052-competitive when the items arrive in a uniformly random order, but unfortunately, their proof is very long and involved. In this work, we present an (arguably) much simpler analysis that provides a slightly better guarantee of 0.5096-competitiveness for the greedy algorithm in the random-arrival model. Moreover, this analysis applies also to a generalization of online SWM in which the sets defining a (simple) partition matroid arrive online in a uniformly random order, and we would like to maximize a monotone submodular function subject to this matroid. Furthermore, for this more general problem, we prove an upper bound of 0.576 on the competitive ratio of the greedy algorithm, ruling out the possibility that the competitiveness of this natural algorithm matches the optimal offline approximation ratio of 1-1/e.

cs.DS↗

arXiv2018

The entropy of lies: playing twenty questions with a liar

`Twenty questions' is a guessing game played by two players: Bob thinks of an integer between $1$ and $n$, and Alice's goal is to recover it using a minimal number of Yes/No questions. Shannon's entropy has a natural interpretation in this context. It characterizes the average number of questions used by an optimal strategy in the distributional variant of the game: let $μ$ be a distribution over $[n]$, then the average number of questions used by an optimal strategy that recovers $x\sim μ$ is between $H(μ)$ and $H(μ)+1$. We consider an extension of this game where at most $k$ questions can be answered falsely. We extend the classical result by showing that an optimal strategy uses roughly $H(μ) + k H_2(μ)$ questions, where $H_2(μ) = \sum_x μ(x)\log\log\frac{1}{μ(x)}$. This also generalizes a result by Rivest et al. for the uniform distribution. Moreover, we design near optimal strategies that only use comparison queries of the form `$x \leq c$?' for $c\in[n]$. The usage of comparison queries lends itself naturally to the context of sorting, where we derive sorting algorithms in the presence of adversarial noise.

cs.DS↗

arXiv2018

A log-Sobolev inequality for the multislice, with applications

Let $κ\in \mathbb{N}_+^\ell$ satisfy $κ_1 + \dots + κ_\ell = n$ and let $\mathcal{U}_κ$ denote the "multislice" of all strings $u$ in $[\ell]^n$ having exactly $κ_i$ coordinates equal to $i$, for all $i \in [\ell]$. Consider the Markov chain on $\mathcal{U}_κ$, where a step is a random transposition of two coordinates of $u$. We show that the log-Sobolev constant $ρ_κ$ for the chain satisfies $$(ρ_κ)^{-1} \leq n \sum_{i=1}^{\ell} \tfrac{1}{2} \log_2(4n/κ_i),$$ which is sharp up to constants whenever $\ell$ 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.

math.PR↗