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### Revision(s):

Revision #2 to TR19-011 | 22nd September 2019 11:10

#### Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error

Revision #2
Authors: Benny Applebaum, Eliran Kachlon
Accepted on: 22nd September 2019 11:10
Keywords:

Abstract:

Suppose that you wish to sample a random graph $G$ over $n$ vertices and $m$ edges conditioned on the event that $G$ does not contain a small'' $t$-size graph $H$ (e.g., clique) as a subgraph. Assuming that most such graphs are $H$-free, the problem can be solved by a simple rejected-sampling algorithm (that tests for $t$-cliques) with an expected running time of $n^{O(t)}$. Is it possible to solve the problem in running time that does not grow polynomially with $n^t$?

In this paper, we introduce the general problem of sampling a random looking'' graph $G$ with a given edge density that avoids some arbitrary predefined $t$-size subgraph $H$. As our main result, we show that the problem is solvable with respect to some specially crafted $k$-wise independent distribution over graphs. That is, we design a sampling algorithm for $k$-wise independent graphs that supports efficient testing for subgraph-freeness in time $f(t)\cdot n^c$ where $f$ is a function of $t$ and the constant $c$ in the exponent is independent of $t$. Our solution extends to the case where both $G$ and $H$ are $d$-uniform hypergraphs.

We use these algorithms to obtain the first probabilistic construction of constant-degree polynomially-unbalanced expander graphs whose failure probability is \emph{negligible} in $n$ (i.e., $n^{-\omega(1)}$). In particular, given constants $d>c$, we output a bipartite graph that has $n$ left nodes, $n^c$ right nodes with right-degree of $d$ so that any right set of size at most $n^{\Omega(1)}$ expands by factor of $\Omega(d)$. This result is extended to the setting of unique expansion as well.

We observe that such a negligible-error construction can be employed in many useful settings, and present applications in coding theory (batch codes and LDPC codes), pseudorandomness (low-bias generators and randomness extractors) and cryptography. Notably, we show that our constructions yield a collection of polynomial-stretch locally-computable cryptographic pseudorandom generators based on Goldreich's one-wayness assumption resolving a central open problem in parallel-cryptography (cf., Applebaum-Ishai-Kushilevitz, FOCS 2004; and Ishai-Kushilevitz-Ostrovsky-Sahai, STOC 2008).

Revision #1 to TR19-011 | 22nd September 2019 10:51

#### Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error

Revision #1
Authors: Benny Applebaum, Eliran Kachlon
Accepted on: 22nd September 2019 10:51
Keywords:

Abstract:

Suppose that you wish to sample a random graph $G$ over $n$ vertices and $m$ edges conditioned on the event that $G$ does not contain a small'' $t$-size graph $H$ (e.g., clique) as a subgraph.
Assuming that most such graphs are $H$-free, the problem can be solved by a
simple rejected-sampling algorithm (that tests for $t$-cliques) with an expected running time of $n^{O(t)}$.
Is it possible to solve the problem in running time that does not grow polynomially with $n^t$?

In this paper, we introduce the general problem of sampling a random looking'' graph $G$ with a given edge density that avoids some arbitrary predefined $t$-size subgraph $H$. As our main result, we show that the problem is solvable with respect to some specially crafted $k$-wise independent distribution over graphs. That is, we design a sampling algorithm for $k$-wise independent graphs that supports efficient testing for subgraph-freeness in time $f(t)\cdot n^c$ where $f$ is a function of $t$ and the constant $c$ in the exponent is independent of $t$. Our solution extends to the case where both $G$ and $H$ are $d$-uniform hypergraphs.

We use these algorithms to obtain the first probabilistic construction of constant-degree polynomially-unbalanced expander graphs whose failure probability is \emph{negligible} in $n$ (i.e., $n^{-\omega(1)}$). In particular, given constants $d>c$, we output a bipartite graph that has $n$ left nodes, $n^c$ right nodes with right-degree of $d$ so that any right set of size at most $n^{\Omega(1)}$ expands by factor of $\Omega(d)$. This result is extended to the setting of unique expansion as well.

We observe that such a negligible-error construction can be employed in many useful settings, and present applications in coding theory (batch codes and LDPC codes), pseudorandomness (low-bias generators and randomness extractors) and cryptography. Notably, we show that our constructions yield a collection of polynomial-stretch locally-computable cryptographic pseudorandom generators based on Goldreich's one-wayness assumption resolving a central open problem in parallel-cryptography (cf., Applebaum-Ishai-Kushilevitz, FOCS 2004; and Ishai-Kushilevitz-Ostrovsky-Sahai, STOC 2008).

### Paper:

TR19-011 | 27th January 2019 08:17

#### Sampling Graphs without Forbidden Subgraphs and Almost-Explicit Unbalanced Expanders

TR19-011
Authors: Benny Applebaum, Eliran Kachlon
Publication: 27th January 2019 19:16
Keywords:

Abstract:

We initiate the study of the following hypergraph sampling problem: Sample a $d$-uniform hypergraph over $n$ vertices and $m$ hyperedges from some pseudorandom distribution $\mathcal{G}$ conditioned on not having some small predefined $t$-size hypergraph $H$ as a subgraph. The algorithm should run in $\mathrm{poly}(n)$-time even when the size of the subgraph $H$ is super-constant.

We solve the problem by carefully designing a sampling algorithm for $k$-wise independent hypergraphs $\mathcal{G}$ that supports efficient testing for subgraph-freeness. We use our algorithm to obtain the first probabilistic construction of constant-degree polynomially-unbalanced expander graphs whose failure probability is negligible in $n$ (i.e., $n^{-\omega(1)}$). In particular, given constants $d>c$, we output a bipartite graph that has $n$ left nodes, $n^c$ right nodes with right-degree of $d$ so that any right set of size at most $n^{\Omega(1)}$ expands by factor of $\Omega(d)$. This result is extended to the setting of unique expansion as well.

We argue that such an almost-explicit'' construction can be employed in many useful settings, and present applications in coding theory (batch codes and LDPC codes), pseudorandomness (low-bias generators and randomness extractors) and cryptography. Notably, we show that our constructions yield a collection of polynomial-stretch locally-computable cryptographic pseudorandom generators based on Goldreich's one-wayness assumption resolving a long-standing open problem.

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