arXiv subjects

Alexander Golovnev

Publications and source records attributed to Alexander Golovnev.

At least 37 records · Page 2Linked to original sources

arXiv2021

Data Structures Meet Cryptography: 3SUM with Preprocessing

This paper shows several connections between data structure problems and cryptography against preprocessing attacks. Our results span data structure upper bounds, cryptographic applications, and data structure lower bounds, as summarized next. First, we apply Fiat--Naor inversion, a technique with cryptographic origins, to obtain a data structure upper bound. In particular, our technique yields a suite of algorithms with space $S$ and (online) time $T$ for a preprocessing version of the $N$-input 3SUM problem where $S^3\cdot T = \widetilde{O}(N^6)$. This disproves a strong conjecture (Goldstein et al., WADS 2017) that there is no data structure that solves this problem for $S=N^{2-δ}$ and $T = N^{1-δ}$ for any constant $δ>0$. Secondly, we show equivalence between lower bounds for a broad class of (static) data structure problems and one-way functions in the random oracle model that resist a very strong form of preprocessing attack. Concretely, given a random function $F: [N] \to [N]$ (accessed as an oracle) we show how to compile it into a function $G^F: [N^2] \to [N^2]$ which resists $S$-bit preprocessing attacks that run in query time $T$ where $ST=O(N^{2-\varepsilon})$ (assuming a corresponding data structure lower bound on 3SUM). In contrast, a classical result of Hellman tells us that $F$ itself can be more easily inverted, say with $N^{2/3}$-bit preprocessing in $N^{2/3}$ time. We also show that much stronger lower bounds follow from the hardness of kSUM. Our results can be equivalently interpreted as security against adversaries that are very non-uniform, or have large auxiliary input, or as security in the face of a powerfully backdoored random oracle. Thirdly, we give non-adaptive lower bounds for 3SUM and a range of geometric problems which match the best known lower bounds for static data structure problems.

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arXiv2021

The (Generalized) Orthogonality Dimension of (Generalized) Kneser Graphs: Bounds and Applications

The orthogonality dimension of a graph $G=(V,E)$ over a field $\mathbb{F}$ is the smallest integer $t$ for which there exists an assignment of a vector $u_v \in \mathbb{F}^t$ with $\langle u_v,u_v \rangle \neq 0$ to every vertex $v \in V$, such that $\langle u_v, u_{v'} \rangle = 0$ whenever $v$ and $v'$ are adjacent vertices in $G$. The study of the orthogonality dimension of graphs is motivated by various applications in information theory and in theoretical computer science. The contribution of the present work is two-fold. First, we prove that there exists a constant $c$ such that for every sufficiently large integer $t$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of an input graph over $\mathbb{R}$ is at most $t$ or at least $3t/2-c$. At the heart of the proof lies a geometric result, which might be of independent interest, on a generalization of the orthogonality dimension parameter for the family of Kneser graphs, analogously to a long-standing conjecture of Stahl (J. Comb. Theo. Ser. B, 1976). Second, we study the smallest possible orthogonality dimension over finite fields of the complement of graphs that do not contain certain fixed subgraphs. In particular, we provide an explicit construction of triangle-free $n$-vertex graphs whose complement has orthogonality dimension over the binary field at most $n^{1-δ}$ for some constant $δ>0$. Our results involve constructions from the family of generalized Kneser graphs and they are motivated by the rigidity approach to circuit lower bounds. We use them to answer a couple of questions raised by Codenotti, Pudlák, and Resta (Theor. Comput. Sci., 2000), and in particular, to disprove their Odd Alternating Cycle Conjecture over every finite field.

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arXiv2021

Optimal Streaming Approximations for all Boolean Max-2CSPs and Max-kSAT

We prove tight upper and lower bounds on approximation ratios of all Boolean Max-2CSP problems in the streaming model. Specifically, for every type of Max-2CSP problem, we give an explicit constant $α$, s.t. for any $ε>0$ (i) there is an $(α-ε)$-streaming approximation using space $O(\log{n})$; and (ii) any $(α+ε)$-streaming approximation requires space $Ω(\sqrt{n})$. This generalizes the celebrated work of [Kapralov, Khanna, Sudan SODA 2015; Kapralov, Krachun STOC 2019], who showed that the optimal approximation ratio for Max-CUT was $1/2$. Prior to this work, the problem of determining this ratio was open for all other Max-2CSPs. Our results are quite surprising for some specific Max-2CSPs. For the Max-DCUT problem, there was a gap between an upper bound of $1/2$ and a lower bound of $2/5$ [Guruswami, Velingker, Velusamy APPROX 2017]. We show that neither of these bounds is tight, and the optimal ratio for Max-DCUT is $4/9$. We also establish that the tight approximation for Max-2SAT is $\sqrt{2}/2$, and for Exact Max-2SAT it is $3/4$. As a byproduct, our result gives a separation between space-efficient approximations for Max-2SAT and Exact Max-2SAT. This is in sharp contrast to the setting of polynomial-time algorithms with polynomial space, where the two problems are known to be equally hard to approximate. Finally, we prove that the tight streaming approximation for \mksat{} is $\sqrt{2}/2$ for every $k\geq2$.

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arXiv2020

Circuit Depth Reductions

The best known size lower bounds against unrestricted circuits have remained around $3n$ for several decades. Moreover, the only known technique for proving lower bounds in this model, gate elimination, is inherently limited to proving lower bounds of less than $5n$. In this work, we propose a non-gate-elimination approach for obtaining circuit lower bounds, via certain depth-three lower bounds. We prove that every (unbounded-depth) circuit of size $s$ can be expressed as an OR of $2^{s/3.9}$ $16$-CNFs. For DeMorgan formulas, the best known size lower bounds have been stuck at around $n^{3-o(1)}$ for decades. Under a plausible hypothesis about probabilistic polynomials, we show that $n^{4-\varepsilon}$-size DeMorgan formulas have $2^{n^{1-Ω(\varepsilon)}}$-size depth-3 circuits which are approximate sums of $n^{1-Ω(\varepsilon)}$-degree polynomials over ${\mathbb F}_2$. While these structural results do not immediately lead to new lower bounds, they do suggest new avenues of attack on these longstanding lower bound problems. Our results complement the classical depth-$3$ reduction results of Valiant, which show that logarithmic-depth circuits of linear size can be computed by an OR of $2^{\varepsilon n}$ $n^δ$-CNFs, and slightly stronger results for series-parallel circuits. It is known that no purely graph-theoretic reduction could yield interesting depth-3 circuits from circuits of super-logarithmic depth. We overcome this limitation (for small-size circuits) by taking into account both the graph-theoretic and functional properties of circuits and formulas. We show that improvements of the following pseudorandom constructions imply new circuit lower bounds: dispersers for varieties, correlation with constant degree polynomials, matrix rigidity, and hardness for depth-$3$ circuits with constant bottom fan-in.

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arXiv2020

Collapsing Superstring Conjecture

In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits $2\frac{11}{23}$-approximation in polynomial time (Mucha, SODA'13). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS. We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph $G$. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph $G$ is the same for all solutions, but that $G$ can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm. While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3. We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation.

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arXiv2019

An Attack on the the Encryption Scheme of the Moscow Internet Voting System

The next Moscow City Duma elections will be held on September 8th with an option of Internet voting. Some source code of the voting system is posted online for public testing. Pierrick Gaudry recently showed that due to the relatively small length of the key, the encryption scheme could be easily broken. This issue has been fixed in the current version of the voting system. In this note we show that the new implementation of the ElGamal encryption system is not semantically secure. We also demonstrate how this newly found security vulnerability can be potentially used for counting the number of votes cast for a candidate.

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arXiv2019

Breaking the encryption scheme of the Moscow Internet voting system

In September 2019, voters for the election at the Parliament of the city of Moscow were allowed to use an Internet voting system. The source code of it had been made available for public testing. In this paper we show two successful attacks on the encryption scheme implemented in the voting system. Both attacks were sent to the developers of the system, and both issues had been fixed after that.The encryption used in this system is a variant of ElGamal over finite fields. In the first attack we show that the used key sizes are too small. We explain how to retrieve the private keys from the public keys in a matter of minutes with easily available resources.When this issue had been fixed and the new system had become available for testing, we discovered that the new implementation was not semantically secure. We demonstrate how this newly found security vulnerability can be used for counting the number of votes cast for a candidate.

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arXiv2019

On the computational complexity of the probabilistic label tree algorithms

Label tree-based algorithms are widely used to tackle multi-class and multi-label problems with a large number of labels. We focus on a particular subclass of these algorithms that use probabilistic classifiers in the tree nodes. Examples of such algorithms are hierarchical softmax (HSM), designed for multi-class classification, and probabilistic label trees (PLTs) that generalize HSM to multi-label problems. If the tree structure is given, learning of PLT can be solved with provable regret guaranties [Wydmuch et.al. 2018]. However, to find a tree structure that results in a PLT with a low training and prediction computational costs as well as low statistical error seems to be a very challenging problem, not well-understood yet. In this paper, we address the problem of finding a tree structure that has low computational cost. First, we show that finding a tree with optimal training cost is NP-complete, nevertheless there are some tractable special cases with either perfect approximation or exact solution that can be obtained in linear time in terms of the number of labels $m$. For the general case, we obtain $O(\log m)$ approximation in linear time too. Moreover, we prove an upper bound on the expected prediction cost expressed in terms of the expected training cost. We also show that under additional assumptions the prediction cost of a PLT is $O(\log m)$.

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arXiv2019

The information-theoretic value of unlabeled data in semi-supervised learning

We quantify the separation between the numbers of labeled examples required to learn in two settings: Settings with and without the knowledge of the distribution of the unlabeled data. More specifically, we prove a separation by $Θ(\log n)$ multiplicative factor for the class of projections over the Boolean hypercube of dimension $n$. We prove that there is no separation for the class of all functions on domain of any size. Learning with the knowledge of the distribution (a.k.a. fixed-distribution learning) can be viewed as an idealized scenario of semi-supervised learning where the number of unlabeled data points is so great that the unlabeled distribution is known exactly. For this reason, we call the separation the value of unlabeled data.

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arXiv2019

String Matching: Communication, Circuits, and Learning

String matching is the problem of deciding whether a given $n$-bit string contains a given $k$-bit pattern. We study the complexity of this problem in three settings. Communication complexity. For small $k$, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large $k$, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol. Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small $k$. Learning. We consider the problem of learning a hidden pattern of length at most $k$ relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem.

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arXiv2019

Static Data Structure Lower Bounds Imply Rigidity

We show that static data structure lower bounds in the group (linear) model imply semi-explicit lower bounds on matrix rigidity. In particular, we prove that an explicit lower bound of $t \geq ω(\log^2 n)$ on the cell-probe complexity of linear data structures in the group model, even against arbitrarily small linear space $(s= (1+\varepsilon)n)$, would already imply a semi-explicit ($\bf P^{NP}\rm$) construction of rigid matrices with significantly better parameters than the current state of art (Alon, Panigrahy and Yekhanin, 2009). Our results further assert that polynomial ($t\geq n^δ$) data structure lower bounds against near-optimal space, would imply super-linear circuit lower bounds for log-depth linear circuits (a four-decade open question). In the succinct space regime $(s=n+o(n))$, we show that any improvement on current cell-probe lower bounds in the linear model would also imply new rigidity bounds. Our results rely on a new connection between the "inner" and "outer" dimensions of a matrix (Paturi and Pudlak, 2006), and on a new reduction from worst-case to average-case rigidity, which is of independent interest.

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arXiv2017

On the Quantitative Hardness of CVP

$ \newcommand{\eps}{\varepsilon} \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\CVP}{\problem{CVP}} \newcommand{\SVP}{\problem{SVP}} \newcommand{\CVPP}{\problem{CVPP}} \newcommand{\ensuremath}[1]{#1} $For odd integers $p \geq 1$ (and $p = \infty$), we show that the Closest Vector Problem in the $\ell_p$ norm ($\CVP_p$) over rank $n$ lattices cannot be solved in $2^{(1-\eps) n}$ time for any constant $\eps > 0$ unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to "almost all" values of $p \geq 1$, not including the even integers. This comes tantalizingly close to settling the quantitative time complexity of the important special case of $\CVP_2$ (i.e., $\CVP$ in the Euclidean norm), for which a $2^{n +o(n)}$-time algorithm is known. In particular, our result applies for any $p = p(n) \neq 2$ that approaches $2$ as $n \to \infty$. We also show a similar SETH-hardness result for $\SVP_\infty$; hardness of approximating $\CVP_p$ to within some constant factor under the so-called Gap-ETH assumption; and other quantitative hardness results for $\CVP_p$ and $\CVPP_p$ for any $1 \leq p < \infty$ under different assumptions.

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arXiv2017

The Minrank of Random Graphs

The minrank of a graph $G$ is the minimum rank of a matrix $M$ that can be obtained from the adjacency matrix of $G$ by switching some ones to zeros (i.e., deleting edges) and then setting all diagonal entries to one. This quantity is closely related to the fundamental information-theoretic problems of (linear) index coding (Bar-Yossef et al., FOCS'06), network coding and distributed storage, and to Valiant's approach for proving superlinear circuit lower bounds (Valiant, Boolean Function Complexity '92). We prove tight bounds on the minrank of random Erdős-Rényi graphs $G(n,p)$ for all regimes of $p\in[0,1]$. In particular, for any constant $p$, we show that $\mathsf{minrk}(G) = Θ(n/\log n)$ with high probability, where $G$ is chosen from $G(n,p)$. This bound gives a near quadratic improvement over the previous best lower bound of $Ω(\sqrt{n})$ (Haviv and Langberg, ISIT'12), and partially settles an open problem raised by Lubetzky and Stav (FOCS '07). Our lower bound matches the well-known upper bound obtained by the "clique covering" solution, and settles the linear index coding problem for random graphs. Finally, our result suggests a new avenue of attack, via derandomization, on Valiant's approach for proving superlinear lower bounds for logarithmic-depth semilinear circuits.

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arXiv2016

Tight Lower Bounds on Graph Embedding Problems

We prove that unless the Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. We also show an exponential-time reduction from Graph Homomorphism to Subgraph Isomorphism. This rules out (subject to ETH) a possibility of $|V(H)|^{o(|V(H)|)}$-time algorithm deciding if graph $G$ is a subgraph of $H$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems. Moreover, as a consequence of our reductions conditional lower bounds follow for other related problems such as Locally Injective Homomorphism, Graph Minors, Topological Graph Minors, Minimum Distortion Embedding and Quadratic Assignment Problem.

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arXiv2015

Tight Bounds for Subgraph Isomorphism and Graph Homomorphism

We prove that unless Exponential Time Hypothesis (ETH) fails, deciding if there is a homomorphism from graph $G$ to graph $H$ cannot be done in time $|V(H)|^{o(|V(G)|)}$. Combined with the reduction of Cygan, Pachocki, and Socała, our result rules out (subject to ETH) a possibility of $|V(G)|^{o(|V(G)|)}$-time algorithm deciding if graph $H$ is a subgraph of $G$. For both problems our lower bounds asymptotically match the running time of brute-force algorithms trying all possible mappings of one graph into another. Thus, our work closes the gap in the known complexity of these fundamental problems.

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arXiv2015

Condensed Unpredictability

We consider the task of deriving a key with high HILL entropy from an unpredictable source. Previous to this work, the only known way to transform unpredictability into a key that was $\eps$ indistinguishable from having min-entropy was via pseudorandomness, for example by Goldreich-Levin (GL) hardcore bits. This approach has the inherent limitation that from a source with $k$ bits of unpredictability entropy one can derive a key of length (and thus HILL entropy) at most $k-2\log(1/ε)$ bits. In many settings, e.g. when dealing with biometric data, such a $2\log(1/ε)$ bit entropy loss in not an option. Our main technical contribution is a theorem that states that in the high entropy regime, unpredictability implies HILL entropy. The loss in circuit size in this argument is exponential in the entropy gap $d$. To overcome the above restriction, we investigate if it's possible to first "condense" unpredictability entropy and make the entropy gap small. We show that any source with $k$ bits of unpredictability can be condensed into a source of length $k$ with $k-3$ bits of unpredictability entropy. Our condenser simply "abuses" the GL construction and derives a $k$ bit key from a source with $k$ bits of unpredicatibily. The original GL theorem implies nothing when extracting that many bits, but we show that in this regime, GL still behaves like a "condenser" for unpredictability. This result comes with two caveats (1) the loss in circuit size is exponential in $k$ and (2) we require that the source we start with has \emph{no} HILL entropy (equivalently, one can efficiently check if a guess is correct). We leave it as an intriguing open problem to overcome these restrictions or to prove they're inherent.

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arXiv2015

Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given $n$-vertex graph $G$ can be mapped to the vertices of a given $h$-vertex graph $H$ such that each edge of $G$ is mapped to an edge of $H$. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the $2$-CSP problem. In this paper, we prove several lower bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main result is a lower bound $2^{Ω\left( \frac{n \log h}{\log \log h}\right)}$. This rules out the existence of a single-exponential algorithm and shows that the trivial upper bound $2^{{\mathcal O}(n\log{h})}$ is almost asymptotically tight. We also investigate what properties of graphs $G$ and $H$ make it difficult to solve HOM$(G,H)$. An easy observation is that an ${\mathcal O}(h^n)$ upper bound can be improved to ${\mathcal O}(h^{\operatorname{vc}(G)})$ where $\operatorname{vc}(G)$ is the minimum size of a vertex cover of $G$. The second lower bound $h^{Ω(\operatorname{vc}(G))}$ shows that the upper bound is asymptotically tight. As to the properties of the "right-hand side" graph $H$, it is known that HOM$(G,H)$ can be solved in time $(f(Δ(H)))^n$ and $(f(\operatorname{tw}(H)))^n$ where $Δ(H)$ is the maximum degree of $H$ and $\operatorname{tw}(H)$ is the treewidth of $H$. This gives single-exponential algorithms for graphs of bounded maximum degree or bounded treewidth. Since the chromatic number $χ(H)$ does not exceed $\operatorname{tw}(H)$ and $Δ(H)+1$, it is natural to ask whether similar upper bounds with respect to $χ(H)$ can be obtained. We provide a negative answer to this question by establishing a lower bound $(f(χ(H)))^n$ for any function $f$. We also observe that similar lower bounds can be obtained for locally injective homomorphisms.

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arXiv2014

Families with infants: a general approach to solve hard partition problems

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with $O^*((2-\varepsilon(d))^n)$ time and exponential space for graphs of average degree $d$. This improves the algorithm by Björklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in $O^*((2-\varepsilon(d))^n)$ time and polynomial space for graphs of average degree $d$. The previously known results of this kind is a polyspace algorithm by Björklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~$d$ we present an algorithm with running time $O^*((2-\varepsilon(d))^{n/2})$ and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.

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