arXiv subjects

Petteri Kaski

Publications and source records attributed to Petteri Kaski.

At least 19 recordsLinked to original sources

arXiv2026

Partition Rank and Algebraic Circuit Lower Bounds

Strassen's theory of bilinear complexity provides a mathematical characterization of the arithmetic complexity of primitives such as matrix multiplication via the rank of tensors. However, the connection to tensor rank is known to break down in higher degrees of multilinearity. In this work, we highlight an unexplored connection between a generalized notion of tensor rank, which can be defined in Naslund's framework of partition ranks (JCTA 2020), and multiplicative complexity. These partition ranks allow us to control the multiplicative complexity, and thus arithmetic complexity, in any constant degree of multilinearity from below, while recovering Strassen's seminal characterization in the bilinear case. This enables novel potential applications of the rank-based approaches to problems in fine-grained algorithms and complexity, such as the hyperclique conjecture of Lincoln-Williams-Vassilevska Williams (SODA 2018). Moreover, we exhibit connections to established notions of rank, such as tensor slice rank (in the sense of Tao and Sawin), as well as its symmetric variant. For computing the latter symmetric variant, we point out a simple NP-hardness proof, contrasting the rather involved NP-hardness proof for ordinary, non-symmetric tensor slice rank by Bl\"aser et al. (SODA 2021).

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arXiv2026

Optimal Union Probability Interval Is NP-Hard

A problem dating back to Boole [Laws of Thought, Walton & Maberly,1854] is what can be computed about the probability of a finite union of events when given as input the probabilities of intersections of some of the events. The modern geometric study of the problem can be traced back to Hailperin [Amer. Math. Monthly 2 (1965) 343--359] who phrased the problem in the language of linear programming and generalized it to logical formulas of the events other than disjunction, heralding a substantial body of work in probabilistic logic [Nilsson, Artif.\ Intell.\ 28 (1986) 71--87], including the probabilistic satisfiability problem of Georgakopoulos, Kavvadis, and Papadimitriou [J.Complexity 4 (1988) 1--11], as well as fundamental connections to the geometry of metrics via cut and correlation polytopes [Deza and Laurent, Geometry of Cuts and Metrics, Springer, 1997] and to the study of marginal polytopes in graphical models of machine learning [Wainwright and Jordan, Found.\ Trends Mach.\ Learn. 1 (2008) 1--305]. This paper (i) describes the pertinent geometry of Boole's problem via coordinate projections of an elementary polytope arising essentially from Hailperin's linear program on the atoms of a Venn diagram, and (ii) shows that computing the optimal interval for the union probability is NP-hard, resolving an apparent gap in the literature highlighted by Pitowsky [Math.\ Programming 50 (1991) 395--414] and Boros et al. [Math.\ Oper.\ Res. 39 (2014) 1311--1329 and 51 (2026) 134--148].

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arXiv2026

Beyond Bilinear Complexity: What Works and What Breaks with Many Modes?

The complexity of bilinear maps (equivalently, of $3$-mode tensors) has been studied extensively, most notably in the context of matrix multiplication. While circuit complexity and tensor rank coincide asymptotically for $3$-mode tensors, this correspondence breaks down for $d \geq 4$ modes. As a result, the complexity of $d$-mode tensors for larger fixed $d$ remains poorly understood, despite its relevance, e.g., in fine-grained complexity. Our paper explores this intermediate regime. First, we give a "graph-theoretic" proof of Strassen's $2\omega/3$ bound on the asymptotic rank exponent of $3$-mode tensors. Our proof directly generalizes to an upper bound of $(d-1)\omega/3$ for $d$-mode tensors. Using refined techniques available only for $d\geq 4$ modes, we improve this bound beyond the current state of the art for $\omega$. We also obtain a bound of $d/2+1$ on the asymptotic exponent of circuit complexity of generic $d$-mode tensors and optimized bounds for $d \in \{4,5\}$. To the best of our knowledge, asymptotic circuit complexity (rather than rank) of tensors has not been studied before. To obtain a robust theory, we first ask whether low complexity of $T$ and $U$ imply low complexity of their Kronecker product $T \otimes U$. While this crucially holds for rank (and thus for circuit complexity in $3$ modes), we show that assumptions from fine-grained complexity rule out such a submultiplicativity for the circuit complexity of tensors with many modes. In particular, assuming the Hyperclique Conjecture, this failure occurs already for $d=8$ modes. Nevertheless, we can salvage a restricted notion of submultiplicativity. From a technical perspective, our proofs heavily make use of the graph tensors $T_H$, as employed by Christandl and Zuiddam ({\em Comput.~Complexity}~28~(2019)~27--56) and [...]

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arXiv2025

Kronecker scaling of tensors with applications to arithmetic circuits and algorithms

We show that sufficiently low tensor rank for the balanced tripartitioning tensor $P_d(x,y,z)=\sum_{A,B,C\in\binom{[3d]}{d}:A\cup B\cup C=[3d]}x_Ay_Bz_C$ for a large enough constant $d$ implies uniform arithmetic circuits for the matrix permanent that are exponentially smaller than circuits obtainable from Ryser's formula. We show that the same low-rank assumption implies exponential time improvements over the state of the art for a wide variety of other related counting and decision problems. As our main methodological contribution, we show that the tensors $P_n$ have a desirable Kronecker scaling property: They can be decomposed efficiently into a small sum of restrictions of Kronecker powers of $P_d$ for constant $d$. We prove this with a new technique relying on Steinitz's lemma, which we hence call Steinitz balancing. As a consequence of our methods, we show that the mentioned low rank assumption (and hence the improved algorithms) is implied by Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)], a bold conjecture that has recently seen intriguing progress.

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arXiv2024

Fast Deterministic Chromatic Number under the Asymptotic Rank Conjecture

In this paper we further explore the recently discovered connection by Björklund and Kaski [STOC 2024] and Pratt [STOC 2024] between the asymptotic rank conjecture of Strassen [Progr. Math. 1994] and the three-way partitioning problem. We show that under the asymptotic rank conjecture, the chromatic number of an $n$-vertex graph can be computed deterministically in $O(1.99982^n)$ time, thus giving a conditional answer to a question of Zamir [ICALP 2021], and questioning the optimality of the $2^n\operatorname{poly}(n)$ time algorithm for chromatic number by Björklund, Husfeldt, and Koivisto [SICOMP 2009]. Viewed in the other direction, if chromatic number indeed requires deterministic algorithms to run in close to $2^n$ time, we obtain a sequence of explicit tensors of superlinear rank, falsifying the asymptotic rank conjecture. Our technique is a combination of earlier algorithms for detecting $k$-colorings for small $k$ and enumerating $k$-colorable subgraphs, with an extension and derandomisation of Pratt's tensor-based algorithm for balanced three-way partitioning to the unbalanced case.

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arXiv2024

A universal sequence of tensors for the asymptotic rank conjecture

The exponent $σ(T)$ of a tensor $T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d$ over a field $\mathbb{F}$ captures the base of the exponential growth rate of the tensor rank of $T$ under Kronecker powers. Tensor exponents are fundamental from the standpoint of algorithms and computational complexity theory; for example, the exponent $ω$ of matrix multiplication can be characterized as $ω=2σ(\mathrm{MM}_2)$, where $\mathrm{MM}_2\in\mathbb{F}^4\otimes\mathbb{F}^4\otimes\mathbb{F}^4$ is the tensor that represents $2\times 2$ matrix multiplication. Our main result is an explicit construction of a sequence $\mathcal{U}_d$ of zero-one-valued tensors that is universal for the worst-case tensor exponent; more precisely, we show that $σ(\mathcal{U}_d)=σ(d)$ where $σ(d)=\sup_{T\in\mathbb{F}^d\otimes\mathbb{F}^d\otimes\mathbb{F}^d}σ(T)$. We also supply an explicit universal sequence $\mathcal{U}_Δ$ localised to capture the worst-case exponent $σ(Δ)$ of tensors with support contained in $Δ\subseteq [d]\times[d]\times [d]$; by combining such sequences, we obtain a universal sequence $\mathcal{T}_d$ such that $σ(\mathcal{T}_d)=1$ holds if and only if Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] holds for $d$. Finally, we show that the limit $\lim_{d\rightarrow\infty}σ(d)$ exists and can be captured as $\lim_{d\rightarrow\infty} σ(D_d)$ for an explicit sequence $(D_d)_{d=1}^\infty$ of tensors obtained by diagonalisation of the sequences $\mathcal{U}_d$. As our second result we relate the absence of polynomials of fixed degree vanishing on tensors of low rank, or more generally asymptotic rank, with upper bounds on the exponent $σ(d)$. Using this technique, one may bound asymptotic rank for all tensors of a given format, knowing enough specific tensors of low asymptotic rank.

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arXiv2024

Another Hamiltonian Cycle in Bipartite Pfaffian Graphs

Finding a Hamiltonian cycle in a given graph is computationally challenging, and in general remains so even when one is further given one Hamiltonian cycle in the graph and asked to find another. In fact, no significantly faster algorithms are known for finding another Hamiltonian cycle than for finding a first one even in the setting where another Hamiltonian cycle is structurally guaranteed to exist, such as for odd-degree graphs. We identify a graph class -- the bipartite Pfaffian graphs of minimum degree three -- where it is NP-complete to decide whether a given graph in the class is Hamiltonian, but when presented with a Hamiltonian cycle as part of the input, another Hamiltonian cycle can be found efficiently. We prove that Thomason's lollipop method~[Ann.~Discrete Math.,~1978], a well-known algorithm for finding another Hamiltonian cycle, runs in a linear number of steps in cubic bipartite Pfaffian graphs. This was conjectured for cubic bipartite planar graphs by Haddadan [MSc~thesis,~Waterloo,~2015]; in contrast, examples are known of both cubic bipartite graphs and cubic planar graphs where the lollipop method takes exponential time. Beyond the lollipop method, we address a slightly more general graph class and present two algorithms, one running in linear-time and one operating in logarithmic space, that take as input (i) a bipartite Pfaffian graph $G$ of minimum degree three, (ii) a Hamiltonian cycle $H$ in $G$, and (iii) an edge $e$ in $H$, and output at least three other Hamiltonian cycles through the edge $e$ in $G$. We also present further improved algorithms for finding optimal traveling salesperson tours and counting Hamiltonian cycles in bipartite planar graphs with running times that are not known to hold in general planar graphs.

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arXiv2023

The Asymptotic Rank Conjecture and the Set Cover Conjecture are not Both True

Strassen's asymptotic rank conjecture [Progr. Math. 120 (1994)] claims a strong submultiplicative upper bound on the rank of a three-tensor obtained as an iterated Kronecker product of a constant-size base tensor. The conjecture, if true, most notably would put square matrix multiplication in quadratic time. We note here that some more-or-less unexpected algorithmic results in the area of exponential-time algorithms would also follow. Specifically, we study the so-called set cover conjecture, which states that for any $ε>0$ there exists a positive integer constant $k$ such that no algorithm solves the $k$-Set Cover problem in worst-case time $\mathcal{O}((2-ε)^n|\mathcal F|\operatorname{poly}(n))$. The $k$-Set Cover problem asks, given as input an $n$-element universe $U$, a family $\mathcal F$ of size-at-most-$k$ subsets of $U$, and a positive integer $t$, whether there is a subfamily of at most $t$ sets in $\mathcal F$ whose union is $U$. The conjecture was formulated by Cygan et al. in the monograph Parameterized Algorithms [Springer, 2015] but was implicit as a hypothesis already in Cygan et al. [CCC 2012, ACM Trans. Algorithms 2016], there conjectured to follow from the Strong Exponential Time Hypothesis. We prove that if the asymptotic rank conjecture is true, then the set cover conjecture is false. Using a reduction by Krauthgamer and Trabelsi [STACS 2019], in this scenario we would also get a $\mathcal{O}((2-δ)^n)$-time randomized algorithm for some constant $δ>0$ for another well-studied problem for which no such algorithm is known, namely that of deciding whether a given $n$-vertex directed graph has a Hamiltonian cycle.

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arXiv2021

The Shortest Even Cycle Problem is Tractable

Given a directed graph, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2) 1999) and, independently, McCuaig (Electron. J. Combin. 2004; announced jointly at STOC 1997) gave an efficiently testable structural characterisation of even-cycle-free directed graphs. Methodologically, our algorithm relies on algebraic fingerprinting and randomized polynomial identity testing over a finite field, and uses a generating polynomial implicit in Vazirani and Yannakakis ( Discrete Appl. Math. 1989) that enumerates weighted cycle covers as a difference of a permanent and a determinant polynomial. The need to work with the permanent is where our main technical contribution occurs. We design a family of finite commutative rings of characteristic 4 that simultaneously (i) give a nondegenerate representation for the generating polynomial identity via the permanent and the determinant, (ii) support efficient permanent computations, and (iii) enable emulation of finite-field arithmetic in characteristic 2. Here our work is foreshadowed by that of Björklund and Husfeldt (SIAM J. Comput. 2019), who used a considerably less efficient ring design to obtain a polynomial-time algorithm for the shortest two disjoint paths problem. Building on work of Gilbert and Tarjan (Numer. Math. 1978) as well as Alon and Yuster (J. ACM 2013), we also show how ideas from the nested dissection technique for solving linear equation systems leads to faster algorithm designs when we have control on the separator structure of the input graph; for example, this happens when the input has bounded genus.

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arXiv2020

The Fine-Grained Complexity of Computing the Tutte Polynomial of a Linear Matroid

We show that computing the Tutte polynomial of a linear matroid of dimension $k$ on $k^{O(1)}$ points over a field of $k^{O(1)}$ elements requires $k^{Ω(k)}$ time unless the \#ETH---a counting extension of the Exponential Time Hypothesis of Impagliazzo and Paturi [CCC 1999] due to Dell {\em et al.} [ACM TALG 2014]---is false. This holds also for linear matroids that admit a representation where every point is associated to a vector with at most two nonzero coordinates. We also show that the same is true for computing the Tutte polynomial of a binary matroid of dimension $k$ on $k^{O(1)}$ points with at most three nonzero coordinates in each point's vector. This is in sharp contrast to computing the Tutte polynomial of a $k$-vertex graph (that is, the Tutte polynomial of a {\em graphic} matroid of dimension $k$---which is representable in dimension $k$ over the binary field so that every vector has two nonzero coordinates), which is known to be computable in $2^k k^{O(1)}$ time [Björklund {\em et al.}, FOCS 2008]. Our lower-bound proofs proceed via (i) a connection due to Crapo and Rota [1970] between the number of tuples of codewords of full support and the Tutte polynomial of the matroid associated with the code; (ii) an earlier-established \#ETH-hardness of counting the solutions to a bipartite $(d,2)$-CSP on $n$ vertices in $d^{o(n)}$ time; and (iii) new embeddings of such CSP instances as questions about codewords of full support in a linear code. We complement these lower bounds with two algorithm designs. The first design computes the Tutte polynomial of a linear matroid of dimension~$k$ on $k^{O(1)}$ points in $k^{O(k)}$ operations. The second design generalizes the Björklund~{\em et al.} algorithm and runs in $q^{k+1}k^{O(1)}$ time for linear matroids of dimension $k$ defined over the $q$-element field by $k^{O(1)}$ points with at most two nonzero coordinates each.

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arXiv2020

Counting Short Vector Pairs by Inner Product and Relations to the Permanent

Given as input two $n$-element sets $\mathcal A,\mathcal B\subseteq\{0,1\}^d$ with $d=c\log n\leq(\log n)^2/(\log\log n)^4$ and a target $t\in \{0,1,\ldots,d\}$, we show how to count the number of pairs $(x,y)\in \mathcal A\times \mathcal B$ with integer inner product $\langle x,y \rangle=t$ deterministically, in $n^2/2^{Ω\bigl(\!\sqrt{\log n\log \log n/(c\log^2 c)}\bigr)}$ time. This demonstrates that one can solve this problem in deterministic subquadratic time almost up to $\log^2 n$ dimensions, nearly matching the dimension bound of a subquadratic randomized detection algorithm of Alman and Williams [FOCS 2015]. We also show how to modify their randomized algorithm to count the pairs w.h.p., to obtain a fast randomized algorithm. Our deterministic algorithm builds on a novel technique of reconstructing a function from sum-aggregates by prime residues, which can be seen as an {\em additive} analog of the Chinese Remainder Theorem. As our second contribution, we relate the fine-grained complexity of the task of counting of vector pairs by inner product to the task of computing a zero-one matrix permanent over the integers.

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arXiv2019

Engineering Boolean Matrix Multiplication for Multiple-Accelerator Shared-Memory Architectures

We study the problem of multiplying two bit matrices with entries either over the Boolean algebra $(0,1,\vee,\wedge)$ or over the binary field $(0,1,+,\cdot)$. We engineer high-performance open-source algorithm implementations for contemporary multiple-accelerator shared-memory architectures, with the objective of time-and-energy-efficient scaling up to input sizes close to the available shared memory capacity. For example, given two terabinary-bit square matrices as input, our implementations compute the Boolean product in approximately 2100 seconds (1.0 Pbop/s at 3.3 pJ/bop for a total of 2.1 kWh/product) and the binary product in less than 950 seconds (2.4 effective Pbop/s at 1.5 effective pJ/bop for a total of 0.92 kWh/product) on an NVIDIA DGX-1 with power consumption at peak system power (3.5 kW). Our contributions are (a) for the binary product, we use alternative-basis techniques of Karstadt and Schwartz [SPAA '17] to design novel alternative-basis variants of Strassen's recurrence for $2\times 2$ block multiplication [Numer. Math. 13 (1969)] that have been optimized for both the number of additions and low working memory, (b) structuring the parallel block recurrences and the memory layout for coalescent and register-localized execution on accelerator hardware, (c) low-level engineering of the innermost block products for the specific target hardware, and (d) structuring the top-level shared-memory implementation to feed the accelerators with data and integrate the results for input and output sizes beyond the aggregate memory capacity of the available accelerators.

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arXiv2018

Tensor network complexity of multilinear maps

We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as $O(n^{ω+ε})$ time matrix multiplication, and in addition many other algorithms such as $O(n \log n)$ time discrete Fourier transform and $O^*(2^n)$ time for computing the permanent of a matrix. However tensor networks sometimes yield faster algorithms than those that follow from low-rank decompositions. For instance the fastest known $O(n^{(ω+ε)t})$ time algorithms for counting $3t$-cliques can be implemented with tensor networks, even though the underlying tensor has border rank $n^{3t}$ for all $t \ge 2$. For counting homomorphisms of a general pattern graph $P$ into a host graph on $n$ vertices we obtain an upper bound of $O(n^{(ω+ε)\operatorname{bw}(P)/2})$ where $\operatorname{bw}(P)$ is the branchwidth of $P$. This essentially matches the bound for counting cliques, and yields small improvements over previous algorithms for many choices of $P$. While powerful, the model still has limitations, and we are able to show a number of unconditional lower bounds for various multilinear maps, including: (a) an $Ω(n^{\operatorname{bw}(P)})$ time lower bound for counting homomorphisms from $P$ to an $n$-vertex graph, matching the upper bound if $ω= 2$. In particular for $P$ a $v$-clique this yields an $Ω(n^{\lceil 2v/3 \rceil})$ time lower bound for counting $v$-cliques, and for $P$ a $k$-uniform $v$-hyperclique we obtain an $Ω(n^v)$ time lower bound for $k \ge 3$, ruling out tensor networks as an approach to obtaining non-trivial algorithms for hyperclique counting and the Max-$3$-CSP problem. (b) an $Ω(2^{0.918n})$ time lower bound for the permanent of an $n \times n$ matrix.

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arXiv2018

An adaptive prefix-assignment technique for symmetry reduction

This paper presents a technique for symmetry reduction that adaptively assigns a prefix of variables in a system of constraints so that the generated prefix-assignments are pairwise nonisomorphic under the action of the symmetry group of the system. The technique is based on McKay's canonical extension framework [J.~Algorithms 26 (1998), no.~2, 306--324]. Among key features of the technique are (i) adaptability---the prefix sequence can be user-prescribed and truncated for compatibility with the group of symmetries; (ii) parallelizability---prefix-assignments can be processed in parallel independently of each other; (iii) versatility---the method is applicable whenever the group of symmetries can be concisely represented as the automorphism group of a vertex-colored graph; and (iv) implementability---the method can be implemented relying on a canonical labeling map for vertex-colored graphs as the only nontrivial subroutine. To demonstrate the practical applicability of our technique, we have prepared an experimental open-source implementation of the technique and carry out a set of experiments that demonstrate ability to reduce symmetry on hard instances. Furthermore, we demonstrate that the implementation effectively parallelizes to compute clusters with multiple nodes via a message-passing interface.

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arXiv2018

A faster subquadratic algorithm for finding outlier correlations

We study the problem of detecting outlier pairs of strongly correlated variables among a collection of $n$ variables with otherwise weak pairwise correlations. After normalization, this task amounts to the geometric task where we are given as input a set of $n$ vectors with unit Euclidean norm and dimension $d$, and for some constants $0<τ<ρ<1$, we are asked to find all the outlier pairs of vectors whose inner product is at least $ρ$ in absolute value, subject to the promise that all but at most $q$ pairs of vectors have inner product at most $τ$ in absolute value. Improving on an algorithm of G. Valiant [FOCS 2012; J. ACM 2015], we present a randomized algorithm that for Boolean inputs ($\{-1,1\}$-valued data normalized to unit Euclidean length) runs in time \[ \tilde O\bigl(n^{\max\,\{1-γ+M(Δγ,γ),\,M(1-γ,2Δγ)\}}+qdn^{2γ}\bigr)\,, \] where $0<γ<1$ is a constant tradeoff parameter and $M(μ,ν)$ is the exponent to multiply an $\lfloor n^μ\rfloor\times\lfloor n^ν\rfloor$ matrix with an $\lfloor n^ν\rfloor\times \lfloor n^μ\rfloor$ matrix and $Δ=1/(1-\log_τρ)$. As corollaries we obtain randomized algorithms that run in time \[ \tilde O\bigl(n^{\frac{2ω}{3-\log_τρ}}+qdn^{\frac{2(1-\log_τρ)}{3-\log_τρ}}\bigr) \] and in time \[ \tilde O\bigl(n^{\frac{4}{2+α(1-\log_τρ)}}+qdn^{\frac{2α(1-\log_τρ)}{2+α(1-\log_τρ)}}\bigr)\,, \] where $2\leqω<2.38$ is the exponent for square matrix multiplication and $0.3<α\leq 1$ is the exponent for rectangular matrix multiplication. The notation $\tilde O(\cdot)$ hides polylogarithmic factors in $n$ and $d$ whose degree may depend on $ρ$ and $τ$. We present further corollaries for the light bulb problem and for learning sparse Boolean functions.

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arXiv2017

Directed Hamiltonicity and Out-Branchings via Generalized Laplacians

We are motivated by a tantalizing open question in exact algorithms: can we detect whether an $n$-vertex directed graph $G$ has a Hamiltonian cycle in time significantly less than $2^n$? We present new randomized algorithms that improve upon several previous works: 1. We show that for any constant $0<λ<1$ and prime $p$ we can count the Hamiltonian cycles modulo $p^{\lfloor (1-λ)\frac{n}{3p}\rfloor}$ in expected time less than $c^n$ for a constant $c<2$ that depends only on $p$ and $λ$. Such an algorithm was previously known only for the case of counting modulo two [Björklund and Husfeldt, FOCS 2013]. 2. We show that we can detect a Hamiltonian cycle in $O^*(3^{n-α(G)})$ time and polynomial space, where $α(G)$ is the size of the maximum independent set in $G$. In particular, this yields an $O^*(3^{n/2})$ time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013]. Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an $O^*(2^k)$-time randomized algorithm for detecting out-branchings with at least $k$ internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Björklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed $k$-Leaf problem, based on a non-standard monomial detection problem.

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arXiv2016

Explicit correlation amplifiers for finding outlier correlations in deterministic subquadratic time

We derandomize G. Valiant's [J. ACM 62 (2015) Art. 13] subquadratic-time algorithm for finding outlier correlations in binary data. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant's randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders in Reingold, Vadhan, and Wigderson [Ann. of Math. 155 (2002) 157--187]. We say that a function $f:\{-1,1\}^d\rightarrow\{-1,1\}^D$ is a correlation amplifier with threshold $0\leqτ\leq 1$, error $γ\geq 1$, and strength $p$ an even positive integer if for all pairs of vectors $x,y\in\{-1,1\}^d$ it holds that (i) $|\langle x,y\rangle|<τd$ implies $|\langle f(x),f(y)\rangle|\leq(τγ)^pD$; and (ii) $|\langle x,y\rangle|\geqτd$ implies $\bigl(\frac{\langle x,y\rangle}{γd}\bigr)^pD \leq\langle f(x),f(y)\rangle\leq \bigl(\frac{γ\langle x,y\rangle}{d}\bigr)^pD$.

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arXiv2016

Sharper Upper Bounds for Unbalanced Uniquely Decodable Code Pairs

Two sets $A, B \subseteq \{0, 1\}^n$ form a Uniquely Decodable Code Pair (UDCP) if every pair $a \in A$, $b \in B$ yields a distinct sum $a+b$, where the addition is over $\mathbb{Z}^n$. We show that every UDCP $A, B$, with $|A| = 2^{(1-ε)n}$ and $|B| = 2^{βn}$, satisfies $β\leq 0.4228 +\sqrtε$. For sufficiently small $ε$, this bound significantly improves previous bounds by Urbanke and Li~[Information Theory Workshop '98] and Ordentlich and Shayevitz~[2014, arXiv:1412.8415], which upper bound $β$ by $0.4921$ and $0.4798$, respectively, as $ε$ approaches $0$.

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