Computational Complexity
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Showing new listings for Friday, 22 November 2024
- [1] arXiv:2411.14267 [pdf, other]
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Title: Truly Supercritical Trade-offs for Resolution, Cutting Planes, Monotone Circuits, and Weisfeiler-LemanComments: 47 pages, 7 figuresSubjects: Computational Complexity (cs.CC); Logic in Computer Science (cs.LO)
We exhibit supercritical trade-off for monotone circuits, showing that there are functions computable by small circuits for which any circuit must have depth super-linear or even super-polynomial in the number of variables, far exceeding the linear worst-case upper bound. We obtain similar trade-offs in proof complexity, where we establish the first size-depth trade-offs for cutting planes and resolution that are truly supercritical, i.e., in terms of formula size rather than number of variables, and we also show supercritical trade-offs between width and size for treelike resolution. Our results build on a new supercritical width-depth trade-off for resolution, obtained by refining and strengthening the compression scheme for the Cop-Robber game in [Grohe, Lichter, Neuen & Schweitzer 2023]. This yields robust supercritical trade-offs for dimension versus iteration number in the Weisfeiler-Leman algorithm, which also translate into trade-offs between number of variables and quantifier depth in first-order logic. Our other results follow from improved lifting theorems that might be of independent interest.
- [2] arXiv:2411.14268 [pdf, html, other]
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Title: Supercritical Tradeoffs for Monotone CircuitsSubjects: Computational Complexity (cs.CC)
We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size-depth tradeoff result for monotone circuits in the so-called supercritical regime. Our proof is based on an analogous result in proof complexity: We introduce a new family of unsatisfiable 3-CNF formulas (called bracket formulas) that admit resolution refutations of quasipolynomial size while any refutation of polynomial depth requires exponential size.
- [3] arXiv:2411.14276 [pdf, html, other]
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Title: A $k^{\frac{q}{q-2}}$ Lower Bound for Odd Query Locally Decodable Codes from Bipartite Kikuchi GraphsSubjects: Computational Complexity (cs.CC); Information Theory (cs.IT)
A code $C \colon \{0,1\}^k \to \{0,1\}^n$ is a $q$-query locally decodable code ($q$-LDC) if one can recover any chosen bit $b_i$ of the message $b \in \{0,1\}^k$ with good confidence by querying a corrupted string $\tilde{x}$ of the codeword $x = C(b)$ in at most $q$ coordinates. For $2$ queries, the Hadamard code is a $2$-LDC of length $n = 2^k$, and this code is in fact essentially optimal. For $q \geq 3$, there is a large gap in our understanding: the best constructions achieve $n = \exp(k^{o(1)})$, while prior to the recent work of [AGKM23], the best lower bounds were $n \geq \tilde{\Omega}(k^{\frac{q}{q-2}})$ for $q$ even and $n \geq \tilde{\Omega}(k^{\frac{q+1}{q-1}})$ for $q$ odd.
The recent work of [AGKM23] used spectral methods to prove a lower bound of $n \geq \tilde{\Omega}(k^3)$ for $q = 3$, thus achieving the "$k^{\frac{q}{q-2}}$ bound" for an odd value of $q$. However, their proof does not extend to any odd $q \geq 5$. In this paper, we prove a $q$-LDC lower bound of $n \geq \tilde{\Omega}(k^{\frac{q}{q-2}})$ for any odd $q$. Our key technical idea is the use of an imbalanced bipartite Kikuchi graph, which gives a simpler method to analyze spectral refutations of odd arity XOR without using the standard "Cauchy-Schwarz trick", a trick that typically produces random matrices with correlated entries and makes the analysis for odd arity XOR significantly more complicated than even arity XOR. - [4] arXiv:2411.14314 [pdf, html, other]
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Title: Switching Graph Matrix Norm Bounds: from i.i.d. to Random Regular GraphsSubjects: Computational Complexity (cs.CC)
In this work, we give novel spectral norm bounds for graph matrix on inputs being random regular graphs. Graph matrix is a family of random matrices with entries given by polynomial functions of the underlying input. These matrices have been known to be the backbone for the analysis of various average-case algorithms and hardness. Previous investigations of such matrices are largely restricted to the \Erdos-\Renyi model, and tight matrix norm bounds on regular graphs are only known for specific examples. We unite these two lines of investigations, and give the first result departing from the \Erdos-\Renyi setting in the full generality of graph matrices. We believe our norm bound result would enable a simple transfer of spectral analysis for average-case algorithms and hardness between these two distributions of random graphs.
As an application of our spectral norm bounds, we show that higher-degree Sum-of-Squares lower bounds for the independent set problem on \Erdos-\Renyi random graphs can be switched into lower bounds on random $d$-regular graphs. Our result is the first to address the general open question of analyzing higher-degree Sum-of-Squares on random regular graphs. - [5] arXiv:2411.14361 [pdf, html, other]
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Title: Improved Lower Bounds for all Odd-Query Locally Decodable CodesSubjects: Computational Complexity (cs.CC); Combinatorics (math.CO)
We prove that for every odd $q\geq 3$, any $q$-query binary, possibly non-linear locally decodable code ($q$-LDC) $E:\{\pm1\}^k \rightarrow \{\pm1\}^n$ must satisfy $k \leq \tilde{O}(n^{1-2/q})$. For even $q$, this bound was established in a sequence of prior works. For $q=3$, the above bound was achieved in a recent work of Alrabiah, Guruswami, Kothari and Manohar using an argument that crucially exploits known exponential lower bounds for $2$-LDCs. Their strategy hits an inherent bottleneck for $q \geq 5$.
Our key insight is identifying a general sufficient condition on the hypergraph of local decoding sets called $t$-approximate strong regularity. This condition demands that 1) the number of hyperedges containing any given subset of vertices of size $t$ (i.e., its co-degree) be equal to the same but arbitrary value $d_t$ up to a multiplicative constant slack, and 2) all other co-degrees be upper-bounded relative to $d_t$. This condition significantly generalizes related proposals in prior works that demand absolute upper bounds on all co-degrees.
We give an argument based on spectral bounds on Kikuchi Matrices that lower bounds the blocklength of any LDC whose local decoding sets satisfy $t$-approximate strong regularity for any $t \leq q$. Crucially, unlike prior works, our argument works despite having no non-trivial absolute upper bound on the co-degrees of any set of vertices. To apply our argument to arbitrary $q$-LDCs, we give a new, greedy, approximate strong regularity decomposition that shows that arbitrary, dense enough hypergraphs can be partitioned (up to a small error) into approximately strongly regular pieces satisfying the required relative bounds on the co-degrees. - [6] arXiv:2411.14431 [pdf, html, other]
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Title: On Optimal Testing of LinearityComments: To appear at SOSA 2025Subjects: Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS)
Linearity testing has been a focal problem in property testing of functions. We combine different known techniques and observations about linearity testing in order to resolve two recent versions of this task.
First, we focus on the online manipulations model introduced by Kalemaj, Raskhodnikova and Varma (ITCS 2022 \& Theory of Computing 2023). In this model, up to $t$ data entries are adversarially manipulated after each query is answered. Ben-Eliezer, Kelman, Meir, and Raskhodnikova (ITCS 2024) showed an asymptotically optimal linearity tester that is resilient to $t$ manipulations per query, but their approach fails if $t$ is too large. We extend this result, showing an optimal tester for almost any possible value of $t$. First, we simplify their result when $t$ is small, and for larger values of $t$ we instead use sample-based testers, as defined by Goldreich and Ron (ACM Transactions on Computation Theory 2016). A key observation is that sample-based testing is resilient to online manipulations, but still achieves optimal query complexity for linearity when $t$ is large. We complement our result by showing that when $t$ is \emph{very} large, any reasonable property, and in particular linearity, cannot be tested at all.
Second, we consider linearity over the reals with proximity parameter $\varepsilon$. Fleming and Yoshida (ITCS 2020) gave a tester using $O(1/\varepsilon\ \cdot log(1/\varepsilon))$ queries. We simplify their algorithms and modify the analysis accordingly, showing an optimal tester that only uses $O(1/\varepsilon)$ queries. This modification works for the low-degree testers presented in Arora, Bhattacharyya, Fleming, Kelman, and Yoshida (SODA 2023) as well, resulting in optimal testers for degree-$d$ polynomials, for any constant degree $d$.
New submissions (showing 6 of 6 entries)
- [7] arXiv:2411.14416 (cross-list from quant-ph) [pdf, html, other]
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Title: QMA vs. QCMA and PseudorandomnessSubjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC)
We study a longstanding question of Aaronson and Kuperberg on whether there exists a classical oracle separating $\mathsf{QMA}$ from $\mathsf{QCMA}$. Settling this question in either direction would yield insight into the power of quantum proofs over classical proofs. We show that such an oracle exists if a certain quantum pseudorandomness conjecture holds. Roughly speaking, the conjecture posits that quantum algorithms cannot, by making few queries, distinguish between the uniform distribution over permutations versus permutations drawn from so-called "dense" distributions.
Our result can be viewed as establishing a "win-win" scenario: \emph{either} there is a classical oracle separation of $\QMA$ from $\QCMA$, \emph{or} there is quantum advantage in distinguishing pseudorandom distributions on permutations.
Cross submissions (showing 1 of 1 entries)
- [8] arXiv:2402.17805 (replaced) [pdf, html, other]
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Title: Graph Neural Networks and Arithmetic CircuitsSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computational Complexity (cs.CC)
We characterize the computational power of neural networks that follow the graph neural network (GNN) architecture, not restricted to aggregate-combine GNNs or other particular types. We establish an exact correspondence between the expressivity of GNNs using diverse activation functions and arithmetic circuits over real numbers. In our results the activation function of the network becomes a gate type in the circuit. Our result holds for families of constant depth circuits and networks, both uniformly and non-uniformly, for all common activation functions.