Proceedings of the 36th Computational Complexity Conference最新文献

{"title":"On query-to-communication lifting for adversary bounds","authors":"Anurag Anshu, S. Ben-David, Srijita Kundu","doi":"10.4230/LIPIcs.CCC.2021.30","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.30","url":null,"abstract":"We investigate query-to-communication lifting theorems for models related to the quantum adversary bounds. Our results are as follows: 1. We show that the classical adversary bound lifts to a lower bound on randomized communication complexity with a constant-sized gadget. We also show that the classical adversary bound is a strictly stronger lower bound technique than the previously-lifted measure known as critical block sensitivity, making our lifting theorem one of the strongest lifting theorems for randomized communication complexity using a constant-sized gadget. 2. Turning to quantum models, we show a connection between lifting theorems for quantum adversary bounds and secure 2-party quantum computation in a certain \"honest-but-curious\" model. Under the assumption that such secure 2-party computation is impossible, we show that a simplified version of the positive-weight adversary bound lifts to a quantum communication lower bound using a constant-sized gadget. We also give an unconditional lifting theorem which lower bounds bounded-round quantum communication protocols. 3. Finally, we give some new results in query complexity. We show that the classical adversary and the positive-weight quantum adversary are quadratically related. We also show that the positive-weight quantum adversary is never larger than the square of the approximate degree. Both relations hold even for partial functions.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131473624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantum complexity of minimum cut","authors":"Simon Apers, Troy Lee","doi":"10.4230/LIPIcs.CCC.2021.28","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.28","url":null,"abstract":"The minimum cut problem in an undirected and weighted graph G is to find the minimum total weight of a set of edges whose removal disconnects G. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If G has n vertices and edge weights at least 1 and at most τ, we give a quantum algorithm to solve the minimum cut problem using [EQUATION] queries and time. Moreover, for every integer 1 ≤ τ ≤ n we give an example of a graph G with edge weights 1 and τ such that solving the minimum cut problem on G requires [EQUATION] queries to the adjacency matrix of G. These results contrast with the classical randomized case where Ω(n2) queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not. In the adjacency array model, when G has m edges the classical randomized complexity of the minimum cut problem is [EQUATION]. We show that the quantum query and time complexity are [EQUATION] and [EQUATION], respectively, where again the edge weights are between 1 and τ. For dense graphs we give lower bounds on the quantum query complexity of Ω(n3/2) for τ ≥ 1 and Ω(τn) for any 1 ≤ τ ≤ n. Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger's tree packing technique (STOC 1996).","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115833804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the cut dimension of a graph","authors":"Troy Lee, Tongyang Li, M. Santha, Shengyu Zhang","doi":"10.4230/LIPIcs.CCC.2021.15","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.15","url":null,"abstract":"Let G = (V, w) be a weighted undirected graph with m edges. The cut dimension of G is the dimension of the span of the characteristic vectors of the minimum cuts of G, viewed as vectors in {0,1}m. For every n ≥ 2 we show that the cut dimension of an n-vertex graph is at most 2n − 3, and construct graphs realizing this bound. The cut dimension was recently defined by Graur et al. [13], who show that the maximum cut dimension of an n-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on n-vertex graphs. For every n ≥ 2, Graur et al. exhibit a graph on n vertices with cut dimension at least 3n/2 − 2, giving the first lower bound larger than n on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of linear queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector x ∈ R(n2) and receives the answer wT x. Our results thus show a lower bound of 2n − 3 on the number of linear queries needed by a deterministic algorithm to solve minimum cut on n-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension. We further introduce a generalization of the cut dimension which we call the ℓ1-approximate cut dimension. The ℓ1-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on n = 3k + 1 vertices with ℓ1-approximate cut dimension 2n − 2, showing that it can be strictly larger than the cut dimension.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"51 12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134536335","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A lower bound for polynomial calculus with extension rule","authors":"Yaroslav Alekseev","doi":"10.4230/LIPIcs.CCC.2021.21","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.21","url":null,"abstract":"A major proof complexity problem is to prove a superpolynomial lower bound on the length of Frege proofs of arbitrary depth. A more general question is to prove an Extended Frege lower bound. Surprisingly, proving such bounds turns out to be much easier in the algebraic setting. In this paper, we study a proof system that can simulate Extended Frege: an extension of the Polynomial Calculus proof system where we can take a square root and introduce new variables that are equivalent to arbitrary depth algebraic circuits. We prove that an instance of the subset-sum principle, the binary value principle 1 + x1 + 2x2 + ... + 2n−1xn = 0 (BVPn), requires refutations of exponential bit size over Q in this system. Part and Tzameret [18] proved an exponential lower bound on the size of Res-Lin (Resolution over linear equations [22]) refutations of BVPn. We show that our system p-simulates Res-Lin and thus we get an alternative exponential lower bound for the size of Res-Lin refutations of BVPn.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130781663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A lower bound on determinantal complexity","authors":"Mrinal Kumar, Ben lee Volk","doi":"10.1007/s00037-022-00228-3","DOIUrl":"https://doi.org/10.1007/s00037-022-00228-3","url":null,"abstract":"","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130745412","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A direct product theorem for one-way quantum communication","authors":"Rahul Jain, Srijita Kundu","doi":"10.4230/LIPIcs.CCC.2021.27","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.27","url":null,"abstract":"We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f⊆X×Y×Z. For any 0 < ε < δ < 1/2 and any k ≥ 1, we show that [EQUATION] where Q1ε(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and fk denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for the quantum communication complexity of a general relation - direct sum theorems were previously known for one-way quantum protocols for general relations, while direct product theorems were only known for special cases. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao [24], and under anchored distributions due to Bavarian, Vidick and Yuen [4], as well as message compression for quantum protocols due to Jain, Radhakrishnan and Sen [29]. In particular, we show that a direct product theorem holds for the distributional one-way quantum communication complexity of f under any distribution q on X × Y that is anchored on one side, i.e., there exists a y* such that q(y*) is constant and q(xy*) = q(x) for all x. This allows us to show a direct product theorem for general distributions, since for any relation f and any distribution p on its inputs, we can define a modified relation [EQUATION] which has an anchored distribution q close to p, such that a protocol that fails with probability at most ε for [EQUATION] under q can be used to give a protocol that fails with probability at most ε + ζ for f under p. Our techniques also work for entangled non-local games which have input distributions anchored on any one side, i.e., either there exists a y* as previously specified, or there exists an x* such that q(x*) is constant and q(yx*) = q(y) for all y. In particular, we show that for any game G = (q,X×Y, A×B, V) where q is a distribution on X × Y anchored on any one side with constant anchoring probability, then [EQUATION] where ω* (G) represents the entangled value of the game G. This is a generalization of the result of [4], who proved a parallel repetition theorem for games anchored on both sides, i.e., where both a special x* and a special y* exist, and potentially a simplification of their proof.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132479204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fractional pseudorandom generators from any fourier level","authors":"Eshan Chattopadhyay, J. Gaitonde, Chin Ho Lee, Shachar Lovett, Abhishek Shetty","doi":"10.4230/LIPIcs.CCC.2021.10","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.10","url":null,"abstract":"We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay et al. [4, 6] that exploit L1 Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the k-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k. This interpolates previous works, which either require Fourier bounds on all levels [4], or have polynomial dependence on the error parameter in the seed length [6], and thus answers an open question in [6]. As an example, we show that for polynomial error, Fourier bounds on the first O(log n) levels is sufficient to recover the seed length in [4], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor's theorem and bounding the degree-k Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the level-k unsigned Fourier sum, which is potentially a much smaller quantity than the L1 notion in previous works. By generalizing a connection established in [5], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for F2 polynomials with seed length close to the state-of-the-art construction due to Viola [26].","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132197862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the complexity of evaluating highest weight vectors","authors":"M. Bläser, Julian Dörfler, Christian Ikenmeyer","doi":"10.4230/LIPIcs.CCC.2021.29","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.29","url":null,"abstract":"Geometric complexity theory (GCT) is an approach towards separating algebraic complexity classes through algebraic geometry and representation theory. Originally Mulmuley and Sohoni proposed (SIAM J Comput 2001, 2008) to use occurrence obstructions to prove Valiant's determinant vs permanent conjecture, but recently Bürgisser, Ikenmeyer, and Panova (Journal of the AMS 2019) proved this impossible. However, fundamental theorems of algebraic geometry and representation theory grant that every lower bound in GCT can be proved by the use of so-called highest weight vectors (HWVs). In the setting of interest in GCT (namely in the setting of polynomials) we prove the NP-hardness of the evaluation of HWVs in general, and we give efficient algorithms if the treewidth of the corresponding Young-tableau is small, where the point of evaluation is concisely encoded as a noncommutative algebraic branching program! In particular, this gives a large new class of separating functions that can be efficiently evaluated at points with low (border) Waring rank. As a structural side result we prove that border Waring rank is bounded from above by the ABP width complexity.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"119 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126869497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The (generalized) orthogonality dimension of (generalized) kneser graphs: bounds and applications","authors":"Alexander Golovnev, I. Haviv","doi":"10.4230/LIPIcs.CCC.2021.8","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.8","url":null,"abstract":"The orthogonality dimension of a graph G = (V, E) over a field F is the smallest integer t for which there exists an assignment of a vector uv ∈ Ft with ⟨uv, uv⟩ ≠ 0 to every vertex v ∈ V, such that ⟨uv, uv'⟩ = 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 NP-hard to decide whether the orthogonality dimension of an input graph over 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 n1−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.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124463423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Proceedings of the 36th Computational Complexity Conference","authors":"","doi":"10.5555/3488669","DOIUrl":"https://doi.org/10.5555/3488669","url":null,"abstract":"","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134156150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}