34 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Improved Hardness of Approximating k-Clique under ETH

    Full text link
    In this paper, we prove that assuming the exponential time hypothesis (ETH), there is no f(k)nko(1/loglogk)f(k)\cdot n^{k^{o(1/\log\log k)}}-time algorithm that can decide whether an nn-vertex graph contains a clique of size kk or contains no clique of size k/2k/2, and no FPT algorithm can decide whether an input graph has a clique of size kk or no clique of size k/f(k)k/f(k), where f(k)f(k) is some function in k1o(1)k^{1-o(1)}. Our results significantly improve the previous works [Lin21, LRSW22]. The crux of our proof is a framework to construct gap-producing reductions for the kk-Clique problem. More precisely, we show that given an error-correcting code C:Σ1kΣ2kC:\Sigma_1^k\to\Sigma_2^{k'} that is locally testable and smooth locally decodable in the parallel setting, one can construct a reduction which on input a graph GG outputs a graph GG' in (k)O(1)nO(logΣ2/logΣ1)(k')^{O(1)}\cdot n^{O(\log|\Sigma_2|/\log|\Sigma_1|)} time such that: \bullet If GG has a clique of size kk, then GG' has a clique of size KK, where K=(k)O(1)K = (k')^{O(1)}. \bullet If GG has no clique of size kk, then GG' has no clique of size (1ε)K(1-\varepsilon)\cdot K for some constant ε(0,1)\varepsilon\in(0,1). We then construct such a code with k=kΘ(loglogk)k'=k^{\Theta(\log\log k)} and Σ2=Σ1k0.54|\Sigma_2|=|\Sigma_1|^{k^{0.54}}, establishing the hardness results above. Our code generalizes the derivative code [WY07] into the case with a super constant order of derivatives.Comment: 48 page

    On streaming approximation algorithms for constraint satisfaction problems

    Full text link
    In this thesis, we explore streaming algorithms for approximating constraint satisfaction problems (CSPs). The setup is roughly the following: A computer has limited memory space, sees a long "stream" of local constraints on a set of variables, and tries to estimate how many of the constraints may be simultaneously satisfied. The past ten years have seen a number of works in this area, and this thesis includes both expository material and novel contributions. Throughout, we emphasize connections to the broader theories of CSPs, approximability, and streaming models, and highlight interesting open problems. The first part of our thesis is expository: We present aspects of previous works that completely characterize the approximability of specific CSPs like Max-Cut and Max-Dicut with n\sqrt{n}-space streaming algorithm (on nn-variable instances), while characterizing the approximability of all CSPs in n\sqrt n space in the special case of "composable" (i.e., sketching) algorithms, and of a particular subclass of CSPs with linear-space streaming algorithms. In the second part of the thesis, we present two of our own joint works. We begin with a work with Madhu Sudan and Santhoshini Velusamy in which we prove linear-space streaming approximation-resistance for all ordering CSPs (OCSPs), which are "CSP-like" problems maximizing over sets of permutations. Next, we present joint work with Joanna Boyland, Michael Hwang, Tarun Prasad, and Santhoshini Velusamy in which we investigate the n\sqrt n-space streaming approximability of symmetric Boolean CSPs with negations. We give explicit n\sqrt n-space sketching approximability ratios for several families of CSPs, including Max-kkAND; develop simpler optimal sketching approximation algorithms for threshold predicates; and show that previous lower bounds fail to characterize the n\sqrt n-space streaming approximability of Max-33AND.Comment: Harvard College senior thesis; 119 pages plus references; abstract shortened for arXiv; formatted with Dissertate template (feel free to copy!); exposits papers arXiv:2105.01782 (APPROX 2021) and arXiv:2112.06319 (APPROX 2022

    Oblivious Algorithms for the Max-kAND Problem

    Get PDF

    On Sketching Approximations for Symmetric Boolean CSPs

    Get PDF

    Sketching Approximability of (Weak) Monarchy Predicates

    Get PDF
    We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every k ? 5, we show that CSPs where the underlying predicate is a pure monarchy function on k variables have no non-trivial sketching approximation algorithm in o(?n) space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by O(log(n)) space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously

    Noisy Boolean Hidden Matching with Applications

    Get PDF
    The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper of Gavinsky et al. [STOC\u2707], has played an important role in lower bounds for graph problems in the streaming model (e.g., subgraph counting, maximum matching, MAX-CUT, Schatten p-norm approximation). The BHM problem typically leads to ?(?n) space lower bounds for constant factor approximations, with the reductions generating graphs that consist of connected components of constant size. The related Boolean Hidden Hypermatching (BHH) problem provides ?(n^{1-1/t}) lower bounds for 1+O(1/t) approximation, for integers t ? 2. The corresponding reductions produce graphs with connected components of diameter about t, and essentially show that long range exploration is hard in the streaming model with an adversarial order of updates. In this paper we introduce a natural variant of the BHM problem, called noisy BHM (and its natural noisy BHH variant), that we use to obtain stronger than ?(?n) lower bounds for approximating a number of the aforementioned problems in graph streams when the input graphs consist only of components of diameter bounded by a fixed constant. We next introduce and study the graph classification problem, where the task is to test whether the input graph is isomorphic to a given graph. As a first step, we use the noisy BHM problem to show that the problem of classifying whether an underlying graph is isomorphic to a complete binary tree in insertion-only streams requires ?(n) space, which seems challenging to show using either BHM or BHH

    Streaming complexity of CSPs with randomly ordered constraints

    Full text link
    We initiate a study of the streaming complexity of constraint satisfaction problems (CSPs) when the constraints arrive in a random order. We show that there exists a CSP, namely Max-DICUT\textsf{Max-DICUT}, for which random ordering makes a provable difference. Whereas a 4/90.4454/9 \approx 0.445 approximation of DICUT\textsf{DICUT} requires Ω(n)\Omega(\sqrt{n}) space with adversarial ordering, we show that with random ordering of constraints there exists a 0.480.48-approximation algorithm that only needs O(logn)O(\log n) space. We also give new algorithms for Max-DICUT\textsf{Max-DICUT} in variants of the adversarial ordering setting. Specifically, we give a two-pass O(logn)O(\log n) space 0.480.48-approximation algorithm for general graphs and a single-pass O~(n)\tilde{O}(\sqrt{n}) space 0.480.48-approximation algorithm for bounded degree graphs. On the negative side, we prove that CSPs where the satisfying assignments of the constraints support a one-wise independent distribution require Ω(n)\Omega(\sqrt{n})-space for any non-trivial approximation, even when the constraints are randomly ordered. This was previously known only for adversarially ordered constraints. Extending the results to randomly ordered constraints requires switching the hard instances from a union of random matchings to simple Erd\"os-Renyi random (hyper)graphs and extending tools that can perform Fourier analysis on such instances. The only CSP to have been considered previously with random ordering is Max-CUT\textsf{Max-CUT} where the ordering is not known to change the approximability. Specifically it is known to be as hard to approximate with random ordering as with adversarial ordering, for o(n)o(\sqrt{n}) space algorithms. Our results show a richer variety of possibilities and motivate further study of CSPs with randomly ordered constraints

    Streaming beyond sketching for Maximum Directed Cut

    Full text link
    We give an O~(n)\widetilde{O}(\sqrt{n})-space single-pass 0.4830.483-approximation streaming algorithm for estimating the maximum directed cut size (Max-DICUT\textsf{Max-DICUT}) in a directed graph on nn vertices. This improves over an O(logn)O(\log n)-space 4/9<0.454/9 < 0.45 approximation algorithm due to Chou, Golovnev, Velusamy (FOCS 2020), which was known to be optimal for o(n)o(\sqrt{n})-space algorithms. Max-DICUT\textsf{Max-DICUT} is a special case of a constraint satisfaction problem (CSP). In this broader context, our work gives the first CSP for which algorithms with O~(n)\widetilde{O}(\sqrt{n}) space can provably outperform o(n)o(\sqrt{n})-space algorithms on general instances. Previously, this was shown in the restricted case of bounded-degree graphs in a previous work of the authors (SODA 2023). Prior to that work, the only algorithms for any CSP were based on generalizations of the O(logn)O(\log n)-space algorithm for Max-DICUT\textsf{Max-DICUT}, and were in particular so-called "sketching" algorithms. In this work, we demonstrate that more sophisticated streaming algorithms can outperform these algorithms even on general instances. Our algorithm constructs a "snapshot" of the graph and then applies a result of Feige and Jozeph (Algorithmica, 2015) to approximately estimate the Max-DICUT\textsf{Max-DICUT} value from this snapshot. Constructing this snapshot is easy for bounded-degree graphs and the main contribution of our work is to construct this snapshot in the general setting. This involves some delicate sampling methods as well as a host of "continuity" results on the Max-DICUT\textsf{Max-DICUT} behaviour in graphs.Comment: 57 pages, 2 figure

    Sketching Approximability of (Weak) Monarchy Predicates

    Get PDF
    We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In~particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first variable (the president) and the sum of the votes of all remaining variables. The pure version of this function is when the president can only be overruled by when all remaining variables agree. For every k5k \geq 5, we show that CSPs where the underlying predicate is a pure monarchy function on kk variables have no non-trivial sketching approximation algorithm in o(n)o(\sqrt{n}) space. We also show infinitely many weaker monarchy functions for which CSPs using such constraints are non-trivially approximable by O(log(n))O(\log(n)) space sketching algorithms. Moreover, we give the first example of sketching approximable asymmetric Boolean CSPs. Our results work within the framework of Chou, Golovnev, Sudan, and Velusamy (FOCS 2021) that characterizes the sketching approximability of all CSPs. Their framework can be applied naturally to get a computer-aided analysis of the approximability of any specific constraint satisfaction problem. The novelty of our work is in using their work to get an analysis that applies to infinitely many problems simultaneously
    corecore