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On Exponential-Time Hypotheses, Derandomization, and Circuit Lower Bounds [Extended Abstract]

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On Exponential-Time Hypotheses, Derandomization, and Circuit Lower Bounds [Extended Abstract] 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) On Exponential-Time Hypotheses, Derandomization, and Circuit Lower Bounds [Extended Abstract] Lijie Chen∗, Ron D. Rothblum†, Roei Tell‡, Eylon Yogev§ ∗Massachusetts Institute of Technology. Email: [email protected] †Technion. Email: [email protected] ‡Massachusetts Institute of Technology. Email: [email protected] §Boston University and Tel Aviv University. Email: [email protected] Abstract—The Exponential-Time Hypothesis (ETH) is a tures that 3-SAT with n variables and m = O(n) clauses strengthening of the P= NP conjecture, stating that 3-SAT ·n ·n cannot be deterministically solved in time less than 2 , for on n variables cannot be solved in (uniform) time 2 ,for > ETH > some constant = m/n 0. The may be viewed some 0. In recent years, analogous hypotheses that P NP are “exponentially-strong” forms of other classical complexity as an “exponentially-strong” version of = , since it conjectures (such as NP BPP or coNP NP) have also conjectures that a specific NP-complete problem requires been introduced, and have become widely influential. essentially exponential time to solve. In this work, we focus on the interaction of exponential-time Since the introduction of ETH many related variants, hypotheses with the fundamental and closely-related questions which are also “exponentially-strong” versions of classical of derandomization and circuit lower bounds. We show that even relatively-mild variants of exponential-time hypotheses have complexity-theoretic conjectures, have also been introduced. far-reaching implications to derandomization, circuit lower For example, the Randomized Exponential-Time Hypoth- bounds, and the connections between the two. Specifically, we esis (rETH), introduced in [4], conjectures that the same prove that: lower bound holds also for probabilistic algorithms (i.e., it is 1) The Randomized Exponential-Time Hypothesis a strong version of NP ⊆BPP). The Non-Deterministic (rETH) BPP implies that can be simulated on “average- Exponential-Time Hypothesis (NETH), introduced (im- case” in deterministic (nearly-)polynomial-time (i.e., in O˜ n n O(1) plicitly) in [5], conjectures that co-3SAT (with n variables time 2 (log( )) = nloglog( ) ). The derandomization and O(n) clauses) cannot be solved by non-deterministic relies on a conditional construction of a pseudorandom ·n generator with near-exponential stretch (i.e., with seed machines running in time 2 for some constant >0 length O˜(log(n))); this significantly improves the state- (i.e., it is a strong version of coNP ⊆NP). The variations of-the-art in uniform “hardness-to-randomness” results, MAETH and AMETH are defined analogously (see [6]1), which previously only yielded pseudorandom generators and other variations conjecture similar lower bounds for with sub-exponential stretch from such hypotheses. SAT 2) The Non-Deterministic Exponential-Time Hypothe- seemingly-harder problems (e.g., for #3 ; see [4]). sis (NETH) implies that derandomization of BPP is These Exponential-Time Hypotheses have been widely completely equivalent to circuit lower bounds against influential across different areas of complexity theory. E, and in particular that pseudorandom generators are Among the numerous fields to which they were applied necessary for derandomization. In fact, we show that the so far are structural complexity (i.e., showing classes of foregoing equivalence follows from a very weak version of NETH, and we also show that this very weak version problems that, conditioned on exponential-time hypotheses, is necessary to prove a slightly stronger conclusion that are “exponentially-hard”), parameterized complexity, com- we deduce from it. munication complexity, and fine-grained complexity; see, Lastly, we show that disproving certain exponential-time hy- e.g., the surveys [7]–[10]. potheses requires proving breakthrough circuit lower bounds. Exponential-time hypotheses focus on conjectured lower CircuitSAT n In particular, if for circuits over bits of size bounds for uniform algorithms. Two other fundamental poly(n) can be solved by probabilistic algorithms in time n/ n 2 polylog( ), then BPE does not have circuits of quasilinear questions in theoretical computer science are those of de- size. randomization, which refers to the power of probabilistic algorithms; and of circuit lower bounds, which refers to the power of non-uniform circuits. Despite the central place Keywords-computational complexity of all three questions, the interactions of exponential-time I. INTRODUCTION hypotheses with derandomization and circuit lower bounds have yet to be systematically studied. The Exponential-Time Hypothesis (ETH), introduced by Impagliazzo and Paturi [2] (and refined in [3]), conjec- 1In [6], the introduction of these variants is credited to a private communication from Carmosino, Gao, Impagliazzo, Mihajlin, Paturi, and A full version of this paper is available online at ECCC [1]. Schneider [5]. 2575-8454/20/$31.00 ©2020 IEEE 13 DOI 10.1109/FOCS46700.2020.00010 A. Our results: Bird’s eye against E. See Section I-C for more details. In this work we focus on the interactions between Lastly, in Section I-D we show that disproving a conjec- exponential-time hypotheses, derandomization, and circuit ture similar to rETH requires proving breakthrough circuit lower bounds. In a nutshell, our main contribution is show- lower bounds. Specifically, we show that if there exists a ing that even relatively-mild variants of exponential-time probabilistic algorithm that solves CircuitSAT for circuits n/ n hypotheses have far-reaching consequences on derandom- with n input bits and of size poly(n) in time 2 polylog( ), ization and circuit lower bounds. then non-uniform circuits of quasilinear size cannot decide def O n Let us now give a brief overview of our specific results, BPE == BPT IME[2 ( )] (see Theorem I.6, and see the before describing them in more detail in Sections I-B, I-C, discussion in Section I-D for a comparison with the state- and I-D. Our two main results are the following: of-the-art). 1) We show that rETH implies a nearly-polynomial- Relation to Strong Exponential Time Hypotheses: time average-case derandomization of BPP. Specifi- The exponential-time hypotheses that we consider also have (1−)·n cally, assuming rETH,2 we construct a pseudorandom “strong” variants that conjecture a lower bound of 2 , > generator for uniform circuits with near-exponential where 0 is arbitrarily small, for solving a correspond- SAT coSAT SAT stretch (i.e., with seed length O˜(log(n))) and with ing problem (e.g., for solving , ,or# ; see, O(1) running time 2O˜(log(n)) = nloglog(n) , and de- e.g., [10]). We emphasize that in this paper we focus only duce that BPP can be decided, in average-case and on the “non-strong” variants that conjecture lower bounds of ·n > infinitely-often, by deterministic algorithms that run 2 for some 0; these are indeed significantly weaker O(1) in time nloglog(n) (see Theorem I.1). This signif- than their “strong” counterparts; in fact, some “strong” icantly improves the state-of-the-art in the long line variants of standard exponential-time hypotheses are simply of uniform “hardness-to-randomness” results, which known to be false (see [6]). previously only yielded pseudorandom generators with We mention that a recent work of Carmosino, Impagli- at most a sub-exponential stretch from worst-case azzo, and Sabin [11] studied the implications of hypotheses lower bounds for uniform probabilistic algorithms in fine-grained complexity on derandomization. These fine- BPT IME grained hypotheses are implied by the “strong” version of (i.e., for ; see Section I-B for details). We rSETH also extend this result to deduce an “almost-always” rETH (i.e., by ), but are not known to follow from derandomization of BPP from an “almost-always” the “non-strong” versions that we consider in this paper. We lower bound (see Theorem I.2), which again improves will refer again to their results in Section I-B. on the state-of-the-art. See Section I-B for details. B. rETH and pseudorandom generators for uniform circuits E 2) Circuit lower bounds against are well-known to The first hypothesis that we study is rETH, which (slightly yield pseudorandom generators for non-uniform cir- changing notation from above) asserts that probabilistic prBPP cuits that can be used to derandomize in the algorithms cannot decide if a given 3-SAT formula with v worst-case. An important open question is whether variables and O(v) clauses is satisfiable in time less than such lower bounds and pseudorandom generators are 2·v, for some constant >0. Note that such a formula can actually necessary for worst-case derandomization of be represented with n = O(v ·log(v)) bits, and therefore the prBPP NETH . We show that a very weak version of conjectured lower bound as a function of the input length is yields a positive answer to the foregoing question; 2·(n/ log(n)). specifically, to obtain a positive answer it suffices to Intuitively, using “hardness-to-randomness” results, we E assume that cannot be computed by small circuits expect that such a strong lower bound would imply a strong that are uniformly generated by a non-deterministic derandomization result. For context, recall that in non- 3 machine. In fact, loosely speaking, we show that this uniform hardness-to-randomness results (following [12]), NETH weak version of is both sufficient and necessary lower bounds for non-uniform circuits yield pseudorandom to show an equivalence between non-deterministic generators (PRGs) that “fool” non-uniform distinguishers. prBPP derandomization of and circuit lower bounds Moreover, these results “scale smoothly” such that lower bounds for larger circuits yield PRGs with longer stretch 2We will in fact consider a hypothesis that is weaker (qualitatively and quantitatively), and conjectures that the specific PSPACE-complete prob- (see [13] for an essentially optimal trade-off); at the extreme, lem Totally Quantified Boolean Formula (TQBF) cannot be solved in prob- if E is hard almost-always for exponential-sized circuits, then n/polylog(n) TQBF SAT abilistic time 2 .
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