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An approximate computation of a Boolean function by a circuit or switching network is a computation in which the function is computed correctly on the majority of the inputs (rather than on all inputs). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many sub areas of complexity theory, s...
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... G does not have a unique source or a unique sink then we can simply add one and connect it to all of the sources or sinks, which is illustrated in Figure 1. The problem GEN is monotone: if we have an instance of GEN given by a set of variables L, and L is an accepting input for GEN, then adding any variable l ∈ L to L will not make L a rejecting input. ...
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Citations
... Reversible pebblings of DAGs have been studied in [LV96,Krá04] and have been employed to shed light on time-space trade-offs in reversible simulation of irreversible computation in [LTV98,LMT00,Wil00,BTV01]. In a different line of work Potechin [Pot10] implicitly used the reversible pebble game for proving lower bounds on monotone space complexity, with the connection made explicit in the follow-up works [CP14,FPRC13]. ...
We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt '70] and the reversible version of this game in [Bennett '89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of eciding whether s~pebbles suffice to reversibly pebble a DAG G is PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan '13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and McKenzie '99], our results apply to the Dymond--Tompa and Raz--McKenzie games as well, and from the same paper it follows that resolution depth is PSPACE-hard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time.
... Reversible pebblings have been studied in [LV96,Krá04,KSS18] and have been used to prove timespace trade-offs in reversible simulation of irreversible computation in [LTV98, LMT00, Wil00, BTV01]. In a different context, Potechin [Pot10] implicitly used reversible pebbling to obtain lower bounds in monotone space complexity, with the connection made explicit in later works [CP14,FPRC13]. The paper [CLNV15] (to which this overview is indebted) studied the relative power of standard and reversible pebblings with respect to space, and also established PSPACE-hardness results for estimating the minimum space required to pebble graphs (reversibly or not). ...
We establish an exactly tight relation between reversible pebblings of graphs and Nullstellensatz refutations of pebbling formulas, showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula over G in size t+1 and degree s (independently of the field in which the Nullstellensatz refutation is made). We use this correspondence to prove a number of strong size-degree trade-offs for Nullstellensatz, which to the best of our knowledge are the first such results for this proof system.
... In a follow-up work, Chan and Potechin (Chan and Potechin 2014) generalized the techniques used here to the iterated indexing and k-clique problems, showing tight monotone lower space bounds, giving an alternate proof of the separation of the monotone NC-hierarchy. Filmus et al. (2013) later showed an average case lower bound on monotone switching networks for directed connectivity over some distribution of inputs. Both of these articles provide an alternate presentation of the results here. ...
... This limitation has been addressed in follow-up work. Filmus et al. (2013) showed an average case lower bound when we take a distribution over minimal YES instances and maximal NO instances. In Potechin (2013), we consider the monotone space complexity of solving directed connectivity on other input graphs. ...
We separate monotone analogues of L and NL by proving that any monotone switching network solving directed connectivity on n vertices must have size at least nΩ(lg n).
... This construction is inspired by discussions in[Rob13]. ...
We investigate monotone circuits with local oracles (K., 2016), i.e., circuits containing additional inputs that can perform unstructured computations on the input string . Let be the locality of the circuit, a parameter that bounds the combined strength of the oracle functions , and be the classical sets of positive and negative inputs considered for k-clique in the approximation method (Razborov, 1985). Our results can be informally stated as follows. 1. For an appropriate extension of depth-2 monotone circuits with local oracles, we show that the size of the smallest circuits separating (triangles) and (complete bipartite graphs) undergoes two phase transitions according to . 2. For , arbitrary depth, and , we prove that the monotone circuit size complexity of separating the sets and is , under a certain technical assumption on the local oracle gates. The second result, which concerns monotone circuits with restricted oracles, extends and provides a matching upper bound for the exponential lower bounds on the monotone circuit size complexity of k-clique obtained by Alon and Boppana (1987).
... In fact, we show that the above bounds hold even if the circuits are allowed to make some errors. In particular, we get a simple proof of an average-case hierarchy theorem within monotone P, similar to a recent result of Filmus et al. [FPRC13]. (Their result was proven using Fourier analytic techniques [Pot10,CP12].) ...
... The original bounds of [RM99] went up to Ω(N δ ) for a small constant δ. This was recently improved by the works [CP12,FPRC13] that prove (among other things) monotone depth bounds of up to Ω(N 1/6−o(1) ) for GEN G type functions. ...
... However, it seems that a precise connection in this direction has not been formalised before. Some related results are known: Filmus et al. [FPRC13] show that the converse of such a connection fails in a certain distributional sense. Raz and Wigderson [RW89] use randomised communication lower bounds for a different purpose, namely, to prove that every sufficiently shallow circuit for a particular function requires many negated inputs. ...
We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordström: if S is a search problem with critical block sensitivity b, then every randomised two-party protocol solving a certain two-party lift of S requires Ω(b) bits of communication. Besides simplicity, our proof has the advantage of generalising to the multi-party setting. We combine these results with new critical block sensitivity lower bounds for Tseitin and Pebbling search problems to obtain the following applications.
• Monotone circuit depth: We exhibit a monotone function on n variables whose monotone circuits require depth Ω(n/log n); previously, a bound of Ω(√n was known (Raz and Wigderson, JACM 1992). Moreover, we prove a tight Θ(√n) monotone depth bound for a function in monotone P. This implies an average-case hierarchy theorem within monotone P similar to a result of Filmus et al. (FOCS 2013).
• Proof complexity: We prove new rank lower bounds as well as obtain the first length--space lower bounds for semi-algebraic proof systems, including Lovász--Schrijver and Lasserre (SOS) systems. In particular, these results extend and simplify the works of Beame et al. (SICOMP 2007) and Huynh and Nordström.
... This paper does not assume prior knowledge of switching networks and the techniques used to analyze them. That said, the paper builds on intuition from previous work, so it is recommended that a reader who is learning about this approach for the first time read either [6], [3], or [4] before reading this paper. ...
In this paper, we extend the work of our previous paper "Bounds on monotone
switching networks for directed connectivity" by further analyzing the monotone
space complexity of directed connectivity. In particular, we analyze the
monotone space compexity of directed connectivity for a variety of input
graphs, not just minimal YES-instances. We show that the monotone space
complexity of input graphs with a very large number of lollipops (vertices v
for which there is an edge from s to v or an edge from v to t) is low
by giving a randomized monotone generalization of Savitch's algorithm which
uses a parity argument. This gives us upper bounds for both monotone switching
networks and monotone circuits. We then give lower bounds on monotone switching
networks for directed connectivity which are tight whenever the input graph is
acyclic and does not have any vertices v which are connected to a large
number of other vertices by short paths.
We establish an exactly tight relation between reversible
pebblings of graphs and Nullstellensatz refutations of pebbling formulas,
showing that a graph G can be reversibly pebbled in time t and space s if and only if there is a Nullstellensatz refutation of the pebbling formula
over G in size t + 1 and degree s (independently of the field in which
the Nullstellensatz refutation is made). We use this correspondence
to prove a number of strong size-degree trade-offs for Nullstellensatz,
which to the best of our knowledge are the first such results for this
proof system.
We study the complexity of approximating monotone Boolean functions with disjunctive normal form (DNF) formulas, exploring two main directions. First, we construct DNF approximators for arbitrary monotone functions achieving one-sided error: we show that every monotone f can be ε-approximated by a DNF g of size satisfying g(x) ≤ f(x) for all x ∈ {0,1}
n
. This is the first non-trivial universal upper bound even for DNF approximators incurring two-sided error.
Next, we study the power of negations in DNF approximators for monotone functions. We exhibit monotone functions for which non-monotone DNFs perform better than monotone ones, giving separations with respect to both DNF size and width. Our results, when taken together with a classical theorem of Quine [1], highlight an interesting contrast between approximation and exact computation in the DNF complexity of monotone functions, and they add to a line of work on the surprising role of negations in monotone complexity [2,3,4].
