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Register a new userOn the complexity of detecting hazards
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Added by
Nitin Saurabh
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Detecting and eliminating logic hazards in Boolean circuits is a fundamental problem in logic circuit design. We show that there is no $O(3^{(1-epsilon)n} text{poly}(s))$ time algorithm, for any $epsilon > 0$, that detects logic hazards in Boolean circuits of size $s$ on $n$ variables under the assumption that the strong exponential time hypothesis is true. This lower bound holds even when the input circuits are restricted to be formulas of depth four. We also present a polynomial time algorithm for detecting $1$-hazards in DNF (or, $0$-hazards in CNF) formulas. Since $0$-hazards in DNF (or, $1$-hazards in CNF) formulas are easy to eliminate, this algorithm can be used to detect whether a given DNF or CNF formula has a hazard in practice.
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The method of partial derivatives is one of the most successful lower bound methods for arithmetic circuits. It uses as a complexity measure the dimension of the span of the partial derivatives of a polynomial. In this paper, we consider this complexity measure as a computational problem: for an input polynomial given as the sum of its nonzero monomials, what is the complexity of computing the dimension of its space of partial derivatives? We show that this problem is #P-hard and we ask whether it belongs to #P. We analyze the trace method, recently used in combinatorics and in algebraic complexity to lower bound the rank of certain matrices. We show that this method provides a polynomial-time computable lower bound on the dimension of the span of partial derivatives, and from this method we derive closed-form lower bounds. We leave as an open problem the existence of an approximation algorithm with reasonable performance guarantees.A slightly shorter version of this paper was presented at STACS17. In this new version we have corrected a typo in Section 4.1, and added a reference to Shitovs work on tensor rank.
Let $fsubseteq{0,1}^ntimesXi$ be a relation and $g:{0,1}^mto{0,1,*}$ be a promise function. This work investigates the randomised query complexity of the relation $fcirc g^nsubseteq{0,1}^{mcdot n}timesXi$, which can be viewed as one of the most general cases of composition in the query model (letting $g$ be a relation seems to result in a rather unnatural definition of $fcirc g^n$). We show that for every such $f$ and $g$, $$mathcal R(fcirc g^n) in Omega(mathcal R(f)cdotsqrt{mathcal R(g)}),$$ where $mathcal R$ denotes the randomised query complexity. On the other hand, we demonstrate a relation $f_0$ and a promise function $g_0$, such that $mathcal R(f_0)inTheta(sqrt n)$, $mathcal R(g_0)inTheta(n)$ and $mathcal R(f_0circ g_0^n)inTheta(n)$ $-$ that is, our composition statement is tight. To the best of our knowledge, there was no known composition theorem for the randomised query complexity of relations or promise functions (and for the special case of total functions our lower bound gives multiplicative improvement of $sqrt{log n}$).
This paper is aimed to investigate some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called {it minimum normalized cuts}/{it isoperimteric numbers} defined through taking minimum of the maximum or the mean of the normalized outgoing flows from a set of subdomains of vertices, where these subdomains constitute a {it partition}/{it subpartition}. Following the main result of [A. Daneshgar, {it et. al.}, {it On the isoperimetric spectrum of graphs and its approximations}, JCTB, (2010)], it is known that the isoperimetric number and the minimum normalized cut both can be described as ${0,1}$-optimization programs, where the latter one does {it not} admit a relaxation to the reals. We show that the decision problem for the case of taking $k$-partitions and the maximum (called the max normalized cut problem {rm NCP}$^M$) as well as the other two decision problems for the mean version (referred to as {rm IPP}$^m$ and {rm NCP}$^m$) are $NP$-complete problems. On the other hand, we show that the decision problem for the case of taking $k$-subpartitions and the maximum (called the max isoperimetric problem {rm IPP}$^M$) can be solved in {it linear time} for any weighted tree and any $k geq 2$. Based on this fact, we provide polynomial time $O(k)$-approximation algorithms for all differe
In this paper we study the family of freezing cellular automata (FCA) in the context of asynchronous updating schemes. A cellular automaton is called freezing if there exists an order of its states, and the transitions are only allowed to go from a lower to a higher state. A cellular automaton is asynchronous if at each time-step only one cell is updated. Given configuration, we say that a cell is unstable if there exists a sequential updating scheme that changes its state. In this context, we define the problem AsyncUnstability, which consists in deciding if a cell is unstable or not. In general AsyncUnstability is in NP, and we study in which cases we can solve the problem by a more efficient algorithm. We begin showing that AsyncUnstability is in NL for any one-dimensional FCA. Then we focus on the family of life-like freezing CA (LFCA), which is a family of two-dimensional two-state FCA that generalize the freezing version of the game of life, known as life without death. We study the complexity of AsyncUnstability for all LFCA in the triangular and square grids, showing that almost all of them can be solved in NC, except for one rule for which the problem is NP-complete.
We study the multiparty communication complexity of high dimensional permutations, in the Number On the Forehead (NOF) model. This model is due to Chandra, Furst and Lipton (CFL) who also gave a nontrivial protocol for the Exactly-n problem where three players receive integer inputs and need to decide if their inputs sum to a given integer $n$. There is a considerable body of literature dealing with the same problem, where $(mathbb{N},+)$ is replaced by some other abelian group. Our work can be viewed as a far-reaching extension of this line of work. We show that the known lower bounds for that group-theoretic problem apply to all high dimensional permutations. We introduce new proof techniques that appeal to recent advances in Additive Combinatorics and Ramsey theory. We reveal new and unexpected connections between the NOF communication complexity of high dimensional permutations and a variety of well known and thoroughly studied problems in combinatorics. Previous protocols for Exactly-n all rely on the construction of large sets of integers without a 3-term arithmetic progression. No direct algorithmic protocol was previously known for the problem, and we provide the first such algorithm. This suggests new ways to significantly improve the CFL protocol. Many new open questions are presented throughout.
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