NP-hardness of circuit minimization for multi-output functions

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions

Solving large-scale linear circuit problems via convex optimization

A broad class of problems in circuits, electromagnetics, and optics can be expressed as finding some parameters of a linear system with a specific type. This paper is concerned with studying this type of circuit using the available control techniques. It is shown that the underlying problem can be recast as a rank minimization problem that is NP-hard in general. In order to circumvent this difficulty, the circuit problem is slightly modified so that the resulting optimization becomes convex. This interesting result is achieved at the cost of complicating the structure of the circuit, which introduces a trade-off between the design simplicity and the implementation complexity. When it is strictly required to solve the original circuit problem, the elegant structure of the proposed rank minimization problem allows for employing a celebrated heuristic method to solve it efficiently

Cluster Before You Hallucinate: Approximating Node-Capacitated Network Design and Energy Efficient Routing

We consider circuit routing with an objective of minimizing energy, in a network of routers that are speed scalable and that may be shutdown when idle. We consider both multicast routing and unicast routing. It is known that this energy minimization problem can be reduced to a capacitated flow network design problem, where vertices have a common capacity but arbitrary costs, and the goal is to choose a minimum cost collection of vertices whose induced subgraph will support the specified flow requirements. For the multicast (single-sink) capacitated design problem we give a polynomial-time algorithm that is O(log^3n)-approximate with O(log^4 n) congestion. This translates back to a O(log ^(4{\alpha}+3) n)-approximation for the multicast energy-minimization routing problem, where {\alpha} is the polynomial exponent in the dynamic power used by a router. For the unicast (multicommodity) capacitated design problem we give a polynomial-time algorithm that is O(log^5 n)-approximate with O(log^12 n) congestion, which translates back to a O(log^(12{\alpha}+5) n)-approximation for the unicast energy-minimization routing problem.Comment: 22 pages (full version of STOC 2014 paper

Approaching MCSP from Above and Below: Hardness for a Conditional Variant and AC^0[p]

The Minimum Circuit Size Problem (MCSP) asks whether a given Boolean function has a circuit of at most a given size. MCSP has been studied for over a half-century and has deep connections throughout theoretical computer science including to cryptography, computational learning theory, and proof complexity. For example, we know (informally) that if MCSP is easy to compute, then most cryptography can be broken. Despite this cryptographic hardness connection and extensive research, we still know relatively little about the hardness of MCSP unconditionally. Indeed, until very recently it was unknown whether MCSP can be computed in AC^0[2] (Golovnev et al., ICALP 2019). Our main contribution in this paper is to formulate a new "oracle" variant of circuit complexity and prove that this problem is NP-complete under randomized reductions. In more detail, we define the Minimum Oracle Circuit Size Problem (MOCSP) that takes as input the truth table of a Boolean function f, a size threshold s, and the truth table of an oracle Boolean function O, and determines whether there is a circuit with O-oracle gates and at most s wires that computes f. We prove that MOCSP is NP-complete under randomized polynomial-time reductions. We also extend the recent AC^0[p] lower bound against MCSP by Golovnev et al. to a lower bound against the circuit minimization problem for depth-d formulas, (AC^0_d)-MCSP. We view this result as primarily a technical contribution. In particular, our proof takes a radically different approach from prior MCSP-related hardness results

Энергосберегающее кодирование частичных состояний параллельного автомата

The problem of a low-power assignment of the partial states of a parallel automaton is considered. A method to solve that problem is suggested that provides minimizing the number of memory elements in the implementing circuit of the automaton and minimization of their switching activity. The problem is reduced to nding a minimal weighted cover of a graph with its complete bipartite sub-graphs (bi-cliques)