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1. Improving the Smoothed Complexity of FLIP for Max Cut Problems

   https://doi.org/10.1145/3454125  

   Bibak, Ali ; Carlson, Charles ; Chandrasekaran, Karthekeyan ( August 2021 , ACM Transactions on Algorithms)

   Finding locally optimal solutions for MAX-CUT and MAX- k -CUT are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is ( O Φ 5 n 15.1 ), where Φ is an upper bound on the random edge-weight density and Φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress toward improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O (Φ n 7.83 ). Our results are based on a carefully chosen matrix whose rank captures themore »run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for MAX-3-CUT in complete graphs is polynomial and for MAX - k - CUT in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest toward showing smoothed polynomial complexity of FLIP for MAX - k - CUT in complete graphs for larger constants k .« less
2. Faster Matchings via Learned Duals

   Dinitz, Michael ; Im, Sungjin ; Lavastida, Thomas ; Moseley, Benjamin ; Vassilvitskii, Sergei ( January 2021 , Advances in neural information processing systems)

   A recent line of research investigates how algorithms can be augmented with machine-learned predictions to overcome worst case lower bounds. This area has revealed interesting algorithmic insights into problems, with particular success in the design of competitive online algorithms. However, the question of improving algorithm running times with predictions has largely been unexplored. We take a first step in this direction by combining the idea of machine-learned predictions with the idea of "warm-starting" primal-dual algorithms. We consider one of the most important primitives in combinatorial optimization: weighted bipartite matching and its generalization to b-matching. We identify three key challenges when using learned dual variables in a primal-dual algorithm. First, predicted duals may be infeasible, so we give an algorithm that efficiently maps predicted infeasible duals to nearby feasible solutions. Second, once the duals are feasible, they may not be optimal, so we show that they can be used to quickly find an optimal solution. Finally, such predictions are useful only if they can be learned, so we show that the problem of learning duals for matching has low sample complexity. We validate our theoretical findings through experiments on both real and synthetic data. As a result we give a rigorous,more »practical, and empirically effective method to compute bipartite matchings.« less
3. Can We Break Symmetry with o(m) Communication?

   https://doi.org/10.1145/3465084  

   Pai, Shreyas ; Pandurangan, Gopal ; Pemmaraju, Sriram ; Robinson, Peter ( July 2021 , PODC'21: Proceedings of the 2021 ACM Symposium on Principles of Distributed Computing)

   We study the communication cost (or message complexity) of fundamental distributed symmetry breaking problems, namely, coloring and MIS. While significant progress has been made in understanding and improving the running time of such problems, much less is known about the message complexity of these problems. In fact, all known algorithms need at least Ω(m) communication for these problems, where m is the number of edges in the graph. We addressthe following question in this paper: can we solve problems such as coloring and MIS using sublinear, i.e., o(m) communication, and if sounder what conditions? In a classical result, Awerbuch, Goldreich, Peleg, and Vainish [JACM 1990] showed that fundamental global problems such asbroadcast and spanning tree construction require at least o(m) messages in the KT-1 Congest model (i.e., Congest model in which nodes have initial knowledge of the neighbors' ID's) when algorithms are restricted to be comparison-based (i.e., algorithms inwhich node ID's can only be compared). Thirty five years after this result, King, Kutten, and Thorup [PODC 2015] showed that onecan solve the above problems using Õ(n) messages (n is the number of nodes in the graph) in Õ(n) rounds in the KT-1 Congest model if non-comparison-based algorithms are permitted. Anmore »important implication of this result is that one can use the synchronous nature of the KT-1 Congest model, using silence to convey information,and solve any graph problem using non-comparison-based algorithms with Õ(n) messages, but this takes an exponential number of rounds. In the asynchronous model, even this is not possible. In contrast, much less is known about the message complexity of local symmetry breaking problems such as coloring and MIS. Our paper fills this gap by presenting the following results. Lower bounds: In the KT-1 CONGEST model, we show that any comparison-based algorithm, even a randomized Monte Carlo algorithm with constant success probability, requires Ω(n 2) messages in the worst case to solve either (△ + 1)-coloring or MIS, regardless of the number of rounds. We also show that Ω(n) is a lower bound on the number ofmessages for any (△ + 1)-coloring or MIS algorithm, even non-comparison-based, and even with nodes having initial knowledge of up to a constant radius. Upper bounds: In the KT-1 CONGEST model, we present the following randomized non-comparison-based algorithms for coloring that, with high probability, use o(m) messages and run in polynomially many rounds.(a) A (△ + 1)-coloring algorithm that uses Õ(n1.5) messages, while running in Õ(D + √ n) rounds, where D is the graph diameter. Our result also implies an asynchronous algorithm for (△ + 1)-coloring with the same message bound but running in Õ(n) rounds. (b) For any constantε > 0, a (1+ε)△-coloring algorithm that uses Õ(n/ε 2 ) messages, while running in Õ(n) rounds. If we increase our input knowledge slightly to radius 2, i.e.,in the KT-2 CONGEST model, we obtain:(c) A randomized comparison-based MIS algorithm that uses Õ(n 1.5) messages. while running in Õ( √n) rounds. While our lower bound results can be viewed as counterparts to the classical result of Awerbuch, Goldreich, Peleg, and Vainish [JACM 90], but for local problems, our algorithms are the first-known algorithms for coloring and MIS that take o(m) messages and run in polynomially many rounds.« less
4. Dynamic Reliability Management in Neuromorphic Computing

   https://doi.org/10.1145/3462330  

   Song, Shihao ; Hanamshet, Jui ; Balaji, Adarsha ; Das, Anup ; Krichmar, Jeffrey L. ; Dutt, Nikil D. ; Kandasamy, Nagarajan ; Catthoor, Francky ( July 2021 , ACM Journal on Emerging Technologies in Computing Systems)

   Neuromorphic computing systems execute machine learning tasks designed with spiking neural networks. These systems are embracing non-volatile memory to implement high-density and low-energy synaptic storage. Elevated voltages and currents needed to operate non-volatile memories cause aging of CMOS-based transistors in each neuron and synapse circuit in the hardware, drifting the transistor’s parameters from their nominal values. If these circuits are used continuously for too long, the parameter drifts cannot be reversed, resulting in permanent degradation of circuit performance over time, eventually leading to hardware faults. Aggressive device scaling increases power density and temperature, which further accelerates the aging, challenging the reliable operation of neuromorphic systems. Existing reliability-oriented techniques periodically de-stress all neuron and synapse circuits in the hardware at fixed intervals, assuming worst-case operating conditions, without actually tracking their aging at run-time. To de-stress these circuits, normal operation must be interrupted, which introduces latency in spike generation and propagation, impacting the inter-spike interval and hence, performance (e.g., accuracy). We observe that in contrast to long-term aging, which permanently damages the hardware, short-term aging in scaled CMOS transistors is mostly due to bias temperature instability. The latter is heavily workload-dependent and, more importantly, partially reversible. We propose a new architectural techniquemore »to mitigate the aging-related reliability problems in neuromorphic systems by designing an intelligent run-time manager (NCRTM), which dynamically de-stresses neuron and synapse circuits in response to the short-term aging in their CMOS transistors during the execution of machine learning workloads, with the objective of meeting a reliability target. NCRTM de-stresses these circuits only when it is absolutely necessary to do so, otherwise reducing the performance impact by scheduling de-stress operations off the critical path. We evaluate NCRTM with state-of-the-art machine learning workloads on a neuromorphic hardware. Our results demonstrate that NCRTM significantly improves the reliability of neuromorphic hardware, with marginal impact on performance.« less
5. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

   https://doi.org/10.1007/s00224-022-10106-8  

   Vyas, Nikhil ; Williams, R. Ryan ( November 2022 , Theory of Computing Systems)

   We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[n^{{\left(\log n\right)}^{O\left(1\right)}}\right]$and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathcal { C}$$C$, by showing that$\mathcal { C}$$C$admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class${\mathcal C}$$C$. Say that a symmetric Boolean functionf(x₁,…,x_n) issparseif it outputs 1 onO(1) values of${\sum }_{i} x_{i}$${\sum }_i{x}_i$. We show that for every sparsef, and for all “typical”$\mathcal { C}$$C$, faster#SAT algorithms for$\mathcal { C}$$C$circuits imply lower bounds against the circuit class$f \circ \mathcal { C}$$f\circ C$, which may bestrongerthan$\mathcal { C}$$C$itself. In particular:

   #SAT algorithms forn^k-size$\mathcal { C}$$C$-circuits running in 2ⁿ/n^ktime (for allk) implyNEXPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

   #SAT algorithms for$2^{n^{{\varepsilon }}}$$2^{n^{\epsilon }}$-size$\mathcal { C}$$C$-circuits running in$2^{n-n^{{\varepsilon }}}$$2^{n-n^{\epsilon }}$time (for someε> 0) implyQuasi-NPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

   Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJ∘ACC⁰∘THRcircuits of polynomialmore »size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC⁰[Williams JACM’14] andACC⁰∘THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

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