scholarly journals Optimal Implementation of Watched Literals and More General Techniques

2013 ◽  
Vol 48 ◽  
pp. 231-252 ◽  
Author(s):  
I. P. Gent
Keyword(s):  
Constraint Satisfaction ◽  
General Framework ◽  
Search Tree ◽  
Constant Factor ◽  
Worst Case ◽  
Backtracking Search ◽  
Run Time ◽  
Unit Propagation ◽  
Implementation Technique ◽  
Optimal Implementation

I prove that an implementation technique for scanning lists in backtracking search algorithms is optimal. The result applies to a simple general framework, which I present: applications include watched literal unit propagation in SAT and a number of examples in constraint satisfaction. Techniques like watched literals are known to be highly space efficient and effective in practice. When implemented in the `circular' approach described here, these techniques also have optimal run time per branch in big-O terms when amortized across a search tree. This also applies when multiple list elements must be found. The constant factor overhead of the worst case is only 2. Replacing the existing non-optimal implementation of unit propagation in MiniSat speeds up propagation by 29%, though this is not enough to improve overall run time significantly.

scholarly journals Improving the Smoothed Complexity of FLIP for Max Cut Problems

2021 ◽  
Vol 17 (3) ◽  
pp. 1-38
Author(s):  
Ali Bibak ◽  
Charles Carlson ◽  
Karthekeyan Chandrasekaran
Keyword(s):  
General Framework ◽  
Edge Weight ◽  
Complete Graphs ◽  
Worst Case ◽  
Weight Density ◽  
Complete Problems ◽  
Substantial Progress ◽  
Run Time ◽  
Optimum Solution ◽  
Arbitrary Graphs

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 the 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 .

scholarly journals The Quest for Strong Inapproximability Results with Perfect Completeness

2021 ◽  
Vol 17 (3) ◽  
pp. 1-35
Author(s):  
Joshua Brakensiek ◽  
Venkatesan Guruswami
Keyword(s):  
Constraint Satisfaction ◽  
Absolute Constant ◽  
Independent Set ◽  
Constant Factor ◽  
Constraint Satisfaction Problems ◽  
Uniform Hypergraph ◽  
Semidefinite Programming Relaxation ◽  
Worst Case ◽  
Unique Games Conjecture ◽  
Unique Games

The Unique Games Conjecture has pinned down the approximability of all constraint satisfaction problems (CSPs), showing that a natural semidefinite programming relaxation offers the optimal worst-case approximation ratio for any CSP. This elegant picture, however, does not apply for CSP instances that are perfectly satisfiable, due to the imperfect completeness inherent in the Unique Games Conjecture. This work is motivated by the pursuit of a better understanding of the approximability of perfectly satisfiable instances of CSPs. We prove that an “almost Unique” version of Label Cover can be approximated within a constant factor on satisfiable instances. Our main conceptual contribution is the formulation of a (hypergraph) version of Label Cover that we call V Label Cover . Assuming a conjecture concerning the inapproximability of V Label Cover on perfectly satisfiable instances, we prove the following implications: • There is an absolute constant c 0 such that for k ≥ 3, given a satisfiable instance of Boolean k -CSP, it is hard to find an assignment satisfying more than c 0 k 2 /2 k fraction of the constraints. • Given a k -uniform hypergraph, k ≥ 2, for all ε > 0, it is hard to tell if it is q -strongly colorable or has no independent set with an ε fraction of vertices, where q =⌈ k +√ k -1/2⌉. • Given a k -uniform hypergraph, k ≥ 3, for all ε > 0, it is hard to tell if it is ( k -1)-rainbow colorable or has no independent set with an ε fraction of vertices.

scholarly journals Angle Bisector Algorithm and Modified Dynamic Programming Algorithm for Dubins Traveling Salesman Problem

2021 ◽  
Author(s):  
Eswara Venkata Kumar Dhulipala
Keyword(s):  
Euclidean Distance ◽  
Travelling Salesman Problem ◽  
Dynamic Programming Algorithm ◽  
Constant Factor ◽  
Programming Algorithm ◽  
Worst Case ◽  
Angle Bisector ◽  
Set Of Points ◽  
The Given ◽  
Tour Length

A Dubin's Travelling Salesman Problem (DTSP) of finding a minimum length tour through a given set of points is considered. DTSP has a Dubins vehicle, which is capable of moving only forward with constant speed. In this paper, first, a worst case upper bound is obtained on DTSP tour length by assuming DTSP tour sequence same as Euclidean Travelling Salesman Problem (ETSP) tour sequence. It is noted that, in the worst case, \emph{any algorithm that uses of ETSP tour sequence} is a constant factor approximation algorithm for DTSP. Next, two new algorithms are introduced, viz., Angle Bisector Algorithm (ABA) and Modified Dynamic Programming Algorithm (MDPA). In ABA, ETSP tour sequence is used as DTSP tour sequence and orientation angle at each point $i_k$ are calculated by using angle bisector of the relative angle formed between the rays $i_{k}i_{k-1}$ and $i_ki_{k+1}$. In MDPA, tour sequence and orientation angles are computed in an integrated manner. It is shown that the ABA and MDPA are constant factor approximation algorithms and ABA provides an improved upper bound as compared to Alternating Algorithm (AA) \cite{savla2008traveling}. Through numerical simulations, we show that ABA provides an improved tour length compared to AA, Single Vehicle Algorithm (SVA) \cite{rathinam2007resource} and Optimized Heading Algorithm (OHA) \cite{babel2020new,manyam2018tightly} when the Euclidean distance between any two points in the given set of points is at least $4\rho$ where $\rho$ is the minimum turning radius. The time complexity of ABA is comparable with AA and SVA and is better than OHA. Also we show that MDPA provides an improved tour length compared to AA and SVA and is comparable with OHA when there is no constraint on Euclidean distance between the points. In particular, ABA gives a tour length which is at most $4\%$ more than the ETSP tour length when the Euclidean distance between any two points in the given set of points is at least $4\rho$.

A General Correspondence Approach to Apparent Motion

Perception ◽  
1993 ◽  
Vol 22 (2) ◽  
pp. 185-192 ◽  
Author(s):  
Terry Caelli ◽  
Mark Manning ◽  
David Finlay
Keyword(s):  
Apparent Motion ◽  
Constraint Satisfaction ◽  
General Framework ◽  
Feature Weights ◽  
General Correspondence ◽  
Perceived Motion

A general framework is considered for how different features of image parts determine the perceived direction of apparent motion between these parts as a function of their internalized feature weights. It is shown how the compatibility and constraints between pairwise part correspondences also play important roles in the types of perceived motion between parts; this process is modelled via a multivariate constraint-satisfaction procedure.

Reliability-Aware Proactive Energy Management in Hard Real-Time Systems

2010 ◽  
Vol 1 (4) ◽  
pp. 1-11
Author(s):  
Satyakiran Munaga ◽  
Francky Catthoor
Keyword(s):  
Energy Management ◽  
Ad Hoc ◽  
Failure Criteria ◽  
Worst Case ◽  
Time Control ◽  
Run Time ◽  
Cost Penalty ◽  
Hard Real Time ◽  
The Cost ◽  
Time Systems

Advanced technologies such as sub-45nm CMOS and 3D integration are known to have more accelerated and increased number of reliability failure mechanisms. Classical reliability assessment methodology, which assumes ad-hoc failure criteria and worst-case for all influencing dynamic aspects, is no longer viable in these technologies. In this paper, the authors advocate that managing temperature and reliability at run-time is necessary to overcome this reliability-wall without incurring significant cost penalty. Nonlinear nature of modern systems, however, makes the run-time control very challenging. The authors suggest that full cost-consciousness requires a truly proactive controller that can efficiently manage system slack with future in perspective. This paper introduces the concept of “gas-pedal,” which enhances the effectiveness of the proactive controller in minimizing the cost without sacrificing the hard guarantees required by the constraints. Reliability-aware dynamic energy management of a processor running AVC motion compensation task is used as a motivational case study to illustrate the proposed concepts.

scholarly journals Power and Area Efficient Clock Stretching and Critical Path Reshaping for Error Resilience

2019 ◽  
Vol 9 (1) ◽  
pp. 5
Author(s):  
Mini Jayakrishnan ◽  
Alan Chang ◽  
Tony Tae-Hyoung Kim
Keyword(s):  
Critical Path ◽  
Error Resilience ◽  
Cost Effective ◽  
Advanced Technology ◽  
Worst Case ◽  
Flip Flop ◽  
Smart Products ◽  
Run Time ◽  
Critical Paths ◽  
And Performance

Energy efficient semiconductor chips are in high demand to cater the needs of today’s smart products. Advanced technology nodes insert high design margins to deal with rising variations at the cost of power, area and performance. Existing run time resilience techniques are not cost effective due to the additional circuits involved. In this paper, we propose a design time resilience technique using a clock stretched flip-flop to redistribute the available slack in the processor pipeline to the critical paths. We use the opportunistic slack to redesign the critical fan in logic using logic reshaping, better than worst case sigma corner libraries and multi-bit flip-flops to achieve power and area savings. Experimental results prove that we can tune the logic and the library to get significant power and area savings of 69% and 15% in the execute pipeline stage of the processor compared to the traditional worst-case design. Whereas, existing run time resilience hardware results in 36% and 2% power and area overhead respectively.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

A Method for Simultaneously Satisfying Important Constraints and Dependencies for Many Different Types of Processes in Embedded Real-Time Systems

2011 ◽  
Author(s):  
Jia Xu
Keyword(s):  
Real Time ◽  
Computation Time ◽  
Real Time Systems ◽  
Worst Case ◽  
Release Times ◽  
Different Types ◽  
High Assurance ◽  
Run Time ◽  
Soft Deadline ◽  
Time Systems

In most embedded, real-time applications, processes need to satisfy various important constraints and dependencies, such as release times, offsets, precedence relations, and exclusion relations. Embedded, real-time systems with high assurance requirements often must execute many different types of processes with such constraints and dependencies. Some of the processes may be periodic and some of them may be asynchronous. Some of the processes may have hard deadlines and some of them may have soft deadlines. For some of the processes, especially the hard real-time processes, complete knowledge about their characteristics can and must be acquired before run-time. For other processes, prior knowledge of their worst case computation time and their data requirements may not be available. It is important for many embedded real-time systems to be able to simultaneously satisfy as many important constraints and dependencies as possible for as many different types of processes as possible. In this paper, we discuss what types of important constraints and dependencies can be satisfied among what types of processes. We also present a method which guarantees that, for every process, no matter whether it is periodic or asynchronous, and no matter whether it has a hard deadline or a soft deadline, as long as the characteristics of that process are known before run-time, then that process will be guaranteed to be completed before predetermined time limits, while simultaneously satisfying many important constraints and dependencies with other processes.

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