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On the Average Similarity Degree between Solutions of Random k-SAT and Random CSPs [article]

Ke Xu, Wei Li
2002 arXiv   pre-print
To study the structure of solutions for random k-SAT and random CSPs, this paper introduces the concept of average similarity degree to characterize how solutions are similar to each other.  ...  For random k-SAT this phenomenon will occur when k>4 .  ... 
arXiv:cs/0008008v2 fatcat:ben6pm2uijb3rlgtgltgwecbye

On the average similarity degree between solutions of random k-SAT and random CSPs

Ke Xu, Wei Li
2004 Discrete Applied Mathematics  
To study the structure of solutions for random k-SAT and random CSPs, this paper introduces the concept of average similarity degree to characterize how solutions are similar to each other.  ...  For random k-SAT this phenomenon will occur when k ¿ 5.  ...  Acknowledgements We wish to thank an anonymous referee and the guest editor Jianer Chen for helpful comments and suggestions which greatly improved the presentation of this paper.  ... 
doi:10.1016/s0166-218x(03)00204-x fatcat:eic5meekwbheziqqj5v6tq4uxa

Counting Solutions of Constraint Satisfiability Problems:Exact Phase Transitions and Approximate Algorithm [article]

Minghao Yin, Ping Huang
2011 arXiv   pre-print
In this paper, we follow this line of research by considering the problem of counting solutions of Constraint Satisfaction Problems (#CSP). We consider the random model, i.e. RB model.  ...  Moreover, we propose an approximate algorithm to estimate the expectation value of the solutions number of a given CSP instance of RB model.  ...  Each number in the table is an average of the solutions numbers of 300 random instances in the SAT region. Conclusion In this paper, we first present a probabilistic analysis of the problem #CSP.  ... 
arXiv:1102.4922v1 fatcat:cwiplexrr5gwnm2qabhkwar7l4

Graph Neural Networks for Maximum Constraint Satisfaction [article]

Jan Toenshoff, Martin Ritzert, Hinrikus Wolf, Martin Grohe
2020 arXiv   pre-print
Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger).  ...  Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific  ...  On G14 and G15, which are random graphs with decaying node degree, the graph cuts produced by RUN-CSP are similar in size to those reported for DSDP.  ... 
arXiv:1909.08387v3 fatcat:gb5kx67bvncfxai2jya5ljudku

Graph Neural Networks for Maximum Constraint Satisfaction

Jan Tönshoff, Martin Ritzert, Hinrikus Wolf, Martin Grohe
2021 Frontiers in Artificial Intelligence  
Training is unsupervised, and it is sufficient to train on relatively small instances; the resulting networks perform well on much larger instances (at least 10-times larger).  ...  Despite being generic, we show that our approach matches or surpasses most greedy and semi-definite programming based algorithms and sometimes even outperforms state-of-the-art heuristics for the specific  ...  Most of the implementation was done by JT, with help and feedback from MR and HW. The work was supervised by MG.  ... 
doi:10.3389/frai.2020.580607 pmid:33733220 pmcid:PMC7959828 fatcat:y5jf5virxvhx5lqcrfbrfuinna

Quiet Planting in the Locked Constraint Satisfaction Problems

Lenka Zdeborová, Florent Krzakala
2011 SIAM Journal on Discrete Mathematics  
We study the planted ensemble of locked constraint satisfaction problems. We describe the connection between the random and planted ensembles.  ...  The use of the cavity method is combined with arguments from reconstruction on trees and first and second moment considerations; in particular the connection with the reconstruction on trees appears to  ...  We thank to Dimitris Achlioptas, Amin Coja-Oghlan, Matti Jarvisalo, Andrea Montanari, Cris Moore and Guilhem Semerjian for fruitful discussions and suggestions.  ... 
doi:10.1137/090750755 fatcat:3jsyuj2h2rerlhmtc3eqxlmboi

Algorithmic Barriers from Phase Transitions

Dimitris Achlioptas, Amin Coja-Oghlan
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
We prove that a completely analogous phase transition also occurs both in random k-SAT and in random hypergraph 2-coloring.  ...  We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of {1,...,k}^n, as edges are added.  ...  In other words, locally, random CSPs are trivial, e.g., random graphs of any finite average degree are locally 2-colorable.  ... 
doi:10.1109/focs.2008.11 dblp:conf/focs/AchlioptasC08 fatcat:aygz6dfyxfhg3b3ativ3o3o2jm

Constructing an asymptotic phase transition in random binary constraint satisfaction problems

Barbara M. Smith
2001 Theoretical Computer Science  
In this paper, an alternative scheme for one of the standard models is proposed in which both the number of values in each variable's domain and the average degree of the constraint graph are increased  ...  This scheme is compared to a similar scheme due to Xu and Li.  ...  Thanks are also due to Alan Slomson for the lemma given in Appendix A:3 and to Martin Dyer and Alan Frieze for the results on the width of random graphs.  ... 
doi:10.1016/s0304-3975(01)00166-9 fatcat:7ydlv6jiefaeblf2huadnfn5sm

The solution space structure of random constraint satisfaction problems with growing domains

Wei Xu, Pan Zhang, Tian Liu, Fuzhou Gong
2015 Journal of Statistical Mechanics: Theory and Experiment  
This picture of phase diagram is different from other classic random CSPs with fixed domain size, such as random K-Satisfiability (K-SAT) and graph coloring problems, where condensation transition exists  ...  Our result verifies the non-rigorous results obtained using cavity method from spin glass theory, and sheds light on the structures of solution spaces of problems with a large number of states.  ...  Note that in some CSPs like K-SAT problem with K = 3 and some combinatorial optimization problems like independent set problem [22, 23] with low average degree, clustering transition and condensation  ... 
doi:10.1088/1742-5468/2015/12/p12006 fatcat:hq6kem34jnewfer7r3lc7kor5y

CRITICALITY AND HETEROGENEITY IN THE SOLUTION SPACE OF RANDOM CONSTRAINT SATISFACTION PROBLEMS

HAIJUN ZHOU
2010 International Journal of Modern Physics B  
For random K-SAT and K-XORSAT, we show that many solution communities start to form in the solution space as the constraint density reaches a critical value alpha_cm, with each community containing a set  ...  of solutions that are more similar with each other than with the outsider solutions.  ...  Application to K-SAT.-We begin with the random K-SAT, a prototypical CSP [3, 4, 9] .  ... 
doi:10.1142/s0217979210056013 fatcat:bm6jousgpvhublwzy5fmtf6m2u

Source coding by efficient selection of ground-state clusters

Demian Battaglia, Alfredo Braunstein, Joël Chavas, Riccardo Zecchina
2005 Physical Review E  
These capabilities can be used to explore efficiently the space of solutions of random NP-complete constraint satisfaction problems, providing a direct experimental evidence of replica symmetry breaking  ...  In this letter, we show how the Survey Propagation algorithm can be generalized to include external forcing messages, and used to address selectively an exponential number of glassy ground states.  ...  In order to have a highly homogeneous distribution of the clusters, we have chosen for our experiments an ensemble of random K-SAT in which variables have fixed degrees and are balanced ͑i.e., have an  ... 
doi:10.1103/physreve.72.015103 pmid:16090025 fatcat:ya27d3nbbzgxxjgeniucxesr7q

Page 10261 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
(PRC-BUAA-SFD; Beijing) On the average similarity degree between solutions of random k- SAT and random CSPs. (English summary) Discrete mathematics and theoretical computer science (DMTCS).  ...  Summary: “To study the structure of solutions for random k-SAT and random CSPs, this paper introduces the concept of average similarity degree to characterize how solutions are similar to each other.  ... 

Faster Algorithms for MAX CUT and MAX CSP, with Polynomial Expected Time for Sparse Instances [chapter]

Alexander D. Scott, Gregory B. Sorkin
2003 Lecture Notes in Computer Science  
An alternative deterministic bound on the solution time, as 2 m/5 , improves upon a series of recent results.  ...  Our method is to show, first, that if a max 2-csp has a connected underlying graph with n vertices and m edges, the solution time can be deterministically bounded by 2 (m−n)/2 .  ...  4 having k neighbors of degree 3 and r − k neighbors of degree 4.  ... 
doi:10.1007/978-3-540-45198-3_32 fatcat:dalzvgoq2nfkhcbspy26vltok4

Constraint satisfaction problems with isolated solutions are hard

Lenka Zdeborová, Marc Mézard
2008 Journal of Statistical Mechanics: Theory and Experiment  
We study the phase diagram and the algorithmic hardness of the random 'locked' constraint satisfaction problems, and compare them to the commonly studied 'non-locked' problems like satisfiability of boolean  ...  On the other hand we show empirically that the clustered phase of these problems is extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions.  ...  Acknowledgments We thank Florent Krzakala, Cris Moore, Thierry Mora and Guilhem Semerjian for very useful discussions.  ... 
doi:10.1088/1742-5468/2008/12/p12004 fatcat:zhqg6m4k5rhxdjo4honfz7kbpe

On the cavity method for decimated random constraint satisfaction problems and the analysis of belief propagation guided decimation algorithms

Federico Ricci-Tersenghi, Guilhem Semerjian
2009 Journal of Statistical Mechanics: Theory and Experiment  
We introduce a version of the cavity method for diluted mean-field spin models that allows the computation of thermodynamic quantities similar to the Franz-Parisi quenched potential in sparse random graph  ...  This method is developed in the particular case of partially decimated random constraint satisfaction problems.  ...  This work has been partially funded by the PHC Galileo program for exchanges between France and Italy. We present in this appendix some details on the computations discussed in Sec. IV C.  ... 
doi:10.1088/1742-5468/2009/09/p09001 fatcat:76d27nbyqrfnbmt66o77ola4wq
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