Partly on the basis of heuristic arguments from physics, it has been suggested that the performance of certain types of algorithms on random @math -SAT formulas is linked to phase transitions that affect the geometry of the set of satisfying assignments. But, beyond intuition, there has been scant rigorous evidence that “practical” algorithms are affected by these phase transitions. In this paper we prove that Walksat, a popular randomized satisfiability algorithm, fails on random @math -SAT formulas not very far above clause variable density, where the set of satisfying assignments shatters into tiny, well-separated clusters. Specifically, we prove that Walksat is ineffective with high probability (w.h.p.) if @math , where @math is the number of clauses, @math is the number of variables, and @math is an absolute constant. By comparison, Walksat is known to find satisfying assignments in linear time w.h.p. if @math [A. Coja-Oghlan and A. Frieze, SIAM J. Comput., 43 (2...
While is not a sequential local algorithm, we critically use one idea of the analysis from @cite_32 , called overlap structures'' in that paper. Specifically, Gamarnik and Sudan prove that for an appropriate integer @math no @math -tuple of NAE-satsifying assignments exist with pairwise distance about @math if the clause variable densities is above @math . However, a coupling argument shows that if a local sequential algorithm were likely to succeed, then there would have to be such an @math -tuple of NAE-satisfying assignments with a non-vanishing probability. Actually the idea of overlap structures originates from the work of Rahman and Virag @cite_1 , who improved the density of an earlier negative result of Gamarnik and Sudan @cite_34 for a more specialised class of algorithms for the independent set problem. The definition of mists'' in the present paper is directly inspired by overlap structures.
@cite_34 @cite_1 @cite_32
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We report on a novel technique called spatial coupling and its application in the analysis of random constraint satisfaction problems (CSP). Spatial coupling was invented as an engineering construction in the area of error correcting codes where it has resulted in efficient capacity-achieving codes for a wide range of channels. However, this technique is not limited to problems in communications, and can be applied in the much broader context of graphical models. We describe here a general methodology for applying spatial coupling to random constraint satisfaction problems and obtain lower bounds for their (rough) satisfiability threshold. The main idea is to construct a distribution of geometrically structured random K-SAT instances - namely the spatially coupled ensemble - which has the same (rough) satisfiability threshold, and is at the same time algorithmically easier to solve. Then by running well-known algorithms on the spatially coupled ensemble we obtain a lower bound on the (rough) satisfiability threshold of the original ensemble. The method is versatile because one can choose the CSP, there is a certain amount of freedom in the construction of the spatially coupled ensemble, and also in the choice of the algorithm. In this work we focus on random K-SAT but we have also checked that the method is successful for Coloring, NAE-SAT and XOR-SAT. We choose Unit Clause propagation for the algorithm which is analyzed over the spatially coupled instances. For K = 3, for instance, our lower bound is equal to 3.67 which is better than the current bounds in the literature. Similarly, for graph 3-colorability we get a bound of 2.22 which is also better than the current bounds in the literature. A novel algorithm based on random forests with surrogate splits is proposed to address the classification problem of incomplete data without imputation.The algorithm allows each tree to cast a vote even the voting process is interrupted by missing attributes.Experimental results on various acknowledged datasets show that the proposed method is robust and efficient. Random forests (RF) is known as an efficient algorithm in classification, however it depends on the integrity of datasets. Conventional methods in dealing with missing values usually employ estimation and imputation approaches whose efficiency is tied to the assumptions of data features. Recently, algorithm of surrogate decisions in RF was developed and this paper proposes a random forests algorithm with modified surrogate splits (Adjusted Weight Voting Random Forest, AWVRF) which is able to address the incomplete data without imputation.Differing from the present surrogate method, in AWVRF algorithm, when the primary splitting attribute and the surrogate attributes of an internal node are all missing, the undergoing instance is allowed to exit at the current node with a vote. Then the weight of the vote is adjusted by the strength of the involved attributes and the final decision is made by weighted voting. AWVRF does not comprise imputation step, thus it is independent of data features.AWVRF is compared with the methods of mean imputation, LeoFill, knnimpute, BPCAfill and conventional RF with surrogate decisions (surrRF) using 50 times repeated 5-fold cross validation on 10 acknowledged datasets. In a total of 22 experiment settings, the method of AWVRF harvests the highest accuracy in 14 settings and the largest AUC in 7 settings, exhibiting its superiority over other methods. Compared with surrRF, AWVRF is significantly more efficient and remain good discrimination of prediction. Experimental results show that the present AWVRF algorithm can successfully handle the classification task for incomplete data. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09]. We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable is there exist two SAT-assignments differing in Nx variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT-assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x=12, but they do not exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). Our method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.