The best current estimates of the thresholds for the existence of solutions in random constraint satisfaction problems ('CSPs') mostly derive from the first and the second moment method. Yet apart from a very few exceptional cases these methods do not quite yield matching upper and lower bounds. According to deep but non-rigorous arguments from statistical mechanics, this discrepancy is due to a change in the geometry of the set of solutions called condensation that occurs shortly before the actual threshold for the existence of solutions (Krzakala, Montanari, Ricci-Tersenghi, Semerjian, Zdeborova: PNAS 2007). To cope with condensation, physicists have developed a sophisticated but non-rigorous formalism called Survey Propagation (Me-zard, Parisi, Zecchina: Science 2002). This formalism yields precise conjectures on the threshold values of many random CSPs. Here we develop a new Survey Propagation inspired second moment method for the random k-NAESAT problem, which is one of the standard benchmark problems in the theory of random CSPs. This new technique allows us to overcome the barrier posed by condensation rigorously. We prove that the threshold for the existence of solutions in random k-NAESAT is 2k-1ln2-(ln 2 2+1 4)+ek, where |ek| ≤ 2-(1-ok(1))k, thereby verifying the statistical mechanics conjecture for this problem.
Parts of our proof require a precise analysis of geometry of the solution space @math . This analysis harnesses some of the ideas that were developed in previous work @cite_30 @cite_25 @cite_18 @cite_26 (e.g., arguments for proving the existence of clusters or of rigid variables''). However, we need to go beyond these previous arguments significantly in two respects. First, we need to generalize them to accommodate the parameter @math that controls the cluster sizes. Second, we need rather precise quantitative information about the cluster structures.
@cite_30 @cite_18 @cite_26 @cite_25
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In our LICS 2004 paper we introduced an approach to the study of the local structure of finite algebras and relational structures that aims at applications in the Constraint Satisfaction Problem (CSP). This approach involves a graph associated with an algebra @math or a relational structure A, whose vertices are the elements of @math (or A), the edges represent subsets of @math such that the restriction of some term operation of @math is ‘good’ on the subset, that is, act as an operation of one of the 3 types: semilattice, majority, or affine. In this paper we significantly refine and advance this approach. In particular, we prove certain connectivity and rectangularity properties of relations over algebras related to components of the graph connected by semilattice and affine edges. We also prove a result similar to 2-decomposition of relations invariant under a majority operation, only here we do not impose any restrictions on the relation. These results allow us to give a new, somewhat more intuitive proof of the bounded width theorem: the CSP over algebra @math has bounded width if and only if @math does not contain affine edges. Actually, this result shows that bounded width implies width (2,3). We also consider algebras with edges from a restricted set of types. In particular, it can be proved that type restrictions are preserved under the standard algebraic constructions. Finally, we prove that algebras without semilattice edges have few subalgebras of powers, that is, the CSP over such algebras is also polynomial time. Given a large data matrix @math , we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution @math , or instead @math contains a principal submatrix @math whose entries have marginal distribution @math . As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided @math for a suitable constant @math . However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when @math . Recently Meka and Wigderson meka2013association , proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree- @math SOS relaxation, and study the construction of meka2013association to prove that SOS fails unless @math . An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random association scheme, i.e. a symmetric random matrix whose rows and columns are indexed by the edges of an Erd "os-Renyi random graph. A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space X carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of X may be obtained by an appropriate finite sample (subset of X), and that persistence of any subset of X is well interleaved with the persistence of X. It follows that the persistence of X is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each (r>0 ) a density (s>0 ), so that for each s-dense sample (S X ) the corresponding fundamental group (and the first homology) of the Rips complex of S is isomorphic to the one of X, leading to an improved reconstruction result. A Semi-Separated Pair Decomposition (SSPD), with parameter s>1, of a set S@?R^d is a set (A"i,B"i) of pairs of subsets of S such that for each i, there are balls D"A"""i and D"B"""i containing A"i and B"i respectively such that d(D"A"""i,D"B"""i)>=s@?min(radius(D"A"""i),radius(D"B"""i)), and for any two points p,q@?S there is a unique index i such that p@?A"i and q@?B"i or vice versa. In this paper, we use the SSPD to obtain the following results: First, we consider the construction of geometric t-spanners in the context of imprecise points and we prove that any set S@?R^d of n imprecise points, modeled as pairwise disjoint balls, admits a t-spanner with O(nlogn (t-1)^d) edges that can be computed in O(nlogn (t-1)^d) time. If all balls have the same radius, the number of edges reduces to O(n (t-1)^d). Secondly, for a set of n points in the plane, we design a query data structure for half-plane closest-pair queries that can be built in O(n^2log^2n) time using O(nlogn) space and answers a query in O(n^1^ ^2^+^@e) time, for any @e>0. By reducing the preprocessing time to O(n^1^+^@e) and using O(nlog^2n) space, the query can be answered in O(n^3^ ^4^+^@e) time. Moreover, we improve the preprocessing time of an existing axis-parallel rectangle closest-pair query data structure from quadratic to near-linear. Finally, we revisit some previously studied problems, namely spanners for complete k-partite graphs and low-diameter spanners, and show how to use the SSPD to obtain simple algorithms for these problems.