The PPSZ Algorithm presented by Paturi, Pudlak, Saks, and Zane in 1998 has the nice feature that the only satisfying solution of a uniquely satisfiable 3-SAT formula can be found in expected running time at most O(1.3071n). Its bound degenerates when the number of solutions increases. In 1999, Schöning proved an O(1.3334n) bound for 3-SAT. In 2003, Iwama and Tamaki combined both algorithms to y...

Let $G$ be a simple graph on $n$ vertices. $L_G \text{ and } \mathcal{I}_G \: $ denote the Lov\'asz-Saks-Schrijver(LSS) ideal parity binomial edge of in polynomial ring $S = \mathbb{K}[x_1,\ldots, x_n, y_1, \ldots, y_n] respectively. We classify graphs whose LSS ideals are complete intersections. also almost intersections, we prove that their Rees algebra is Cohen-Macaulay. compute second grade...

We give an n O (log log ) -time membership query algorithm for properly and agnostically learning decision trees under the uniform distribution over { ± 1} . Even in realizable setting, previous fastest runtime was , a consequence of classic Ehrenfeucht Haussler. Our shares similarities with practical heuristics trees, which we augment additional ideas to circumvent known lower bounds against t...

The rectangle covering number of an n × n Boolean matrix M is the smallest number of 1-rectangles which are needed to cover all the 1-entries of M . Its binary logarithm is the Nondeterministic Communication Complexity, and it equals the chromatic number of a graph G (M) obtained from M by a construction of Lovász and Saks. We determine the rectangle covering number and related parameters (cliq...

For SAT and CIRCUIT -SAT there is an obvious brute force algorithm that requires 2m time where n is the number of variables and m is a measure of the overall input size. How much better we can seems to depend on the input structure. In the case of kSAT there are some improvements. Theorem 1.1 (Paturi-Pudlak,Schöning,Paturi-Pudlak-Saks-Zane,Hertli). For constants 1 ≤ c ≤ 2, there are (randomized...

Suppose that the vertices of a graph G are colored with two colors in an unknown way. The color occurs on more than half is called majority (if it exists), and any vertex this . We study problem finding (or show none if we can query edges to learn whether their endpoints have same or different colors. Denote least number queries needed worst case by m ( ) It was shown Saks Werman K n = − b , wh...

Some Schur, Vitali-Hahn-Saks and Nikodým convergence theorems for (l)-group-valued measures are given in the context of (D)-convergence. We consider both the σ-additive and the finitely additive case. Here the notions of strong boundedness, countable additivity and absolute continuity are formulated not necessarily with respect to a same regulator, while the pointwise convergence of the measure...

By the Central Element Theorem of Linial and Saks, it follows that for the problem of (generalised) sea.rching in posets, the information-theoretic lower bound of log N comparisons (where N is the number of order-ideals in the poset) is tight asymptotically. We observe that this implies that the problem of (generalised) sorting in posets has complexity 9( n . log N) (where n is the number of de...

We give a simple proof, via query elimination, of a result due to O’Donnell, Saks, Schramm, and Servedio, which shows a lower bound on the zero-error expected query complexity of a function f in terms of the maximum influence of any variable of f . Our lower bound also applies to the two-sided error expected query complexity of f , and it allows an immediate extension which can be used to prove...