On solving the MAX-SAT using sum of squares

We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability problem (MAX-SAT) and the weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate the (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems, by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS) based SDP bounds and an efficient parser within a branch & bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems, and show that a member of this family provides the rank two guarantee. We also provide a parametric family of semidefinite relaxations for the MAX-SAT, and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for the (MAX)-SAT, in an elegant way. Moreover, we relate our SOS-SDP relaxations for the partial MAX-SAT to the known SAT relaxations.Comment: 26 pages, 5 figures, 8 tables, 2 appendix page

On solving MAX-SAT using sum of squares

We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability (MAX-SAT) problem and weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS)–based SDP bounds and an efficient parser within a branch-and-bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems and show that a member of this family provides the rank-two guarantee. We also provide a parametric family of semidefinite relaxations for MAX-SAT and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for (MAX)-SAT in an elegant way. Moreover, we relate our SOS-SDP relaxations for partial MAX-SAT to the known SAT relaxations

On the generalized ϑ-number and related problems for highly symmetric graphs

This paper is an in-depth analysis of the generalized ϑ-number of a graph. The generalized ϑ-number, ϑk(G), serves as a bound for both the k-multichromatic number of a graph and the maximum k-colorable subgraph problem. We present various properties of ϑk(G), such as that the sequence (ϑk(G))k is increasing and bounded from above by the order of the graph G. We study ϑk(G) when G is the strong, disjunction, or Cartesian product of two graphs. We provide closed form expressions for the generalized ϑ-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs, and orthogonality graphs. Our paper provides bounds on the product and sum of the k-multichromatic number of a graph and its complement graph, as well as lower bounds for the k-multichromatic number on several graph classes including the Hamming and Johnson graphs

Dormant barley aleurone shows heterogeneity and a specific cytodifferentiation

In response to gibberellic acid, aleurone layers isolated from dormant barley (Hordeum distichum L. cv. Triumph) kernels produced significantly less alpha-amylase than aleurones from non-dormant kernels. Light microscopical investigations using the dye acridine orange as well as electron microscopical studies showed that the relatively low alpha-amylase production by aleurones from dormant kernels appears to be due to the majority of the cells remaining quiescent after GA3 incubation. The cells that did show ultrastructural signs of GA3-induced activation occurred in clusters that spanned all three cell layers of the dormant aleurone. These cells showed a characteristic ultrastructure, which was different from that described previously for GA3 incubated non-dormant aleurone layers. Dictyosomes, which are extremely difficult to visualise in non-dormant barley aleurone, were abundant. The presence of alpha-amylase in the Golgi could be demonstrated by immuno-gold labelling in these cells. Remarkably, most endosperm-facing aleurone cells in the activated clusters showed no cell wall degradation. This cytodifferentiation pattern could be mimicked in aleurone cells from non-dormant grains by imbibition of the grains in abscisic acid (ABA) prior to GA3 incubation. These data support the notion of heterogeneity of aleurone cells with respect to GA3 sensitivity, and suggest that ABA is involved in the impaired GA3 response of aleurone cells from dormant barley. © 1997 Academic Press Limited

The tobacco homolog of mammalian calreticulin is present in protein complexes in vivo.

The analysis of protein sorting signals responsible for the retention of reticuloplasmins (RPLs), a group of soluble proteins that reside in the lumen of the endoplasmic reticulum (ER), has revealed a structural similarity between mammalian and plant ER retention signals. We present evidence that the corresponding epitope is conserved in a vast family of soluble ER resident proteins. Microsequences of RPL60 and RPL90, two abundant members of this family, show high sequence similarity with mammalian calreticulin and endoplasmin. RPL60/calreticulin cofractionates and costains with the lumenal binding protein (BiP). Both proteins were detected in the nuclear envelope and the ER, and in mitotic cells in association with the spindle apparatus and the phragmoplast. Immunoprecipitation of proteins from in vivo-labeled cells demonstrated that RPL60/calreticulin is associated with other polypeptides in a stress- and ATP-dependent fashion. RPL60/calreticulin transcript levels increased rapidly in abundance during the proliferation of the secretory apparatus and the onset of hydrolase secretion in gibberellic acid-treated barley aleurone cells. This induction profile is identical to that of the well-characterized ER chaperones BiP and endoplasmin. However, expression patterns in response to different stress conditions as well as tissue-specific expression patterns indicate that these genes are differentially regulated and may not act in concert