We have been developing a theory for the generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a directed, acyclic shock graph, and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar. The grammar permits a reduction of a shock graph to a unique rooted shock tree. We introduce a novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time. Using a diverse database of shapes, we demonstrate our system's performance under articulation, occlusion, and changes in viewpoint. Keywords: shape representation; shape matching; shock graph; shock graph grammar; subgraph isomorphism. 1 I...

We introduce a novel view-based object representation, called the saliency map graph (SMG), which captures the salient regions of an object view at multiple scales using a wavelet transform. This compact representation is highly invariant to translation, rotation (image and depth), and scaling, and offers the locality of representation required for occluded object recognition. To compare two saliency map graphs, we introduce two graph similarity algorithms. The first computes the topological similarity between two SMG's, providing a coarse-level matching of two graphs. The second computes the geometrical similarity between two SMG's, providing a fine-level matching of two graphs. We test and compare these two algorithms on a large database of model object views. Keywords: View-Based Object Recognition, Shape Representation and Recovery, Graph Matching. 1 Introduction The view-based approach to 3-D object recognition represents an object as a collection of 2-D views, sometimes called...

It is shown that determining whether a quantum computation has a nonzero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness

We investigate the computational complexity of the Boolean Isomorphism problem (BI): on input of two Boolean formulas F and G decide whether there exists a permutation of the variables of G such that F and G become equivalent. Our main result is a one-round interactive proof for BI, where the verifier has access to an NP oracle. To obtain this, we use a recent result from learning theory by Bshouty et.al. that Boolean formulas can be learned probabilistically with equivalence queries and access to an NP oracle. As a consequence, BI cannot be \Sigma p 2 complete unless the Polynomial Hierarchy collapses. This solves an open problem posed in [BRS95]. Further properties of BI are shown: BI has And- and Or-functions, the counting version, #BI, can be computed in polynomial time relative to BI, and BI is self-reducible.

We study containment and equivalence of (unions of) conjunctive queries on relations annotated with elements of a commutative semiring. Such relations and the semantics of positive relational queries on them were introduced in a recent paper as a generalization of set semantics, bag semantics, incomplete databases, and databases annotated with various kinds of provenance information. We obtain positive decidability results and complexity characterizations for databases with lineage, why-provenance, and provenance polynomial annotations, for both conjunctive queries and unions of conjunctive queries. At least one of these results is surprising given that provenance polynomial annotations seem “more expressive ” than bag semantics and under the latter, containment of unions of conjunctive queries is known to be undecidable. The decision procedures rely on interesting variations on the notion of containment mappings. We also show that for any positive semiring (a very large class) and conjunctive queries without self-joins, equivalence is the same as isomorphism. 1.

The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers. Despite the worst-case exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification [29–31, 228], automatic test pattern generation [138, 221], planning [129, 197], scheduling [103], and even challenging problems from algebra [238]. Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13,

We prove that the graph isomorphism problem restricted to trees and to colored graphs with color multiplicities 2 and 3 is many-one complete for several complexity classes within NC². In particular we show that tree isomorphism, when trees are encoded as strings, is NC¹-hard under AC0-reductions. NC¹-completeness thus follows from Buss's NC¹ upper bound. By contrast, we prove that testing isomorphism of two trees encoded as pointer lists is L-complete. Concerning colored graphs we show that the isomorphism problem for graphs with color multiplicities 2 and 3 is complete for symmetric logarithmic space SL under many-one reductions. This result improves the existing upper bounds for the problem. We also show that the graph automorphism problem for colored graphs with color classes of size 2 is equivalent to deciding whether a graph has more than a single connected component and we prove that for color classes of size 3 the graph automorphism problem is contained in SL.

We present a novel low-overhead framework for encoding and utilizing structural symmetry in propositional satisfiability algorithms (SAT solvers). We use the notion of complete multi-class symmetry and demonstrate the efficacy of our technique through a solver SymChaff that achieves exponential speedup by using simple tags in the specification of problems from both theory and practice. Efficient implementations of DPLL-based SAT solvers are routinely used in areas as diverse as planning, scheduling, design automation, model checking, verification, testing, and algebra. A natural feature of many application domains is the presence of symmetry, such as that amongst all trucks at a certain location in logistics planning and all wires connecting two switch boxes in an FPGA circuit. Many of these problems turn out to have a concise description in many-sorted first order logic. This description can be easily specified by the problem designer and almost as easily inferred automatically. SymChaff, an extension of the popular SAT solver zChaff, uses information obtained from the “sorts” in the first order logic constraints to create symmetry sets that are used to partition variables into classes and to maintain and utilize symmetry information dynamically. Current approaches designed to handle symmetry include: (A) symmetry breaking predicates (SBPs), (B) pseudo-Boolean solvers with implicit representation for counting, (C) modifications of DPLL that handle symmetry dynamically, and (D) techniques based on ZBDDs. SBPs are prohibitively many, often large, and expensive to compute for problems such as the ones we report experimental results for. Pseudo-Boolean solvers are provably exponentially slow in certain symmetric situations and their implicit counting representation is not always appropriate. Suggested modifications of DPLL either work on limited global symmetry and are difficult to extend, or involve expensive algebraic group computations. Finally, techniques based on ZBDDs often do not compare well even with ordinary DPLL-based solvers. Sym-Chaff addresses and overcomes most of these limitations.

The problem of canonically labeling a graph is studied. Within the general framework of backtracking algorithms based on individualization and refinement, data structures, subroutines, and pruning heuristics especially for fast handling of large and sparse graphs are developed. Experiments indicate that the algorithm implementation in most cases clearly outperforms existing state-of-the-art tools.

This paper presents an efficient approach to address the task of learning from large number of learning examples in structural domains. While in attribute-value representations only one mapping is possible between descriptions, in first order logic representations there are potentially many mappings. Classic approaches consider all mappings and then define a restricted hypothesis space to cope with the intractability of exploring all mappings. Our approach is to select one particular type of mapping at a time and use it as a basis to define a new hypothesis space. We show that such a hypothesis space, called a Matching Space, may be represented using attribute-value pairs. In a Matching Space, it is therefore possible to use propositional learners. The concept descriptions found may then be mapped back into the initial first order logic representation. It appears that characterizing a Matching Space is equivalent to shifting the representation of examples: the new ...