Symmetries are intrinsic to many combinatorial problems including Boolean Satisfiability (SAT) and Constraint Programming (CP). In SAT, the identification of symmetry breaking predicates (SBPs) is a well-known, often effective, technique for solving hard problems. The identification of SBPs in SAT has been the subject of significant improvements in recent years, resulting in more compact SBPs and more effective algorithms. The identification of SBPs has also been applied to pseudo-Boolean (PB) constraints, showing that symmetry breaking can also be an effective technique for PB constraints. This paper extends further the application of SBPs, and shows that SBPs can be identified and used in Maximum Satisfiability (MaxSAT), as well as in its most well-known variants, including partial MaxSAT, weighted MaxSAT and weighted partial MaxSAT. As with SAT and PB, symmetry breaking predicates for MaxSAT and variants are shown to be effective for a representative number of problem domains, allowing solving problem instances that current state of the art MaxSAT solvers could not otherwise solve.
Symmetries are a well-known research topic, that serve to tackle complexity in many combinatorial problems. The first ideas on symmetry breaking were developed in the 90s @cite_11 @cite_13 , by relating symmetries with the graph automorphism problem, and by proposing the first approach for generating symmetry breaking predicates. This work was later extended and optimized for propositional satisfiability @cite_10 .
@cite_10 @cite_13 @cite_11
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Identifying and breaking the symmetries of CNF formulae has been shown to lead to significant reductions in search times. In this paper we describe a more systematic and efficient construction of symmetry-breaking predicates (SBPs). In particular, we use the cycle structure of symmetry generators, which typically involve very few variables, to drastically reduce the size of SBPs. Furthermore, our new SBP construction grows linearly with the number of relevant variables as opposed to the previous quadratic constructions. Our empirical data suggest that these improvements reduce search run times by one to two orders of magnitude on a wide variety of benchmarks with symmetries. We investigate the role of symmetry detection and symmetry breaking in answer set programming to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We reduce symmetry detection to a graph automorphism problem which allows us to extract symmetries of a logic program from the symmetries of the constructed coloured graph. The correctness of our reduction is proven. We also propose an encoding of symmetry-breaking constraints in terms of permutation cycles and use only generators in this process to implicitly represent symmetries with exponential compression. These ideas are formulated as preprocessing and implemented in a completely automated flow that first detects symmetries from a given answer set program, adds symmetry-breaking constraints, and can be applied to any existing answer set solver. We demonstrate computational impact on benchmarks versus direct application of the solver. Symmetry-breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this article we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry-breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes the following: —An MIS algorithm running in O(log2D Δ L 2√log n, and comes close to the Ω(flog Δ log log Δ lower bound of Kuhn, Moscibroda, and Wattenhofer. —A maximal matching algorithm running in O(log Δ + log 4log n) time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on Δ is nearly optimal. —A (Δ + 1)-coloring algorithm requiring O(log Δ + 2o(√log log n) time, improving on an O(log Δ + √log n)-time algorithm of Schneider and Wattenhofer. —A method for reducing symmetry-breaking problems in low arboricity degeneracy graphs to low-degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an O(√log n)-time maximal matching algorithm for graphs with arboricity up to 2√log n and an O(log 2 3n)-time MIS algorithm for graphs with arboricity up to 2(log n)1 3. Each of our algorithms is based on a simple but powerful technique for reducing a randomized symmetry-breaking task to a corresponding deterministic one on a poly(log n)-size graph. "Symmetry is a complexity-reducing concept [...]; seek it every-where." - Alan J. PerlisMany natural and man-made objects exhibit significant symmetries or contain repeated substructures. This paper presents a new algorithm that processes geometric models and efficiently discovers and extracts a compact representation of their Euclidean symmetries. These symmetries can be partial, approximate, or both. The method is based on matching simple local shape signatures in pairs and using these matches to accumulate evidence for symmetries in an appropriate transformation space. A clustering stage extracts potential significant symmetries of the object, followed by a verification step. Based on a statistical sampling analysis, we provide theoretical guarantees on the success rate of our algorithm. The extracted symmetry graph representation captures important high-level information about the structure of a geometric model which in turn enables a large set of further processing operations, including shape compression, segmentation, consistent editing, symmetrization, indexing for retrieval, etc. We introduce the Symmetry Factored Embedding (SFE) and the Symmetry Factored Distance (SFD) as new tools to analyze and represent symmetries in a point set. The SFE provides new coordinates in which symmetry is "factored out," and the SFD is the Euclidean distance in that space. These constructions characterize the space of symmetric correspondences between points -- i.e., orbits. A key observation is that a set of points in the same orbit appears as a clique in a correspondence graph induced by pairwise similarities. As a result, the problem of finding approximate and partial symmetries in a point set reduces to the problem of measuring connectedness in the correspondence graph, a well-studied problem for which spectral methods provide a robust solution. We provide methods for computing the SFE and SFD for extrinsic global symmetries and then extend them to consider partial extrinsic and intrinsic cases. During experiments with difficult examples, we find that the proposed methods can characterize symmetries in inputs with noise, missing data, non-rigid deformations, and complex symmetries, without a priori knowledge of the symmetry group. As such, we believe that it provides a useful tool for automatic shape analysis in applications such as segmentation and stationary point detection. The interest shown recently in the algorithmic treatment of symmetries, also known as Computational Symmetry, covers several application areas among which textile and tiles design are some of the most significant. Designers make new creations based on the symmetric repetition of motifs that they find in databases of wallpaper images. The existing methods for dealing with these images have several drawbacks because the use of heuristics achieves low recovery rates when images exhibit imperfections due to the fabrication or the handmade process. The presence of symmetry in Boolean satisfiability (SAT) problem instances often poses challenges to solvers. Currently, the most effective approach to handle symmetry is by static symmetry breaking, which generates asymmetric constraints to add to the instance. An alternative way is to handle symmetry dynamically during solving. As modern SAT solvers can be viewed as propositional proof generators, adding a symmetry rule in a solver’s proof system would be a straightforward technique to handle symmetry dynamically. However, none of these proposed symmetrical learning techniques are competitive to static symmetry breaking. In this paper, we present symmetric explanation learning, a form of symmetrical learning based on learning symmetric images of explanation clauses for unit propagations performed during search. A key idea is that these symmetric clauses are only learned when they would restrict the current search state, i.e., when they are unit or conflicting. We further provide a theoretical discussion on symmetric explanation learning and a working implementation in a state-of-the-art SAT solver. We also present extensive experimental results indicating that symmetric explanation learning is the first symmetrical learning scheme competitive with static symmetry breaking. Abstract A simple method of analyzing symmetries of a line drawing is presented. The procedure consists of three stages: (1) computing the axes of ‘microsymmetry’ for pairs of microsegments based on the mean square error criterion, (2) mapping the axes into two-dimensional parameter space, and (3) detecting the peaks in the parameter space to obtain the global axes of symmetry. We generalize Krishnamurthy’s well-studied symmetry rule for resolution systems by considering homomorphisms instead of symmetries; symmetries are injective maps of literals whichpreserve complements and clauses; homomorphisms arise from symmetries by releasing the constraint of being injective. We prove that the use of homomorphisms yields a strictly more powerful system than the use of symmetries by exhibiting an infinite sequence of sets of clauses for which the consideration of global homomorphisms allows exponentially shorter proofs than the consideration of local symmetries. It is known that local symmetries give rise to a strictly more powerful system than global symmetries; we prove a similar result for local and global homomorphisms. Finally, we obtain an exponential lower bound for the resolution system enhanced by the local homomorphism rule. Fundamental local symmetry breaking problems such as Maximal Independent Set (MIS) and coloring have been recognized as important by the community, and studied extensively in (standard) graphs. In particular, fast (i.e., logarithmic run time) randomized algorithms are well-established for MIS and Δ + 1-coloring in both the LOCAL and CONGEST distributed computing models. On the other hand, comparatively much less is known on the complexity of distributed symmetry breaking in hypergraphs. In particular, a key question is whether a fast (randomized) algorithm for MIS exists for hypergraphs. This paper addresses the problem of detecting axes of bilateral symmetry in images. In order to achieve robustness to variation in illumination, only edge-gradient information is used. To overcome the problem of edge breaks, a potential field is developed from the edge map which spreads the information in the image plane. Pairs of points in the image plane are made to vote for their axes of symmetry with some confidence values. To make the method robust to overlapping objects, only local features in the form of Taylor coefficients are used for quantifying symmetry. We define an axis of symmetry histogram, which is used to accumulate the weighted votes for all possible axes of symmetry. To reduce the computational complexity of voting, a hashing scheme is proposed, wherein pairs of points, whose potential fields are too asymmetric, are pruned by not being counted for the vote. Experimental results indicate that the proposed method is fairly robust to edge breaks and is able to detect symmetries even when only 0.05 of the possible pairs are used for voting. A novel and efficient method is presented for grouping feature points on the basis of their underlying symmetry and characterising the symmetries present in an image. We show how symmetric pairs of features can be efficiently detected, how the symmetry bonding each pair is extracted and evaluated, and how these can be grouped into symmetric constellations that specify the dominant symmetries present in the image. Symmetries over all orientations and radii are considered simultaneously, and the method is able to detect local or global symmetries, locate symmetric figures in complex backgrounds, detect bilateral or rotational symmetry, and detect multiple incidences of symmetry. Previous work identifying depth-optimal n-channel sorting networks for 9 a#x2264; n a#x2264; 16 is based on exploiting symmetries of the first two layers. However, the naive generate-and-test approach typically applied does not scale. This paper revisits the problem of generating two-layer prefixes modulo symmetries. An improved notion of symmetry is provided and a novel technique based on regular languages and graph isomorphism is shown to generate the set of non-symmetric representations. An empirical evaluation demonstrates that the new method outperforms the generate-and-test approach by orders of magnitude and easily scales until n = 40. Ideas for higher-order array processing are introduced, focusing on fourth-order cumulant statistics. They are expressed in an index-free formalism allowing the exploitation of all their symmetry properties. It is shown that, when dealing with 4-index quantities, symmetries are related to rank properties. The rich symmetry structure yields a class of identification algorithms. An algebraic technique for blind identification is included, and a few others are briefly indicated. > Instances of the Boolean satisfiability problem (SAT) arise in many areas of circuit design and verification. These instances are typically constructed from some human-designed artifact, and thus are likely to possess much inherent symmetry and sparsity. Previous work[4] has shown that exploiting symmetries results in vastly reduced SAT solver run times, often with the search for the symmetries themselves dominating the total SAT solving time. Our contribution is twofold. First, we dissect the algorithms behind the venerable NAUTY[9] package, particularly the partition refinement procedure responsible for the majority of search space pruning as well as the majority of run time overhead. Second, we present a new symmetry-detection tool, SAUCY, which outperforms NAUTY by several orders of magnitude on the large, structured CNF formulas generated from typical EDA problems. Symmetry, as one of the key components of Gestalt theory, provides an important mid-level cue that serves as input to higher visual processes such as segmentation. In this work, we propose a complete approach that links the detection of curved reflection symmetries to produce symmetry-constrained segments of structures regions in real images with clutter. For curved reflection symmetry detection, we leverage on patch-based symmetric features to train a Structured Random Forest classifier that detects multiscaled curved symmetries in 2D images. Next, using these curved symmetries, we modulate a novel symmetry-constrained foreground-background segmentation by their symmetry scores so that we enforce global symmetrical consistency in the final segmentation. This is achieved by imposing a pairwise symmetry prior that encourages symmetric pixels to have the same labels over a MRF-based representation of the input image edges, and the final segmentation is obtained via graph-cuts. Experimental results over four publicly available datasets containing annotated symmetric structures: 1) SYMMAX-300 [38], 2) BSD-Parts, 3) Weizmann Horse (both from [18]) and 4) NY-roads [35] demonstrate the approach's applicability to different environments with state-of-the-art performance. We present an algorithm for multi-scale partial intrinsic symmetry detection over 2D and 3D shapes, where the scale of a symmetric region is defined by intrinsic distances between symmetric points over the region. To identify prominent symmetric regions which overlap and vary in form and scale, we decouple scale extraction and symmetry extraction by performing two levels of clustering. First, significant symmetry scales are identified by clustering sample point pairs from an input shape. Since different point pairs can share a common point, shape regions covered by points in different scale clusters can overlap. We introduce the symmetry scale matrix (SSM), where each entry estimates the likelihood two point pairs belong to symmetries at the same scale. The pair-to-pair symmetry affinity is computed based on a pair signature which encodes scales. We perform spectral clustering using the SSM to obtain the scale clusters. Then for all points belonging to the same scale cluster, we perform the second-level spectral clustering, based on a novel point-to-point symmetry affinity measure, to extract partial symmetries at that scale. We demonstrate our algorithm on complex shapes possessing rich symmetries at multiple scales. For many random constraint satisfaction problems, by now there exist asymptotically tight estimates of the largest constraint density for which solutions exist. At the same time, for many of these problems, all known polynomial-time algorithms stop finding solutions at much smaller densities. For example, it is well-known that it is easy to color a random graph using twice as many colors as its chromatic number. Indeed, some of the simplest possible coloring algorithms achieve this goal. Given the simplicity of those algorithms, one would expect room for improvement. Yet, to date, no algorithm is known that uses (2 - epsiv)chi colors, in spite of efforts by numerous researchers over the years. In view of the remarkable resilience of this factor of 2 against every algorithm hurled at it, we find it natural to inquire into its origin. We do so by analyzing the evolution of the set of k-colorings of a random graph, viewed as a subset of 1,...,k n, as edges are added. We prove that the factor of 2 corresponds in a precise mathematical sense to a phase transition in the geometry of this set. Roughly speaking, we prove that the set of k-colorings looks like a giant ball for k ges 2chi, but like an error-correcting code for k les (2 - epsiv)chi. We also prove that an analogous phase transition occurs both in random k-SAT and in random hypergraph 2-coloring. And that for each of these three problems, the location of the transition corresponds to the point where all known polynomial-time algorithms fail. To prove our results we develop a general technique that allows us to establish rigorously much of the celebrated 1-step replica-symmetry-breaking hypothesis of statistical physics for random CSPs. From art to science, ornaments constructed by repeating a base motif (tiling) have been a part of human culture. These ornaments exhibit various kinds of symmetries depending on the construction process as well as the symmetries of the base motif. The scientific study of the ornaments is the study of symmetry, i.e., the repetition structure. There is, however, an artistic side of the problem too: intriguing color permutations, clever choices of asymmetric interlocking forms, several symmetry breaking ideas, all that come with the artistic freedom. In this paper, in the context of Escher's Euclidean ornaments, we study ornaments without reference to fixed symmetry groups. We search for emergent categorical relations among a collection of tiles. We explore how these relations are affected when new tiles are inserted to the collection. We ask and answer whether it is possible to code symmetry group information implicitly without explicitly extracting the repetition structure, grids and motifs. Symmetry is a pervasive phenomenon presenting itself in all forms and scales in natural and manmade environments. Its detection plays an essential role at all levels of human as well as machine perception. The recent resurging interest in computational symmetry for computer vision and computer graphics applications has motivated us to conduct a US NSF funded symmetry detection algorithm competition as a workshop affiliated with the Computer Vision and Pattern Recognition (CVPR) Conference, 2013. This competition sets a more complete benchmark for computer vision symmetry detection algorithms. In this report we explain the evaluation metric and the automatic execution of the evaluation workflow. We also present and analyze the algorithms submitted, and show their results on three test sets of real world images depicting reflection, rotation and translation symmetries respectively. This competition establishes a performance baseline for future work on symmetry detection.