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Network motif discovery using subgraph enumeration and symmetry breaking
 IN PROCEEDINGS OF THE 11TH INTERNATIONAL CONFERENCE ON RESEARCH IN COMPUTATIONAL MOLECULAR BIOLOGY (RECOMB
, 2007
"... The study of biological networks and network motifs can yield significant new insights into systems biology. Previous methods of discovering network motifs – networkcentric subgraph enumeration and sampling – have been limited to motifs of 6 to 8 nodes, revealing only the smallest network componen ..."
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Cited by 23 (1 self)
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The study of biological networks and network motifs can yield significant new insights into systems biology. Previous methods of discovering network motifs – networkcentric subgraph enumeration and sampling – have been limited to motifs of 6 to 8 nodes, revealing only the smallest network components. New methods are necessary to identify larger network substructures and functional motifs. Here we present a novel algorithm for discovering large network motifs that achieves these goals, based on a novel symmetrybreaking technique, which eliminates repeated isomorphism testing, leading to an exponential speedup over previous methods. This technique is made possible by reversing the traditional networkbased search at the heart of the algorithm to a motifbased search, which also eliminates the need to store all motifs of a given size and enables parallelization and scaling. Additionally, our method enables us to study the clustering properties of discovered motifs, revealing even larger network elements. We apply this algorithm to the proteinprotein interaction network and transcription regulatory network of S. cerevisiae, and discover several large network motifs, which were previously inaccessible to existing methods, including a 29node cluster of 15node motifs corresponding to the key transcription machinery of S. cerevisiae.
The Perfect Binary OneErrorCorrecting Codes of Length 15: Part II  Properties
, 2009
"... A complete classification of the perfect binary oneerrorcorrecting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. Östergård and O. Pottonen, “The perfect binary oneerrorcorrecting codes of length 15: Part I—Classification, ” submitted for publica ..."
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Cited by 20 (3 self)
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A complete classification of the perfect binary oneerrorcorrecting codes of length 15 as well as their extensions of length 16 was recently carried out in [P. R. J. Östergård and O. Pottonen, “The perfect binary oneerrorcorrecting codes of length 15: Part I—Classification, ” submitted for publication]. In the current accompanying work, the classified codes are studied in great detail, and their main properties are tabulated. The results include the fact that 33 of the 80 Steiner triple systems of order 15 occur in such codes. Further understanding is gained on fullrank codes via icomponents, as it turns out that all but two fullrank codes can be obtained through a series of transformations from the Hamming code. Other topics studied include (non)systematic codes, embedded oneerrorcorrecting codes, and defining sets of codes. A classification of certain mixed perfect codes is also obtained.
Orbital branching
 in Proceedings of IPCO XII
, 2007
"... Orbital branching is a method for branching on variables in integer programming that reduces the likelihood of evaluating redundant, isomorphic nodes in the branchandbound procedure. In this work, the orbital branching methodology is extended so that the branching disjunction can be based on an ar ..."
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Cited by 19 (3 self)
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Orbital branching is a method for branching on variables in integer programming that reduces the likelihood of evaluating redundant, isomorphic nodes in the branchandbound procedure. In this work, the orbital branching methodology is extended so that the branching disjunction can be based on an arbitrary constraint. Many important families of integer programs are structured such that small instances from the family are embedded in larger instances. This structural information can be exploited to define a group of strong constraints on which to base the orbital branching disjunction. The symmetric nature of the problems is further exploited by enumerating nonisomorphic solutions to instances of the small family and using these solutions to create a collection of typically easytosolve integer programs. The solution of each integer program in the collection is equivalent to solving the original large instance. The effectiveness of this methodology is demonstrated by computing the optimal incidence width of Steiner Triple Systems and minimum cardinality covering designs.
Minimum diameters of plane integral point sets
"... ABSTRACT. Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its ..."
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Cited by 16 (15 self)
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ABSTRACT. Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2, n) of a plane integral point set consisting of n points arises. We give some new exact values and describe stateoftheart algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2, n) achieving the known upper bound nc2 log log n up to a constant in the exponent. A famous question of Erdős asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by ˙ d(2, n). Recently ˙ d(2, 7) = 22 270 could be determined via an exhaustive search. 1.
Characteristicindependence of Betti numbers of graph ideals
 J. Combinatorial Theory, Series A
, 2006
"... Throughout this paper K will denote a field. For any homogeneous ideal I of a polynomial ring R = K [x1,..., xn] there exists a graded minimal finite free resolution 0 → ⊕ ..."
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Cited by 16 (1 self)
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Throughout this paper K will denote a field. For any homogeneous ideal I of a polynomial ring R = K [x1,..., xn] there exists a graded minimal finite free resolution 0 → ⊕
Small Covering Designs by BranchandCut
, 2000
"... A BranchandCut algorithm for nding covering designs is presented. Its originality resides in the use of isomorphism pruning of the enumeration tree. A proof that no 4 (10; 5; 1)covering design with less than 51 sets exists is obtained together with all non isomorphic 4 (10; 5; 1)covering desi ..."
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Cited by 14 (2 self)
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A BranchandCut algorithm for nding covering designs is presented. Its originality resides in the use of isomorphism pruning of the enumeration tree. A proof that no 4 (10; 5; 1)covering design with less than 51 sets exists is obtained together with all non isomorphic 4 (10; 5; 1)covering designs with 51 sets. 1 Introduction Let V be a set of elements of cardinality v and let k and t be integers such that v k t 0. Let K be the set of all ksubsets of V and T be the set of all tsubsets of V . For 1, a t (v; k; )covering design is a collection C of sets in K such that each t 2 T is contained in at least sets of C. A t (v; k; )covering design C is minimum if the cardinality of C is as small as possible. This cardinality is denoted by C (v; k; t). Covering designs have a long history and have applications in statistics, coding theory and combinatorics, among others. Numerous theorems give the value of a minimal covering design under certain assumptions on the p...
An Orderly Algorithm and Some Applications in Finite Geometry
 Discrete Math
, 1996
"... An algorithm for generating combinatorial structures is said to be an orderly algorithm if it produces precisely one representative of each isomorphism class. In this paper we describe a way to construct an orderly algorithm that is suitable for several common searching tasks in combinatorics. We ..."
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Cited by 12 (2 self)
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An algorithm for generating combinatorial structures is said to be an orderly algorithm if it produces precisely one representative of each isomorphism class. In this paper we describe a way to construct an orderly algorithm that is suitable for several common searching tasks in combinatorics. We illustrate this with examples of searches in finite geometry, and an extended application where we classify all the maximal partial flocks of the hyperbolic and elliptic quadrics in PG(3; q) for q 13.
Small latin squares, quasigroups and loops
 Journal of Combinatorial Designs
, 2007
"... We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, 1990), quasigroups of order 6 (Bower, 2000) and loops of ..."
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Cited by 11 (4 self)
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We present the numbers of isotopy classes and main classes of Latin squares, and the numbers of isomorphism classes of quasigroups and loops, up to order 10. The best previous results were for Latin squares of order 8 (Kolesova, Lam and Thiel, 1990), quasigroups of order 6 (Bower, 2000) and loops of order 7 (Brant and Mullen, 1985). The loops of order 8 have been independently found by \QSCGZ" and Guerin (unpublished, 2001). We also report on the most extensive search so far for a triple of mutually orthogonal Latin squares (MOLS) of order 10. Our computations show that any such triple must have only squares with trivial symmetry groups. 1
Classification of regular twographs on 36 and 38 vertices
, 2001
"... In a previous paper an incomplete investigation into regular twographs on 36 vertices established the existence of at least 227. Using a more efficient algorithm, the two authors have independently verified that in fact these 227 comprise the complete set. An immediate consequence of this is that a ..."
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Cited by 11 (1 self)
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In a previous paper an incomplete investigation into regular twographs on 36 vertices established the existence of at least 227. Using a more efficient algorithm, the two authors have independently verified that in fact these 227 comprise the complete set. An immediate consequence of this is that all strongly regular graphs with parameters (35,16,6,8), (36,14,4,6), (36,20,10,12) and their complements are now known. Similar techniques were attempted in the case of regular twographs on 38 vertices, but without success on account of the vast amount of computer time required. Instead a different approach was used which managed to increase the known number of such regular twographs from 11 to 191.
There are 526,915,620 nonisomorphic onefactorizations of K 12
 J. COMBIN. DES
, 1994
"... We enumerate the nonisomorphic and the distinct onefactorizations of K 12 . We also describe the algorithm used to obtain the result, and the methods we used to verify these numbers. ..."
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Cited by 10 (1 self)
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We enumerate the nonisomorphic and the distinct onefactorizations of K 12 . We also describe the algorithm used to obtain the result, and the methods we used to verify these numbers.