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ON THE COMBINATORIAL PROBLEMS WHICH I WOULD MOST LIKE TO SEE SOLVED
, 1979
"... I was asked to write a paper about the major unsolved problems in combinatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no re ..."
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I was asked to write a paper about the major unsolved problems in combinatorial mathematics. After some thought it seemed better to modify the title to a less pretentious one. Combinatorial mathematics has grown enormously and a genuine survey would have to include not only topics where I have no real competence but also topics about which I never seriously thought, e.g. algorithmic combinatorics, coding theory and matroid theory. There is no doubt that the proof of the conjecture that several simply stated problems have no good algorithm is fundamental and may have important consequences for many other branches of mathematics, but unfortunately I have no real feeling for these questions and I feel I should leave the subject to those who are more competent. I just heard that Khachiyan [59], has a polynomial algorithm for linear programming. (See also [50].) This is considered a sensational result and during my last stay in the U.S. many of my friends were greatly impressed by it.
Software and Hardware Testing Using Combinatorial Covering Suites
, 2003
"... In the 21 st century our society is becoming more and more dependent on software systems. The safety of these systems and the quality of our lives is increasingly dependent on the quality of such systems. A key element in the manufacture and quality assurance process in software engineering is the ..."
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Cited by 28 (1 self)
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In the 21 st century our society is becoming more and more dependent on software systems. The safety of these systems and the quality of our lives is increasingly dependent on the quality of such systems. A key element in the manufacture and quality assurance process in software engineering is the testing of software and hardware systems. The construction of efficient combinatorial covering suites has important applications in the testing of hardware and software. In this paper we define the general problem, discuss the lower bounds on the size of covering suites, and give a series of constructions that achieve these bounds asymptotically. These constructions include the use of finite field theory, extremal set theory, group theory, coding theory, combinatorial recursive techniques, and other areas of computer science and mathematics. The study of these combinatorial covering suites is a fascinating example of the interplay between pure mathematics and the applied problems generated by software and hardware engineers. The wide range of mathematical techniques used, and the often unexpected applications of combinatorial covering suites make for a rewarding study.
Linearity and solvability in multicast networks
 IEEE TRANS. INF. THEORY
, 2004
"... It is known that for every solvable multicast network, there exists a large enough finitefield alphabet such that a scalar linear solution exists. We prove: i) every binary solvable multicast network with at most two messages has a binary scalar linear solution; ii) for more than two messages, not ..."
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It is known that for every solvable multicast network, there exists a large enough finitefield alphabet such that a scalar linear solution exists. We prove: i) every binary solvable multicast network with at most two messages has a binary scalar linear solution; ii) for more than two messages, not every binary solvable multicast network has a binary scalar linear solution; iii) a multicast network that has a solution for a given alphabet might not have a solution for all larger alphabets.
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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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
A Model Seeker: Extracting Global Constraint Models From Positive Examples
"... Abstract. We describe a system which generates finite domain constraint models from positive example solutions, for highly structured problems. The system is based on the global constraint catalog, providing the library of constraints that can be used in modeling, and the Constraint Seeker tool, whi ..."
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Abstract. We describe a system which generates finite domain constraint models from positive example solutions, for highly structured problems. The system is based on the global constraint catalog, providing the library of constraints that can be used in modeling, and the Constraint Seeker tool, which finds a ranked list of matching constraints given one or more sample call patterns. We have tested the modeler with 230 examples, ranging from 4 to 6,500 variables, using between 1 and 7,000 samples. These examples come from a variety of domains, including puzzles, sportsscheduling, packing & placement, and design theory. When comparing against manually specified “canonical ” models for the examples, we achieve a hit rate of 50%, processing the complete benchmark set in less than one hour on a laptop. Surprisingly, in many cases the system finds usable candidate lists even when working with a single, positive example. 1
Mutually Orthogonal Latin Squares: A Brief Survey of Constructions
, 1999
"... In the two centuries since Euler first asked about mutually orthogonal latin squares, substantial progress has been made. The biggest breakthroughs came in 1960 with the celebrated theorems of Bose, Shrikhande, and Parker, and in 1974 in the research of Wilson. Current efforts have concentrated on r ..."
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In the two centuries since Euler first asked about mutually orthogonal latin squares, substantial progress has been made. The biggest breakthroughs came in 1960 with the celebrated theorems of Bose, Shrikhande, and Parker, and in 1974 in the research of Wilson. Current efforts have concentrated on refining these approaches, and finding new applications of the substantial theory opened. This paper provides a detailed list of constructions for MOLS, concentrating on the uses of pairwise balanced designs and transversal designs in recursive constructions as pioneered in the papers of Bose, Shrikhande, and Parker. In addition, several new lower bounds for MOLS are given and an uptodate table of lower bounds for MOLS is provided. 1 An Historical Introduction In 1779, Euler began a study of a simple mathematical puzzle, the 36 Officers Problem. Thirtysix officers drawn from six different ranks and six different regiments (one of each rank from each regiment) are to be arranged in a squar...
Switching of edges in strongly regular graphs. I. A family of partial difference sets on 100 vertices
 ELECTRON. J. COMBIN., 10(1):RESEARCH PAPER
, 2003
"... We present 15 new partial difference sets over 4 nonabelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,2 ..."
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We present 15 new partial difference sets over 4 nonabelian groups of order 100 and 2 new strongly regular graphs with intransitive automorphism groups. The strongly regular graphs and corresponding partial difference sets have the following parameters: (100,22,0,6), (100,36,14,12), (100,45,20,20), (100,44,18,20). The existence of strongly regular graphs with the latter set of parameters was an open question. Our method is based on combination of Galois correspondence between permutation groups and association schemes, classical Seidel's switching of edges and essential use of computer algebra packages. As a byproduct, a few new amorphic association schemes with 3 classes on 100 points are discovered.
Repairable replicationbased storage systems using resolvable designs
 in Proc. 50th Annual Allerton conference on commun. control and computing
, 2012
"... Abstract—We consider the design of regenerating codes for distributed storage systems at the minimum bandwidth regeneration (MBR) point. The codes allow for a repair process that is exact and uncoded, but tablebased. These codes were introduced in prior work and consist of an outer MDS code followe ..."
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Abstract—We consider the design of regenerating codes for distributed storage systems at the minimum bandwidth regeneration (MBR) point. The codes allow for a repair process that is exact and uncoded, but tablebased. These codes were introduced in prior work and consist of an outer MDS code followed by an inner fractional repetition (FR) code where copies of the coded symbols are placed on the storage nodes. The main challenge in this domain is the design of the inner FR code. In our work, we consider generalizations of FR codes, by establishing their connection with a family of combinatorial structures known as resolvable designs. Our constructions based on affine geometries, Hadamard designs and mutually orthogonal Latin squares allow the design of systems where a new node can be exactly regenerated by downloading β ≥ 1 packets from a subset of the surviving nodes (prior work only considered the case of β = 1). Our techniques allow the design of systems over a large range of parameters. Specifically, the repetition degree of a symbol, which dictates the resilience of the system can be varied over a large range in a simple manner. Moreover, the actual table needed for the repair can also be implemented in a rather straightforward way. Furthermore, we answer an open question posed in prior work by demonstrating the existence of codes with parameters that are not covered by Steiner systems. I.
Candelabra Systems And Designs
, 1999
"... Combinatorial structures called candelabra systems can be used in recursive constructions to build Steiner 3designs. We introduce a new closure operation on natural numbers involving candelabra systems. This new closure operation makes it possible to generalize various constructions for Steiner ..."
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Combinatorial structures called candelabra systems can be used in recursive constructions to build Steiner 3designs. We introduce a new closure operation on natural numbers involving candelabra systems. This new closure operation makes it possible to generalize various constructions for Steiner 3designs and to create new infinite families of Steiner 2designs and 3designs. We provide an independent proof for Wilson's "product theorem" for Steiner 3designs. We also construct new group divisible designs of strength 2 and 3.
Selforthogonal latin squares of all orders n 6
 Bulletin of the American Mathematical Society
, 1974
"... latin square " to denote a latin square orthogonal to its transpose. We have proved that there exists a selforthogonal latin square of order n if and only if n^29 3, 6. Previous literature on this problem ([3], [6][10], [12] [15], [17][19]) had constructed such squares for certain infinite ..."
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latin square " to denote a latin square orthogonal to its transpose. We have proved that there exists a selforthogonal latin square of order n if and only if n^29 3, 6. Previous literature on this problem ([3], [6][10], [12] [15], [17][19]) had constructed such squares for certain infinite