IN MEMORIAM PETER HAMMER (1936 - 2006)

On Dec. 27, 2006, four days after the celebration of his 70th birthday, Peter Ladislau (Laci) Hammer suffered a fatal car accident while driving home from Washington, D.C. He leaves behind his wife of 45 years, Anca, his sons Maxim and Alex Hammer, and his four grandchildren. This is an irreparable loss not only to his family and close friends, but also to his professional colleagues, the Discrete Mathematics and Operations Research communities, to Rutgers University and to many others.

HUNGARIAN ROOTS

Peter was born on Dec. 23, 1936, into a middle class family of Hungarian Jews of Timisoara, Romania. He went to school in his hometown, then earned his degree in mathematics at the University of Bucharest, and in 1959 started working at the Institute of Mathematics of the Academy under the direction of the famous Romanian algebraist Grigore Moisil.

It was at this point that our life trajectories crossed each other. I had just been excommunicated because of a heretical book on economic theory, and was in the process of retraining myself as a mathematician, interested in optimization. Peter and I were brought together by Moisil, who wanted to see operations research developed in Romania and thought that Peter and I could team up to start this development. And so we did.

Peter (at that time Laci) Hammer was 23 years old when I met him. He was extremely bright and mature far beyond his years. I soon befriended him, and despite the 14 years of age difference between us, I was able to relate to him and discuss matters of daily life and politics with him as I would with my older friends.

Early in his childhood, Laci had contracted polio, and that terrible disease had left him with a partial paralysis of his legs. He could walk only with crutches and with great difficulty. As if to compensate for this, nature had endowed him with unlimited resources of energy and vitality; rarely if ever in my life have I encountered a person with such stamina.

For about three years,between the fall of 1959 and the second half of 1962, Laci and I worked closely together and launched what was then considered the beginnings of operations research in Romania.In order to garner some official support for this activity – operations research was a notion completely unknown at the time in Bucharest – we started by solving some practical problems that could demonstrate the usefulness of our approach. Thus, we developed an optimal transportation plan for a category of lumber that had to be shipped regularly throughout the year in large quantities from 36 production sites to seven different consumption centers, and we were able to show that an optimal transportation plan could save about 8 percent of the transportation costs. This created some legitimacy for our work in the eyes of the officialdom and also gave us access to the one computer existing at the time in Romania.

Along with this and some other practical applications, we also started doing research on solution methods for new variants and generalizations of the transportation problem. Between the spring of 1960 and the winter of 1961-62, we wrote a sequence of five papers, developing new solution methods for various models in this area. The papers were first published in Romanian mathematical journals, but we then collected our results into a longer English-language paper, which appeared in two parts in 1962 under the title “On the transportation problem. Part I – Part II” in the journal Cahiers du Centre de Recherche Operationnelle published in Brussels. This paper, among other things, gave efficient methods for solving parametric transportation problems of several types. As it appeared in English and in a Western journal with some circulation in the O.R. community, it “put us on the map,” as they say.

Laci and I wrote our last joint paper in 1962 under the title,“On the generalized transportation problem,” which appeared in Management Science. This paper gave a solution procedure for the model that later became known under the name of minimum-cost flows in networks with gains.

In the midst of our rather intense and often adventurous professional undertakings, I vividly remember the event that changed Laci’s life forever: In the winter of 1960-61, he married Anca Ivanescu, a very bright, lovely young lady whom he had been tutoring in mathematics for a while. 

He also took her family name, as permitted by Romanian law. The reason? Although there was no official discrimination against Jews at the time in Romania, there was a sustained effort to “improve” the ethnic composition of educational and cultural institutions, by reducing the number of Jews in such positions. Thus, as far as a young professional’s career prospects were concerned, Ivanescu, a typical Romanian name, was much preferable to Hammer, a name that only Germans and Jews had. As a result of this name change, all our joint papers published from 1961 on appeared under the authorship of E. Balas and P. L. Ivanescu.

Seven or eight years later,when Laci and Anca defected to the West,he changed back his family name to Hammer. I remember that in the late 1960s, Bill Cooper, at the time a colleague of mine at Carnegie Mellon, asked me what had happened to my former collaborator and co-author, Ivanescu. I told him that Ivanescu was the same as Peter L. Hammer, whose name by that time everybody knew, and explained the reasons for the double name-change. Bill’s reaction was,“What a pity!”

“Why?” I asked.

“Well,”Bill said,“Ivanescu was so exotic, so romantic. Hammer? There are thousands of them.”

There may have been thousands, but few were like this one, as it soon became clear to everybody.

CROSSING THE IRON CURTAIN

Soon after earning his doctorate from the University of Bucharest in 1966, Peter used the opportunity offered by a conference in Bulgaria that he was able to attend together with Anca, to cross the Iron Curtain to Istanbul and from there to Israel. He worked for three years at the Technion in Haifa, then after a brief sojourn at the University of Montreal, in 1972 he accepted a position at the University of Waterloo where he was a professor for more than a decade. In 1983, he moved to Rutgers University, where he became a distinguished professor and director of the newly founded Center for Operations Research, RUTCOR. He worked in this capacity up until the time of his death.

Peter Hammer was first of all a prolific researcher, whose work has influenced numerous areas of discrete mathematics and operations research during the past 45 years. His strongest impact was on nonlinear 0-1 programming, with some striking results for the quadratic case; but he has also obtained many interesting results in graph theory, logic and data mining, to mention just a few of the areas affected by his research. In the last 15 years, he and his collaborators have developed a Boolean approach to discriminant analysis called Logical Analysis of Data, which has had numerous practical applications in areas as diverse as cancer treatment and risk analysis in financial markets. I will say more about Peter’s mathematical discoveries later on.

But Peter Hammer’s impact on the development of operations research theory is not confined to his own discoveries. For the last 35 years, he was a uniquely effective, resourceful and dynamic organizer of scientific activities and disseminator of knowledge. Dr. Hammer is not only the author of more than 220 research papers and scientific monographs, but he is also the founder and editor in chief of several prominent professional journals that he managed to turn into major vehicles for the advancement of our science. The list includes Discrete Mathematics, Discrete Applied Mathematics, Annals of Operations Research, Annals of Discrete Mathematics, and, for the last three years, Discrete Optimization, his latest creation that he was most proud of. He was also a master organizer of timely, well-received and well-attended professional conferences, like Discrete Optimization 77, a number of European Operations Research workshops, summer schools, the ARIDAM series of meetings at RUTCOR, as well as some joint ventures with DIMACS. This fantastically intensive professional activity with its 12-hour workdays and frequent traveling on the part of a person unable to walk or perform any physical activity without the help of another person would have obviously been impossible without the extraordinary self-effacing devotion of Peter’s wife and life companion, Anca.

Peter developed research relations with numerous internationally wellknown institutions and individuals. Apart from his close scientific collaborators whose list includes Sergiu Rudeanu, Dominique de Werra, Bruno Simeone, Uri Peled, Yves Crama, Pierre Hansen, Endre Boros, Stephan Foldes, Toshidide Ibaraki, Alexander Kogan and numerous other well-known researchers, the list of his co-authors ranges over scientists at many major American and Canadian universities, literally all over Europe, from Romania to Italy, from Belgium to Switzerland, from Hungary to Germany, as well as Israel, Japan and China. As a result of the wide recognition that he enjoys through his research results and all his other scientific activities, he was the recipient of honorary doctorates from three distinguished European universities.

THEORETICAL CONTRIBUTIONS

I will now try to summarize some of Peter’s main theoretical contributions. Pseudo-Boolean optimization, Peter Hammer’s main object of study, is coextensive with nonlinear 0-1 programming. It works with set functions represented by closed algebraic expressions or,to put it differently, with real-valued functions of 0-1 variables.A pivotal result of the early 1970s due to Ivo Rosenberg (formerly a colleague of Peter’s in Bucharest), according to which nonlinear 0-1 optimization is polynomially reducible to quadratic 0-1 optimization,moved the latter problem to the center of attention. Using the tools of Boolean algebra, Peter and his co-workers developed a machinery for describing and analyzing the properties of such functions.

Some of their results paralleled those obtained by combinatorialists through different techniques, but they also found numerous new results, which could not be established, or would have been harder to establish, by other means. Some of these results characterize various graphtheoretic structures. For instance, given a graph with n vertices, the problems of finding a maximum stable set or a maximum clique, of finding a minimum vertex cover or a minimum coloring, of finding a maximum cut, and others, can all be expressed as the problem of finding the (unconstrained) minimum of a quadratic pseudo-boolean function of n variables.

While this does not, of course, make these NP-complete problems polynomially solvable, the different viewpoint and set of tools have led to new insights and have sometimes produced strong results that had eluded combinatorial optimizers using polyhedral techniques. For example, after several studies by eminent combinarorialists devoted to the cut polytope, in the early 90’s Peter and his coworkers established a 1-1 correspondence between facets of the latter and certain quadratic pseudo-Boolean functions, that led them to the discovery of a large, new family of facets, with important polynomially separable subfamilies.

As another example,the concept of roof duality,introduced by Hammer and his coworkers in the early 1980s,while on the one hand (nontransparently) equivalent to some earlier duality concepts, has on the other hand led to the discovery several years later of a hierarchy of increasingly tighter lower bounds on the minimum of a quadratic function in 0-1 variables.

To put it succinctly, Peter Hammer has pioneered a new approach to the study of combinatorial objects, off the beaten track, that has led to some remarkable discoveries.

Peter Hammer’s work has touched over the years to varying degrees many areas of operations research: graph theory (split graphs, threshold graphs, domination-graphs, matroid-producing graphs, threshold sequences of graphs, etc.), logic (Horn functions, satisfiability, knowledge compression, establishing cause-effect relationship, disjunctive analogs of submodular and supermodular functions), statistics/data mining (distance-based classification methods, convexity and logical analysis of data, Pareto-optimal patterns and their detection, pattern-based clustering and attribute analysis) and others.

The enormous impact that Peter Hammer’s research and research-related activities have had on our field will live for a long time after his tragically unexpected departure from among us. We will all dourly miss him.