拉兹洛·洛瓦兹László Lovász

拉兹洛·洛瓦兹在布达佩斯接受教育，在中学时就展现出杰出的数学才能。十四岁时，洛瓦兹在Mathematical and Physical Journal for Secondary Schools（1962年）上读到埃尔德什的一篇文章，为之着迷，读了“至少二十遍”。第二年他得以当面见到自己的数学英雄[3]：-

我有幸在1963年作为一名高中生见到了埃尔德什。那时冷战正稍有缓和，他开始越来越频繁地访问匈牙利。……说我从他那里学到了很多，这还是一种保守的说法，不仅是技术意义上的数学，甚至不仅是解题这门精妙艺术的基本要素，还有他追求知识的方式：对问题和部分结果完全开放，这必然带来合作和更广阔的视野。

受埃尔德什的激励，洛瓦兹在1964年、1965年和1966年连续三年赢得国际数学奥林匹克竞赛金牌。洛瓦兹的第一篇论文On graphs not containing independent circuits发表于1965年，当时他十七岁。在这篇论文中，他对任意两个回路都有公共节点的图进行了分类。这不是一篇孤立的论文，因为在接下来的几年里，他发表了On decomposition of graphs（1966年）、Operations with structures（1967年）、Über die starke Multiplikation von geordneten GraphenⓉ（《论有序图的强乘法》）（1967年）、On connected sets of points（1967年）和On chromatic number of finite set-systems（1968年）。Frank Harary对最后提到的这篇论文描述如下：-

埃尔德什在1959年用概率方法证明了对于任意正整数$n, g ≥ 3$，存在一个色数为$n$且围长≥$g$的图，但未能构造出这样的图。通过将问题从图推广到有限集合系统，作者成功地为图获得了这样的构造。

高中毕业后，洛瓦兹在布达佩斯的厄特沃什·罗兰大学学习，并于1970年由布达佩斯的匈牙利科学院授予数学科学候选学位（C.Sc.）。当时，由匈牙利科学院授予的候选学位被认为比大学授予的博士学位更高，因此他先获得这个学位可能看起来令人费解。原因是大学规定不允许学生用在本科期间取得的成果申请博士学位。然而，科学院不存在这样的规定，因为在制定规则时，人们认为本科生永远无法做出足够深度的研究，使他有权申请C.Sc.学位。值得注意的是，到1970年获得候选学位时，洛瓦兹已有十五篇论文发表。在获得任何学位之前，他已在多个国际会议上演讲，并在会议论文集中发表论文，例如：1966年9月在匈牙利巴拉顿湖北岸的蒂豪尼举行的Theory of graphs；1967年5月在德国马内巴赫举行的Beiträge zur GraphentheorieⓉ（图论贡献）；1969年在加拿大卡尔加里大学举行的Combinatorial Structures and their Applications；以及1969年在匈牙利巴拉顿湖北岸的巴拉顿菲赖德举行的Combinatorial Theory and its Applications。

因其杰出成就，洛瓦兹于1970年获得了鲍耶学会颁发的Grünwald Géza奖。次年，他凭借学位论文Factors of Graphs在布达佩斯的厄特沃什·罗兰大学获得了博士学位（Dr.Rher.Nat.）。他的学位论文导师是高洛伊·蒂博尔。随后，他被任命为厄特沃什·罗兰大学的研究助理，从1971年到1975年担任该职位四年。在此期间，他于1972-73年在美国田纳西州纳什维尔的范德堡大学度过了一年。1975年，他转到塞格德的József Attila大学，被任命为讲师（Docent），但三年后，他晋升为教授，担任几何学讲席。1977年，Hungarian Academy of Sciences授予他Dr.Math.Sci.学位。他在1978-79学年在加拿大滑铁卢大学度过，然后在1979年，31岁时成为匈牙利科学院的成员，使他成为匈牙利科学院有史以来最年轻的成员。1983年，他离开塞格德，被任命为布达佩斯厄特沃什·罗兰大学的计算机科学讲席。

1993年，洛瓦兹放弃了厄特沃什·罗兰大学的计算机科学讲席，但保留了那里的教授职位。他离开布达佩斯的讲席，前往美国耶鲁大学担任William K Lanman计算机科学与数学教授。1999年，他离开学术界，担任微软研究院的高级研究员，但于2006年回到匈牙利，成为厄特沃什·罗兰大学数学研究所所长。

他的研究涉及组合优化、算法、复杂性、图论和随机游走方面的深刻结果，这些领域跨越数学和理论计算机科学。他最著名的可能是广泛使用的“LLL算法”，该算法以Arjen K Lenstra、Hendrik W Lenstra和洛瓦兹命名，他们首次在联合论文Factoring Polynomials with Rational Coefficients（1982）中给出了该算法。该算法为点格提供了一种有效的基约化方法。我们通过引用他于1999年获得的沃尔夫奖的颁奖词[2]来说明洛瓦兹的贡献：-

洛瓦兹在离散数学中取得了突破性的成果，这些成果对纯数学和应用数学的其他领域以及理论计算机科学都有重要应用。他通过引入依赖于几何多面体和拓扑技术的深刻数学方法，解决了几个悬而未决的问题，包括完美图猜想、Kneser猜想以及五边形的克劳德·香农容量的确定。他的算法思想——包括椭球方法在组合优化中的应用、格基约化算法、拟阵奇偶算法以及改进的体积计算程序——都对理论计算机科学产生了深远影响。洛瓦兹还对NP的PCP刻画及其与近似困难性的联系做出了贡献。他的“局部引理”是概率方法发展中的主要早期成果之一。他全面的著作和引人入胜的讲座激发了世界各地的数学研究。

在考察授予洛瓦兹的进一步荣誉和奖项之前，我们详细介绍他撰写的杰出著作，包括研究专著和教学教材。1979年，他出版了Combinatorial problems and exercises（1979）。我们引用这部经典文本的前言：-

组合数学在数学科学的边缘沉寂了几个世纪之后，如今已蓬勃发展成数学中增长最快的分支之一——如果我们考虑这一领域的出版物数量、它在数学其他分支和其他科学中的应用，以及科学家、经济学家和工程师对组合结构的兴趣，这无疑是如此。数学界曾被代数与分析的成功所吸引，直到近年来才清楚，主要是由于经济学、统计学、电气工程和其他应用科学中产生的问题，组合数学，即对有限集合和有限结构的研究，有它自己的问题和原理。这些原理独立于代数与分析中的原理，但在难度、实践和理论意义以及美感上可与它们匹敌。然而，许多一流数学家对组合数学的看法仍然带有贬义。他们虽然承认它的趣味性和难度，却否认它的深度。人们常常有力地宣称，组合数学是一堆问题，这些问题本身可能有趣，但彼此没有联系，也不构成一个理论。在组合数学或图论中很容易获得新结果，因为需要学习的技术很少，这导致出版物数量快速增长。上述指责显然是任何科学领域在其发展早期——即收集数据阶段——的特征。只要主要问题尚未被表述出来，向一般层次的抽象尚未完成，就无法区分有趣的结果和不太有趣的结果——除非基于审美标准，而这当然太主观了。上述被指责缺失的那些技术，正等待着它们的发现者。所以不发达不是反对、反而是支持引导年轻科学家进入某一领域的理由。在我看来，组合数学现在正在走出这个早期阶段。有技术需要学习：枚举技术、拟阵论、概率方法、线性规划、区组设计构造等。有些分支由构成层次结构的定理组成，并包含构成研究骨干的中心结构定理：图的连通性（网络流）或图的因子，仅从图论中举两个例子。有些概念从许多非平凡结果中抽象出来，统一了理论的大部分，例如拟阵或良好刻画的概念。我的感觉是，不了解这些事实、概念和技术，已不再可能获得重要结果。（当然，例外可能发生，因为这个领域注定要覆盖数学世界中如此大的一部分，以至于全新的问题仍可能出现。）

Frank Harary 对这本书的评论是这样开始的：-

这本书是一部经典。作者精妙地分析了607个不同结果中所使用的证明技巧，这些结果被划分为……十五节。

1993年，洛瓦兹 推出了第二版，并在前言中写道：-

当本书的出版商要求我修订和更新我的问题集以出版第二版时，我不得不决定要改动多少，考虑到该领域的快速发展（但也考虑到第一版已经绝版）。组合数学在过去十年中发展了很多，尤其是在那些与其他数学分支相互作用的领域，如多面体组合数学、代数组合数学、组合几何、随机结构，以及最显著的是算法组合数学和复杂性理论。（计算理论在组合数学中有如此多的应用，反之亦然，以至于有时很难划清它们之间的界限。）但组合数学是一门独立的学科，这使得这本习题集（经过一些更新）仍然有效。我决定不改变本书的结构和主要主题。任何概念上的改变（例如一致地引入算法问题，连同算法分析和算法问题的复杂性分类）都意味着写一本新书。然而，我无法抗拒编写一系列关于图上随机游走及其与特征值、膨胀性质和电阻的关系的习题（这一领域有经典根源，但在过去几年中爆炸性增长）。

洛瓦兹 的其他书有（与 Michael D Plummer 合著）Matching theory（1986），关于这本书 Knut Richter 写道：-

这本优秀的书主要面向图论、组合学和离散优化的业内人士。

同样在1986年，An algorithmic theory of numbers, graphs and convexity出版了。Arjen Lenstra写道：-

这本书对图论、数论和组合优化等领域中计算问题高效求解的一些近期进展作了很好的综述。

两年后，洛瓦兹（与Martin Grötschel和Alexander Schrijver合作）出版了Geometric algorithms and combinatorial optimization（1988）；第二版于1993年问世。Jürgen Köhler评论了第一版：-

作者们综述了数学优化的现代发展，讨论了经典结果并给出了许多新结果。……作者们在高层次上给出了该主题的大量结果，并赋予它们最佳形式：这是一部最优的专著。

接下来，他与Bernhard Korte和Rainer Schrader合作，出版了Greedoids（1991）。Ulrich Faigle解释道：-

拟阵有兩個根本不同的方面。一方面，可以将拟阵视为满足拟阵独立性公理之松弛的集合系统，即并非独立集的每个子集都必然独立。从这个角度看，拟阵的结构理论归结为在这样的松弛公理假设下，拟阵理论还能保留多少。第二个方面将拟阵视为字母表上的有限语言，这些语言对取词的前缀封闭，并具有这样的性质：任何非极大词都可以用任何更长词中的某个字母来扩充。

2003年，他与Josef Pelikan和Katalin L Vesztergombi合著了教科书Discrete mathematics。Robin Wilson写道：-

近年来出现了大量离散数学教科书，旨在平衡学院和大学中对微积分的过度强调。这本书是这些教科书中的一个受欢迎的补充，由该领域三位知名实践者以令人愉快的非正式风格写成。

除了上面提到的Grünwald Géza奖之外，洛瓦兹还获得了：1979年的乔治·波利亚奖；1981年IEEE的最佳信息论论文奖；Ray D Fulkerson奖，与Lovasz、Grötschel和Schrijver共同因其论文The ellipsoid method and its consequences in combinatorial optimisation（1981）而获得；1985年匈牙利国家奖；1992年鲍耶学会的Tibor Szele奖章；1993年荷兰数学会的勒伊岑·布劳威尔奖章；1998年匈牙利国家功勋勋章；1998年捷克数学会的伯纳德·波尔查诺奖章；1999年以色列沃尔夫奖；以及1999年的高德纳奖。该奖项的引文如下：-

洛瓦兹对算法理论产生了巨大影响。他做出了基础性的发现，这些发现已成为理论计算机科学中的标准工具。洛瓦兹局部引理、格基约化——在格中寻找短向量，以及椭球方法在各种凸规划问题中的应用，都已成为算法和复杂性广泛领域中的标准工具。洛瓦兹在近似困难性与概率证明之间联系方面的贡献至关重要。除了在算法方面的基础性贡献外，Laci 洛瓦兹还撰写了许多优美的著作，所有这些著作都强调各种主题中的算法。

继续本千年所授予的荣誉，他于2001年获得匈牙利政府颁发的Corvin链，并因与Uriel Feige、Shafi Goldwasser、Shmuel Safra和Mario Szegedy合写的论文Interactive Proofs and the Hardness of Approximating Cliques而于2001年获得库尔特·弗雷德里希·哥德尔奖。2006年，运筹学与管理科学研究所将冯·诺伊曼理论奖授予洛瓦兹。该奖项附有以下引文：-

洛瓦兹 曾在匈牙利和美国的大学以及微软研究院任职，目前是布达佩斯 厄特沃什·罗兰 大学数学研究所所长。他于1972年证明完美图猜想时年仅24岁，由此首次广为人知。1979年，他通过为图的顶点分配向量并构建相关的半定规划问题，解决了编码理论中 克劳德·香农 的一个长期未解问题；该方法此后成为攻克组合优化问题的有力工具。1991年，他与 Schrijver 展示了提升与投影方法在0-1整数规划问题中的威力，以及半定规划技术获得紧松弛的潜力。洛瓦兹 还利用新颖技术对图论的许多主题、随机算法以及子模函数最小化做出了关键贡献。洛瓦兹 将于明年成为国际数学联盟主席。他是匈牙利、欧洲、俄罗斯和荷兰科学院的院士。

洛瓦兹于2007年获得鲍耶研究奖，同年当选为瑞典科学院会士，并于2008年3月15日获得匈牙利塞切尼大奖。他于2008年8月获得欧洲研究理事会的高级资助。2008年11月，洛瓦兹获得鲍耶基金会的鲍耶大奖，并于2009年7月3日当选为伦敦数学会的名誉成员。2010年，他获得稻盛基金会颁发的京都奖，2012年获得美国数学会富尔克森奖，2019年获得巴塞罗那希帕蒂娅欧洲科学奖。

见THIS LINK。

最后我们提到，1990年，他是京都举行的国际数学家大会的全会报告人。他的演讲Geometric algorithms and algorithmic geometry出现在会议论文集中，也以视频形式发布。1988年，他还在埃克塞特举行的英国数学讨论会上作全会报告，当时他演讲的主题是Lattice points in convex bodies。

自2007年1月1日起，他一直担任国际数学联盟的主席。他已婚，有四个孩子。

Laszlo Lovász was educated in Budapest where he showed outstanding ability in mathematics at secondary school. When he was fourteen years old, Lovász came across an article by Paul Erdős in the Mathematical and Physical Journal for Secondary Schools (1962) and was so enchanted that he read it "at least twenty times". In the following year he was able to meet his mathematical hero in person [3]:-

I had the great fortune to meet Paul Erdős as a high school student in 1963. In those days the cold war was quieting down a little, and he began to visit Hungary more and more often. ... It is an understatement to say that I have learned a lot from him, not only mathematics in the technical sense, and not even only elements of the fine art of problem solving, but also his way of pursuing knowledge: complete openness in problems and partial results, which necessarily leads to collaboration and a wider perspective.

Inspired by Erdős, Lovász won gold medals in the International Mathematical Olympiad competition in each of the three years 1964, 1965 and 1966. Lovász's first paper On graphs not containing independent circuits was published in 1965 when he was seventeen years old. In this paper he classified graphs in which any two circuits have a common node. This was not an isolated paper for, in the next couple of years, he published On decomposition of graphs (1966), Operations with structures (1967), Über die starke Multiplikation von geordneten Graphen Ⓣ (1967), On connected sets of points (1967), and On chromatic number of finite set-systems (1968). Frank Harary describes this last mentioned paper as follows:-

P Erdős proved by probabilistic methods [in 1959] that for any positive integers $n, g ≥ 3$, there exists a graph with chromatic number $n$ and girth ≥ $g$, but was unable to construct such graphs. By generalizing the question from graphs to finite set-systems, the author succeeds in obtaining such a construction for graphs.

After graduating from high school, Lovász studied at Eötvös Loránd University in Budapest and he was awarded a Candidate Degree of Mathematical Science (C.Sc.) in 1970 by the Hungarian Academy of Sciences in Budapest. At this time the Candidate Degree awarded by the Hungarian Academy of Sciences was considered a higher degree than a doctorate awarded by a university so it may appear puzzling that he received this degree first. The reason was that university regulations did not allowed a student to apply for a Ph.D. with results achieved during their undergraduate years. No such rules existed in the Academy of Sciences, however, because when the rules were drawn up it was assumed that an undergraduate would never be able to produce research of sufficient depth to entitle him to apply for the C.Sc. degree. Remarkably, Lovász had fifteen papers in print by the time he was awarded his Candidate Degree in 1970. Before the award of any degree, he had lectured at several international conferences and published papers in conference proceedings such as: Theory of graphs held in Tihany, on the northern shore of Lake Balaton, in Hungary in September 1966; Beiträge zur Graphentheorie Ⓣ held in Manebach, Germany in May 1967; Combinatorial Structures and their Applications held at the University of Calgary in Canada in 1969; and Combinatorial Theory and its Applications held in Balatonfüred on the northern shore of Lake Balaton, in Hungary in 1969.

For his outstanding achievements, Lovász received the Grünwald Géza Prize from the Bolyai Society in 1970. In the following year he was awarded a doctorate (Dr.Rher.Nat.) from Eötvös Loránd University in Budapest for his thesis Factors of Graphs. His thesis advisor was Tibor Gallai. He was then appointed as a research assistant at Eötvös Loránd University, an appointment he held for four years from 1971 to 1975. During this time he spent the year 1972-73 at Vanderbilt University in Nashville, Tennessee in the United States. In 1975 he moved to the József Attila University in Szeged where he was appointed as a Docent but, three years later, he was promoted to professor filling the Chair of Geometry. In 1977 the Hungarian Academy of Sciences awarded him the degree Dr.Math.Sci. He spent the academic year 1978-79 at the University of Waterloo in Canada then in 1979, at the age of thirty-one, he became a member of the Hungarian Academy of Sciences making him by far the youngest member ever of the Hungarian Academy of Sciences. He left Szeged in 1983 when appointed to the Chair of Computer Science at Eötvös Loránd University in Budapest.

In 1993 Lovász gave up the Chair of Computer Science at Eötvös Loránd University but retained a professorship there. He left the Chair in Budapest to take up the William K Lanman professorship of Computer Science and Mathematics at Yale University in the United States. In 1999 he left academia to take up the position of Senior Researcher at Microsoft Research but he returned to Hungary in 2006 to become Director of the Mathematical Institute of Eötvös Loránd University.

His research involves deep results on combinatorial optimization, algorithms, complexity, graph theory, and random walks, areas which span mathematics and theoretical computer science. He is perhaps best known for the widely used 'LLL algorithm' named after Arjen K Lenstra, Hendrik W Lenstra and László Lovász who first gave the algorithm in their joint paper Factoring Polynomials with Rational Coefficients (1982). The algorithm gives an efficient basis reduction method for point lattices. We give an indication of Lovász's contributions by quoting from the citation for the Wolf Prize which he received in 1999 [2]:-

László Lovász has obtained ground-breaking results in discrete mathematics that have had significant applications to other areas of pure and applied mathematics as well as to theoretical computer science. He solved several outstanding problems, including the perfect graph conjecture, Kneser's conjecture, and the determination of the Shannon capacity of the pentagon, by introducing deep mathematical methods relying on geometric polyhedral and topological techniques. His algorithmic ideas - including applications of the ellipsoid method in combinatorial optimization, the lattice basis reduction algorithm, the matroid parity algorithm, and the improved procedures for volume computation - all had profound influence on theoretical computer science. Lovász also contributed to the PCP characterization of NP and its connection to the hardness of approximation. His "Local Lemma" is one of the main early results in the development of the probabilistic method. His comprehensive books and fascinating lectures have stimulated mathematical research around the world.

Before looking at further honours and prizes awarded to Lovász we give details of the outstanding books, both research monographs and teaching texts, he has written. In 1979 he published Combinatorial problems and exercises (1979). We quote from the Preface of this classic text:-

Having vegetated on the fringes of mathematical science for centuries, combinatorics has now burgeoned into one of the fastest growing branches of mathematics - undoubtedly so if we consider the number of publications in this field, its applications in other branches of mathematics and in other sciences, and also the interest of scientists, economists and engineers in combinatorial structures. The mathematical world was attracted by the successes of algebra and analysis and only in recent years has it become clear, due largely to problems arising from economics, statistics, electrical engineering and other applied sciences, that combinatorics, the study of finite sets and finite structures, has its own problems and principles. These are independent of those in algebra and analysis but match them in difficulty, practical and theoretical interest and beauty. Yet the opinion of many first-class mathematicians about combinatorics is still in the pejorative. While accepting its interest and difficulty, they deny its depth. It is often forcefully stated that combinatorics is a collection of problems which may be interesting in themselves but are not linked and do not constitute a theory. It is easy to obtain new results in combinatorics or graph theory because there are few techniques to learn, and this results in a fast-growing number of publications. The above accusations are clearly characteristic of any field of science at an early stage of its development - at the stage of collecting data. As long as the main questions have not been formulated and the abstractions to a general level have not been carried through, there is no way to distinguish between interesting and less interesting results - except on an aesthetic basis, which is, of course, too subjective. Those techniques whose absence has been disapproved of above await their discoverers. So underdevelopment is not a case against, but rather for, directing young scientists toward a given field. In my opinion, combinatorics is now growing out of this early stage. There are techniques to learn: enumeration techniques, matroid theory, the probabilistic method, linear programming, block design constructions, etc. There are branches which consist of theorems forming a hierarchy and which contain central structure theorems forming the backbone of study: connectivity of graphs (network flows) or factors of graphs, just to pick two examples from graph theory. There are notions abstracted from many nontrivial results, which unify large parts of the theory, such as matroids or the concept of good characterization. My feeling is that it is no longer possible to obtain significant results without the knowledge of these facts, concepts and techniques. (Of course, exceptions may occur, since the field is destined to cover such a large part of the world of mathematics that entirely new problems may still arise.)

Frank Harary began a review of the book as follows:-

This book is a classic. The author masterfully analyzes the proof techniques utilized in 607 different results partitioned into ... fifteen sections.

In 1993 Lovász produced a second edition, writing in the Preface:-

When the publishers of this book asked me to revise and update my problem book for a second edition, I had to decide how much to change, taking into consideration the fast development of the field (but also that the first edition was out of print). Combinatorics has grown a lot in the last decade, especially in those fields interacting with other branches of mathematics, like polyhedral combinatorics, algebraic combinatorics, combinatorial geometry, random structures and, most significantly, algorithmic combinatorics and complexity theory. (The theory of computing has so many applications in combinatorics, and vice versa, that sometimes it is difficult to draw the border between them.) But combinatorics is a discipline in its own right, and this makes this collection of exercises (subject to some updating) still valid. I decided not to change the structure and main topics of the book. Any conceptual change (like introducing algorithmic issues consistently, together with an analysis of the algorithms and the complexity classification of the algorithmic problems) would have meant writing a new book. I could not resist, however, working out a series of exercises on random walks on graphs, and their relations to eigenvalues, expansion properties, and electrical resistance (this area has classical roots but has grown explosively in the last few years).

Other books by Lovász are (with Michael D Plummer) Matching theory (1986) about which Knut Richter writes:-

This excellent book is mainly addressed to insiders of graph theory, combinatorics and discrete optimization.

Also in 1986, An algorithmic theory of numbers, graphs and convexity was published. Arjen Lenstra writes:-

This book presents a nice survey of some recent developments towards the efficient solution of computational problems in areas like graph theory, number theory, and combinatorial optimization.

Two years later Lovász published (with Martin Grötschel and Alexander Schrijver) Geometric algorithms and combinatorial optimization (1988); a second edition appeared in 1993. Jürgen Köhler reviewed the first edition:-

The authors review modern developments in mathematical optimization, discuss classical results and present many new results. ... The authors present, on a high level, a great number of results on this topic and give them their best form: this is an optimal monograph.

Next, written in collaboration with Bernhard Korte and Rainer Schrader, he published Greedoids (1991). Ulrich Faigle explains:-

There are two fundamentally different aspects of greedoids. One may think of greedoids as set systems obeying a relaxation of the matroid independence axioms in that not necessarily every subset of an independent set is independent. Viewed from this angle, structural theory of greedoids amounts to the question of how much of matroid theory is retained under this relaxed axiomatic assumption. The second aspect sees greedoids as finite languages over alphabets that are closed with respect to taking prefixes of words and have the property that any nonmaximal word may be augmented with some letter in any longer word.

In 2003 he wrote the textbook (with Josef Pelikan and Katalin L Vesztergombi) Discrete mathematics. Robin Wilson writes:-

In recent years there has been a plethora of textbooks on discrete mathematics, designed as a counter-balance to the over-emphasis on calculus in colleges and universities. This book is a welcome addition to these, being written in a delightfully informal style by three well-known practitioners in the field.

In addition to the Grünwald Géza Prize mentioned above, Lovász was awarded: the George Pólya Prize in 1979; the Best Information Theory Paper Award from the IEEE in 1981; the Ray D Fulkerson Prize awarded jointly to Lovasz, Grötschel and Schrijver for their paper The ellipsoid method and its consequences in combinatorial optimisation (1981); the State Prize, Hungary in 1985; the Tibor Szele Medal from the Bolyai Society in 1992; the Brouwer Medal from the Dutch Mathematical Society in 1993; the National Order of Merit of Hungary in 1998; the Bolzano Medal from the Czech Mathematical Society in 1998; the Wolf Prize, Israel in 1999; and the Knuth Prize in 1999. The citation for this award reads:-

Lovász had an enormous influence on the theory of algorithms. He has made fundamental discoveries that have became standard tools in theoretical computer science. The Lovász Local lemma, Lattice Basis Reduction - finding short vectors in lattices, and the application of the ellipsoid method for various convex programming problems have all become standard tools in a wide range of areas of algorithms and complexity. Lovász's contribution to the connection between hardness of approximation and probabilistic proofs was essential. In addition to his fundamental contributions in algorithms Laci Lovász has also written a number of beautiful books all emphasizing algorithms in a variety of topics.

Continuing to honours given in the present millennium, he received the Corvin Chain from the Hungarian Government in 2001 and the Gödel Prize in 2001 for the paper Interactive Proofs and the Hardness of Approximating Cliques written with Uriel Feige, Shafi Goldwasser, Shmuel Safra and Mario Szegedy. In 2006 the Institute for Operations Research and the Management Sciences awarded Lovász its John von Neumann Theory Prize. The award came with the following citation:-

László Lovász has held positions at universities in Hungary and the US as well as at Microsoft Research and is presently the director of the Mathematical Institute of the Eötvös Loránd University in Budapest. He first became well known when he proved the Perfect Graph Conjecture in 1972, at the age of 24. In 1979 he solved a long-standing problem of C Shannon in coding theory by assigning vectors to the vertices of a graph and formulating an associated semidefinite programming problem; the approach has since become a powerful tool in attacking combinatorial optimization problems. In 1991, he and Schrijver showed the power of lift-and-project methods in 0-1 integer programming problems and the potential of semidefinite programming techniques to obtain tight relaxations. Lovász has also made key contributions to many topics in graph theory using novel techniques, to randomized algorithms and to submodular function minimization. Lovász will become the President of the International Mathematical Union next year. He is a member of the Hungarian, European, Russian and Dutch Academies of Sciences.

Lovász was awarded the János Bolyai Research Prize in 2007, elected to the Swedish Academy of Sciences in the same year, and awarded Hungary's Széchenyi Grand Prize on 15 March 2008. He received the Advanced Grant of the European Research Council in August 2008. In November 2008 Lovász was awarded the Bolyai Grand Prize of the János Bolyai Foundation and, on 3 July 2009, he was elected an honorary member of the London Mathematical Society. In 2010 he was awarded the Kyoto Prize by the Inamori Foundation, in 2012 he was awarded the American Mathematical Society Fulkerson prize and in 2019 he was awarded the Barcelona Hypatia European Science Prize.

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Finally we mention that, in 1990, he was a plenary speaker at the International Congress of Mathematicians held in Kyoto. His lecture Geometric algorithms and algorithmic geometry appeared in the conference proceeding and also as a video. He was also a plenary speaker at the British Mathematical Colloquium at Exeter in 1988 when he lectured on Lattice points in convex bodies.

He has served as president of the International Mathematical Union since January 1, 2007. He is married with four children.