numerative ombinatorics

A pplications Enumerative and Applications ECA 1:1 (2021) Interview #S3I4 ecajournal.haifa.ac.il

Interview with Toufik Mansour

Maria Chudnovsky received a B.Sc. in mathematics from the Technion, and her Ph.D. in 2003 from un- der the supervision of Paul Seymour. After postdoctoral re- search at the Clay Mathematics Institute, she became an assis- tant professor at Princeton University in 2005, and moved to in 2006. By 2014, she was the Liu Family Professor of Industrial Engineering and Operations Research at Columbia. In 2015, she returned to Princeton as a professor of Photo by Theresa T-R McLaughlin mathematics. In 2004, Chudnovsky was named one of the “Brilliant 10” by Popular Science magazine. Her work on the strong theorem won for her and her co-authors the 2009 . In 2013 she was awarded the MacArthur grant. Professor Chudnovsky is a leading expert on and contributions to the field including the proof of the strong (with Neil Robertson, Paul Seymour, and Robin Thomas), the first polynomial-time algorithm for recognizing perfect graphs, and of a structural characterization of the claw-free graphs.

Mansour: Professor Chudnovsky, first of all ory. What that means is that you give me a we would like to thank you for accepting this class of graphs with certain properties, and I interview. Would you tell us broadly what try to come up with a description of its mem- combinatorics is? bers. If I succeed, then I can answer questions Chudnovsky: I think there are almost as about your class of graphs, because I now com- many answers to this question as there are pletely understand how they are built. How- combinatorialists... I usually say that combi- ever, often coming up with explicit structure is natorics is the study of discrete patterns. too hard (or maybe impossible), so then there Mansour: What do you think about the de- are compromises. First you ask me a question velopment of the relations between combina- about a class of graphs, then I try to under- torics and the rest of mathematics? stand enough about how they are built-in or- Chudnovsky: I think this is a wide two-way der to answer your question. I mostly work on street. Obviously many questions in all areas classes of graphs that are defined by forbidding of math lead to understanding a pattern of ob- certain induced subgraphs. jects; and conversely combinatorial questions Mansour: We would like to ask you about can be encoded as instances of other problems, your formative years. What were your early to which solutions are known by methods from experiences with mathematics? Did that hap- different fields. Examples of that are topolog- pen under the influence of your family, or some ical methods, algebraic methods, the proba- other people? bilistic method, and many others. Chudnovsky: I grew up in a home where Mansour: What have been some of the main mathematics was treated with utmost respect; goals of your research? it was the most important subject in school, Chudnovsky: I study structural graph the- and in the world... I also went to a special The authors: Released under the CC BY-ND license (International 4.0), Published: November 3, 2020 Toufik Mansour is a professor of mathematics at the University of Haifa, . His email address is [email protected] Interview with Maria Chudnovsky math school in St Petersburg (school #30; I achieved for all induced subgraphs of G, then G am guessing more than a few of your readers is perfect). Perfect graphs were defined by the are its graduates), which was a fantastic place French mathematician in 1961, to get an introduction to the subject. After my and they have some amazing properties and family came to Israel (I was 13 at the time) connections. First of all, many seemingly un- I studied math in the Columbia program at related theorems in graph theory and combina- the Leo Baeck High School in Haifa; I also at- torics can be rephrased as stating that graphs tended the Technion Math Circle. All of these in a certain family are perfect. Secondly, there were amazing experiences, each taught me dif- are algorithmic problems that are hard (NP- ferent kinds of math. More importantly, each complete) in general, but that can be solved taught me how to see the beauty of mathemat- efficiently (in polynomial time) if the input ics and gave me the first taste of the experience graph is perfect. It also turns out that be- of thinking about a problem. ing perfect is invariant under taking comple- Chudnovsky: Were there specific problems ments (the complement Gc of G is the graph that made you first interested in combina- with the same vertex set as G, and two ver- torics? tices are adjacent in G if and only if they are Chudnovsky: Absolutely! If six people come non-adjacent in Gc). Berge conjectured this in- to a party, then either there are three who variance when he defined perfect graphs, call- know each other or three who do not. Five ing it “The Weak Perfect Graph Conjecture”; are not enough. I heard this problem when I L´aszl´oLov´aszproved it in 19721. In addition was about 14 or 15; and I have never been the to this conjecture, Berge also proposed a list same person again. of minimal (under taking induced subgraphs) Mansour: What was the reason you chose graphs that are not perfect. This list consisted Princeton University for your Ph.D. and your of all odd cycles of length at least five, and their advisor, Paul Seymour? complements; and the “Strong Perfect Graph Chudnovsky: I asked my M.Sc. adviser Ron Conjecture” of Berge states that a graph if per- Aharoni where to apply for my Ph.D. He rec- fect if and only if it does not contain (as an in- ommended a few places. I applied there, and duced subgraph) any of the graphs in this list. got into some and not others. The rest is his- One direction of this conjecture is an easy exer- tory... cise: it is not hard to see that for each graph in Mansour: How was the mathematics at the list, the chromatic number is strictly big- Princeton at that time? ger than the number, and therefore they Chudnovsky: It was (and I hope still is) an cannot appear as in a perfect unbelievably vibrant and active place. There graph. The other direction turned out to be a was a charge in the air (or at least that is how lot harder; it remained open for over 40 years, it felt to me as a first-year graduate student) attracting a lot of attention, and leading to the of all these amazing conjectures about to be development of a lot of beautiful directions in conquered. I was thrilled to be allowed to be graph theory and combinatorial optimization. a part of it. In 2003 Paul Seymour, Neil Robertson, Robin Mansour: Your Ph.D. work was on perfect Thomas and I proved this conjecture and that graphs and the result, known as the strong per- is the Strong Perfect Graph Theorem. fect graph theorem, is considered by many ex- Mansour: Would you comment about why perts in the field as one of the most important the strong perfect graph theorem has been so recent developments in mathematics. Would influential? you tell us about perfect graphs and your re- Chudnovsky: I think it is because so much sult? theory was built motivated by finding its so- Chudnovsky: A graph G is perfect if for ev- lution. Also, as I said earlier, many facts in ery induced subgraph H of G, the chromatic several fields can be stated in the language of number of H equals the clique number of H. perfection, so having an equivalent, and eas- (The clique number is an obvious lower bound ier to verify notion is very helpful under many for the chromatic number; if this lower bound circumstances. 1See https://www.sciencedirect.com/science/article/pii/0095895672900457?via%3Dihub

ECA 1:1 (2021) Interview #S3I4 2 Interview with Maria Chudnovsky

Mansour: What would guide you in your re- Mansour: Are you also interested in some search? A general theoretical question or a enumerative combinatorial questions? Do enu- specific problem? merative techniques play a role in your re- Chudnovsky: There is no one answer, and search? most of the time it is somewhere in between. Chudnovsky: There are many beautiful ques- Mansour: When you are working on a prob- tions there, but I have not worked on them. lem, do you feel that something is true even Mansour: What do you think about the dis- before you have the proof? tinction between pure and applied mathemat- Chudnovsky: Definitely. Sometimes it then ics that some people focus on? Is it meaningful turns out to be false ... at all in your own case? How do you see the Mansour: What are the top three open ques- relationship between so-called “pure” and “ap- tions in your list? plied” mathematics? - The Erdos-Hajnal¨ Conjecture: This is a con- Chudnovsky: I do not really believe in pure jecture that states that for every graph H there and applied mathematics; only in mathematics is an (H) ∈ (0, 1) such that every n-vertex and its applications. graph with no induced subgraph isomorphic to Mansour: What advice would you give to H has a clique or a stable set of size at least young people thinking about pursuing a re- n(H). search career in mathematics? - Hadwigger’s Conjecture: This one says that Chudnovsky: There is zero reasons not to a graph that does not contains the complete try. If you succeed, you will have a wonder- graph on t+1 vertices as a minor is t-colorable. ful professional life. If not, you can get a great - A combinatorial algorithm to find the job outside of the pure research world using the largest clique in a perfect graph that runs skills you have accumulated, and again be very in time polynomial in the number of ver- happy with your career. Doing mathematics tices of the input. This problem of finding a teaches one to think clearly and critically, and largest clique in a perfect graph is known to that is a sought after commodity in every area be polynomial-time solvable (this is a result of intellectual pursuit. of Grotschel, Lov´aszand Schrijver), but the Mansour: While we see that there are more algorithm uses techniques from combinatorial women in science and technology fields today optimization, such as the . I than ever before, bias still affects women in would like to find an algorithm that can be their scientific careers. What do you think formulated entirely in terms of the graph. about this issue? Mansour: What kind of mathematics would Chudnovsky: I think things have been you like to see in the next ten-to-twenty years steadily getting better, and I am very hopeful. as the continuation of your work? It is a pleasure to see more and more women at Chudnovsky: I would like to see more graph conferences and workshops; I hope that within structure theorems, and algorithms based on my lifetime a day will come when we will be them. I also hope that we will be able to un- able to take it for granted. The other day my derstand better the behavior of the chromatic seven-year-old son asked me: Mommy, why did number of graphs with certain induced sub- people once think that girls are not good in graphs forbidden. math? I hope this is a reflection of what we Mansour: What would you say about some will see 20 years from now. of the major directions in graph theory for the Mansour: Would you tell us about your in- next two decades? terests besides mathematics? Chudnovsky: This is too hard to answer; I Chudnovsky: I love art and reading. I love will pass on this one. walking around beautiful cities. Mansour: Do you think that there are core or Mansour: Before we close this interview with mainstream areas in mathematics? Are some one of the foremost experts in combinatorics, topics more important than others? we would like to ask some more specific mathe- Chudnovsky: I definitely think that some matical questions. You have worked on several topics are more important than other, but only graph theoretic problems throughout your ca- time will tell which ones they are. reer. What kind of graph questions are your

ECA 1:1 (2021) Interview #S3I4 3 Interview with Maria Chudnovsky favorite? picture is correct. Sometimes it is not, so I Chudnovsky: I like questions where there is try to modify the world I have been imagining. an underlying phenomenon that you can un- And so I go around in what feels like “concen- derstand, and then use this understanding to tric circles” until either I find a solution, or I derive answers to questions. But I suspect that give up. Sometimes I cannot prove everything this is true for most people doing research. I imagined about A, but enough to get B. Mansour: How would you describe doing re- Mansour: problems enjoy search in graph theory? Does someone need to many practical applications as well as theoreti- have a strong intuition or formidable technical cal challenges and it is still a very active field of abilities to solve difficult questions in the field? research. Would you briefly describe the main Chudnovsky: I will answer this in short: yes; points in such problems and comment on some you need both. future directions? Mansour: Settling algorithmic complexity of Chudnovsky: One aspect of graph coloring some graph-theoretic questions is also very im- is something called “χ-boundedness”. A class portant and difficult. Would you tell us about C of graphs, closed under taking induced sub- your related works and point out some future graphs, is called χ-bounded if there is a func- directions? tion fC such that for every G ∈ C we have Chudnovsky: I see many algorithmic results χ(G) ≤ fC(ω(G)) (here χ(G) denotes the chro- as a good concise way to “package” a lot of matic number of G, and ω(G) is the size of information about a certain class of graphs. I largest clique in G). The function fC is called know this, and this, and that.. and you can a χ-bounding function for C. Thus perfect not really write it as a theorem. But because graphs are a χ-bounded class with the iden- of this, this and that I can efficiently approx- tity being a χ-bounding function. The class imate the clique number—and suddenly you of all graph is not χ-bounded: there are sev- have an interesting result to tell people about. eral constructions of graphs with no clique of Most of my algorithmic result have that flavor: size three, and with arbitrarily large chromatic I can prove enough about the structure of the numbers. So it is a deep and elegant question input that certain algorithms follow. to ask: which graph classes are χ-bounded? Mansour: One of the main tools you used to There has been a lot of progress on this re- prove the strong perfect graph theorem was a cently, and I hope there is more to come. technical result, as you called the “wonderful There are also beautiful algorithmic ques- lemma” (first proved by F. Roussel and P. Ru- tions. In general, it is NP-complete to compute bio2). In 2018, you presented a very elegant (or even approximate) the chromatic number yet shorter proof of this lemma. Why do you of a graph, and even to answer if a graph is call it “wonderful lemma”? colorable with a fixed number k of colors. But Chudnovsky: Because it is a wonderful pow- what if some graph H is forbidden as an in- erful tool that allows you to generalize some duced subgraph? It turns out that usually very simple facts about perfect graphs to much that does not help: as long as some compo- deeper statements. It is also used all the time nent of H is not a path, the problem remains in the proof of The Strong Perfect Graph The- NP-complete. So what about taking H to be orem. a path on t vertices? Now for every k and t Mansour: Would you tell us about your there is a problem: what is the complexity de- thought process for the proof of one of your ciding if a graph with no induced path on t favorite results? How did you become inter- vertices can be k-colored? And we know the ested in that problem? How long did it take answer if k ≥ 4 or t ≤ 7. The only open case you to figure out a proof? Did you have a “eu- is: what is the complexity of 3-coloring graphs reka moment”? with no induced t-vertex path when t ≥ 8? Chudnovsky: When I try to prove that A Now there are variations: instead of excluding implies B, I usually try to come up with the one graph H, exclude several. Instead of ask- description of the world where A holds, and ing for an efficient coloring algorithm, ask for where B follows. Then I try to prove that my a list of minimal obstructions. There are a lot 2See https://www.sciencedirect.com/science/article/pii/S0095895601920441

ECA 1:1 (2021) Interview #S3I4 4 Interview with Maria Chudnovsky of lovely questions there. to play with the function in the conclusion? Mansour: Do you think that one day some- Ask for something slightly smaller that n(H), one or a group of researchers will be able to but still big enough that it will be asymp- present a non-computer-assisted proof for the totically different from the regime when noth- four-color theorem? ing is excluded (then the correct function is Chudnovsky: I really would not bet one way log(n)). Or we can formulate a stronger con- or another. On the one hand, I think there clusion, and ask what needs to be excluded to are more and more methods available that may achieve it. These are all beautiful and inter- help. On the other hand, I think our ac- esting questions; trying to think about them ceptance of computer-assisted proofs is grow- deepens our understanding of graphs with for- ing, and so I am not sure how much moti- bidden induced subgraphs. It is quite likely vation there will be to continue looking for a that what we learn from this is much more im- computer-free proof. portant than the exact asymptotic behavior of Mansour: What is the Erd˝os-Hajnal conjec- the function... ture? Is there an exciting recent development Mansour: Is there a specific problem you related to this conjecture? have been working on for many years? What Chudnovsky: This conjecture states that for progress have you made? every graph H there is an (H) ∈ (0, 1) such that every n-vertex graph with no induced sub- Chudnovsky: The Erd˝os-Hajnal conjecture, graph isomorphic to H has a clique or a stable and the problem of computing the maximum set of size at least n(H). There have been a clique size in a perfect graph. lot of developments on variants of this conjec- Mansour: Professor Maria Chudnovsky, I ture. What if instead of excluding one graph would like to thank you for this very interesting H, we excluded H and its complement Hc? interview on behalf of the journal Enumerative Or exclude a family of graphs? What if we try Combinatorics and Applications.

ECA 1:1 (2021) Interview #S3I4 5