PERFECT GRAPHS SUMS OF SQUARES AND CEMIL DIBEK A DISSERTATION PRESENTED TO THE FACULTY RECOMMENDED FOR ACCEPTANCE

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P ^ERFECT G ^RAPHS

AND

S ^{UMS OF} S ^QUARES

CEMILDIBEK

A DISSERTATION

PRESENTED TO THEFACULTY OF PRINCETON UNIVERSITY INCANDIDACY FOR THEDEGREE

OF DOCTOR OF PHILOSOPHY

RECOMMENDED FORACCEPTANCE BY THEDEPARTMENT OF

OPERATIONSRESEARCH AND FINANCIAL ENGINEERING

ADVISERS: AMIRALIAHMADI AND MARIACHUDNOVSKY

NOVEMBER2021

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Abstract

A graph is perfect if the clique number and the chromatic number coincide for any of its induced subgraphs. A polynomial is a sum of squares (sos) if it can be written as a sum of squares of some other polynomials. In the first technical chapter of this thesis, we bring these two notions together by presenting an algebraic characterization of perfect graphs. We show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sos. As a byproduct, we obtain several infinite families of nonnegative polynomials that are not sos through graph-theoretic constructions. We also establish a number of other connections between structural graph theory and real algebra.

In the subsequent chapters, we focus on problems that are relevant to sos polynomials and perfect graphs individually.

In one chapter, we study separable plus quadratic (SPQ) polynomials. Motivated by the fact that nonnegative separable and nonnegative quadratic polynomials are sos, we study whether nonnegative SPQ polynomials are (i) the sum of a nonnegative separable and a nonnegative quadratic polynomial, and (ii) a sum of squares. We establish that the answer to question (i) is positive for univariate plus quadratic polynomials and for convex SPQ polynomials, but negative already for bivariate quartic SPQ polynomials. For question (ii), we provide a complete characterization of the answer based on the degree and the number of variables of the SPQ polynomial. We also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. We end the chapter by presenting applications of SPQ polynomials to problems in statistics and nonlinear optimization.

In the last two chapters, we focus on the study of graphs that are closely related to the class of perfect graphs, namely even-hole-free graphs and strongly perfect graphs. Al- though the class of even-hole-free graphs is a widely studied class of graphs, the complexity of the maximum independent set problem in this class is a long-standing open problem. We take a step forward in this direction by showing that there is a polynomial-time algorithm to solve the maximum independent set problem in the class of (pyramid, even hole)-free

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graphs. Regarding strongly perfect graphs, although their characterization by a set of forbidden induced subgraphs remains open, we provide several new infinite families of minimal non-strongly-perfect graphs. We also present a new proof of the characterization of claw-free strongly perfect graphs, which is shorter and quite different from the original proof.

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Acknowledgements

I would first like to thank my advisors, Amir Ali Ahmadi and Maria Chudnovsky. Amir Ali, I am greatly indebted to you for your constant guidance and encouragement to achieve my professional goals, for helping me become a better writer and presenter, for never letting me take the easy way out, for teaching me why and how to reach beyond my grasp, and for giving me different opportunities to improve my teaching skills. You are a kind, caring, and a truly remarkable advisor. Thank you for being incredibly flexible in meeting with me any time of the day and any day of the week. It is impossible not to be impressed with your outstanding ability to understand and teach mathematical concepts profoundly.

Thank you for being a great scholar to look up to. Maria, I am deeply grateful to you for sharing your distinguished expertise and knowledge with me, for being friendly and supportive while maintaining your professional attitude, and for your phenomenal speed in responding emails and giving valuable feedback. The creativity in your thinking has amazed me so many times in these past few years. Thank you for your patience in our meetings when I was usually trying to keep up with your superior intelligence. I do not remember how many times I took pictures of the board in your office. (I was worried when I realized all the nostalgic videos automatically generated by my phone’s photo library were slideshows of some crazy graphs on your board, arranged around a theme with a music in the background.) Thank you for providing an inspiring and challenging working environment. I feel fortunate for working with such an excellent researcher and having such a great advisor.

I am proud to have been advised by Amir Ali and Maria. It has been an honor and a pleasure to work with both. What I have learned from them will remain a source of immense inspiration while guiding me throughout my academic career.

My sincere gratitude goes to Noga Alon and Ramon van Handel for serving on my thesis committee, and to Bartolomeo Stellato for being a thesis reader. Thank you for being there to answer my questions and offering valuable comments on the dissertation.

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During my studies at Princeton, I have had the opportunity to work with wonderful researchers. Thank you to my co-authors: Tara Abrishami, Matthew Andrews, Sepehr Hajebi, Georgina Hall, Karina Palyutina, Paweł Rz ˛a˙zewski, Paul Seymour, Sophie Spirkl, Stéphan Thomassé, Nicolas Trotignon, and Kristina Vuškovi´c. Working with you all has been a great inspiration and pleasure to me, and I look forward to collaborating with you again in the future. A special thank you to Matthew Andrews and Iraj Saniee for hosting me at Bell Labs for the summer of 2019, where I had the chance to do research within the Mathematics & Algorithms Research Team. I am also grateful to Martin Loebl and Irena Penev for hosting me at Charles University in Prague for a month.

This is also a fitting opportunity to thank my mentors from my undergrad years who have always helped me along the way. Tınaz Ekim, for introducing me to the field of graph theory and for being an amazing advisor during my master’s studies. Pınar Heggernes, for encouraging me to go to Princeton to work with Maria Chudnovsky and for inviting me to the University of Bergen for a research visit. Mustafa Akan, for being a fantastic mentor guiding me towards the right path through life and career advices. I will always be grateful to all of you for your support and kindness.

Special thanks are due to all the members of the ORFE administrative staff who have always helped me have a smooth experience in the department. Thank you to Kim for being an excellent program administrator and for her friendly smile, to Tabitha for making events and organizational tasks easy and enjoyable, and to Melissa, Michael, and Tiffany for answering all my questions with patience.

I feel very fortunate to have had the chance to work within two different research groups. Abraar, Bachir, Georgina, and Jeff from Amir Ali’s group, and Sophie and Tara from Maria’s group have all been great friends, colleagues, and guiding lights at times. It has been a pleasure to work alongside them and talk (gossip?) about Amir Ali or Maria from time to time.

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A special thanks to Malte for his true companionship. Thank you for listening and supporting me whenever I needed help. I owe gratitude to Murat (Bozluolcay) and Cecilia for their sincere friendship and for always making me feel warm. I also thank Sophie and Logan for their continuous support and for always creating time to listen. Many thanks go to a group of wonderful colleagues and friends for providing an enjoyable environment within and outside Princeton: Carla, Egecan, Elahe, Gökçe, Levon, Linda, Matheus, Murat (Özatay), Oylum, Samy, Sinem, Suqi, ¸Sirag, Talha Can, and Yair. I would also like to thank my childhood friends who live in Turkey: Edip, Erdem, Görkem, Sabit, and Umutcan. No matter the distance, our friendships have never faltered. Thank you for the thoughtful video calls when I was in the US, for lots of good times and delightful memories whenever I went to Turkey, and for being a shoulder I can always lean on regardless of where I am.

I would like to thank Irmak for filling me with compassion, for making me more sen- sitive and affectionate than I ever thought I could be, but mostly for reminding me of what matters. I look forward to making many more memories with you. I would also like to thank Irmak’s parents, Nevgül and Semir, and her sister Ilgaz, for their gracious hospitality throughout my stay at their house and for being a family to me whenever I am in Istanbul.

I finally want to thank my parents, Semira and Adnan, and my loving sisters, Fulya, Melda, and Mona Lisa, who have never failed to give me their unconditional love. Mom and Dad, thank you for always supporting my pursuits and for never hesitating to say “we are proud of you”. Hearing those words is the best reward one can get. Thank you for providing a support system for the entire family at all times by giving of yourselves in a myriad of ways. I am grateful to be your son and I feel blessed to be part of this family.

The work in this thesis was partially supported by an AFOSR MURI award, the DARPA Young Faculty Award, the Princeton SEAS Innovation Award, the NSF CAREER Award, the Google Faculty Award, the Sloan Fellowship, the NSF Grant DMS-1763817, and the U.S. Army Research Laboratory.

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To my parents, Semira and Adnan.

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List of Tables

2.1 A lower bound on the number of times that 100 randomly generated graphs G_n,p satisfy ω(G_n,p) < ϑ⁰(G_n,p) (or equivalently make the (nonnegative) polynomial p_G_n,p(x) not sos) . . . 33 3.1 Minimal examples showing the strictness of the inclusions in (3.4) . . . 78 3.2 Comparison of running times (in seconds) of the semidefinite programs

(3.11) and (3.12) for minimizing a convex SPQ polynomial . . . 89 3.3 Pairwise comparison of lower bounds p1, p^u_fixed, p^u_free on p0 (left) and pair-

wise comparison of their ceilings (right): in each row, the number of times (out of 100 randomly generated instances of (P₀)) that equalities/strict in- equalities hold between two quantities are recorded . . . 100 3.4 The four regression problems considered in our experiments . . . 102 3.5 Comparison of running times (in seconds) averaged over 5 instances for

sos-convex and SPQ convex polynomial regression on problems of increas- ing size with degree 2d = 4 . . . 104

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List of Figures

2.1 The graph C5 and its max-clique matrix C . . . 21

2.2 The Paley graphs on 5 and 13 vertices . . . 26

2.3 The graph G is obtained from the graph H by adding new vertices with arbitrary adjacencies such that ω(G) = ω(H), making p_Ginherit the property of being not sos from p_H . . . 29

2.4 Since pG1 is not sos, the polynomial pG associated with the join G of G1 and G2 is not sos . . . 29

2.5 Since pG1 is not sos, the polynomial pGassociated with the strong product G of G₁and G2is not sos . . . 31

2.6 The graph P₃ (left) and its complement P₃ (right) . . . 44

2.7 The graph associated with Remark 2.6.1 . . . 49

2.8 The graphs G (left) and G⁰(right) associated with Remark 2.6.2 . . . 50

3.1 The optimal input-independent penalty polynomials for degrees 2d = 6, 10, 14, 18, together with the `₀-pseudonorm and the `1-norm in dimen- sion one . . . 98

3.2 Comparison of optimal input-independent and input-dependent penalty polynomials of degree 2d = 6 for an input A ∈ R^5×10, b ∈ R⁵ to (P₀) for which dp^u_freee > dp^u_fixede . . . 99

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3.3 Histograms demonstrating test set performance of the different regression approaches over 100 functions randomly chosen from the family of convex

functions fa,bgiven in (3.21) . . . 103

3.4 Basins of attraction (in light gray) of the global minimum of f in (3.23) for the Newton’s method (left) and for the Newton-SPQ method (right) . . . . 106

4.1 Theta, pyramid, prism, and turtle . . . 111

4.2 k-theta, k-pyramid, k-prism, k-turtle, and k-ladder . . . 112

4.3 The cube graph . . . 119

4.4 MNC configurations (dashed lines represent possible edges) . . . 123

5.1 The claw graph . . . 155

5.2 Some forbidden induced subgraphs for strongly perfect graphs . . . 157

5.3 A closer look at Graph V shown in Figure 5.2 . . . 158

5.4 The graphs A₁, A₂, A₃, A₄, A₅, A₆are strongly perfect . . . 160

5.5 A larva, a pupa, and a butterfly . . . 161

5.6 New minimal non-strongly-perfect graphs . . . 162

5.8 A mutated pupa and a mutated butterfly . . . 167

5.10 Handcuffs and eye masks (the dotted lines represent odd paths) . . . 169

5.11 A clown . . . 175

5.12 Bicycles and thetas (the dotted lines represent even paths) . . . 190

5.13 Thetas in ˆB and in B . . . 193

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Chapter 1 Introduction

In this thesis, we consider several problems that are related to “perfect graphs” and “sum of squares polynomials”. While the very first part of the thesis builds a bridge between these two seemingly unrelated mathematical structures, the subsequent chapters focus on problems that are relevant to these notions individually.

The problem of optimizing over the set of nonnegative polynomials is a fundamental problem in computational mathematics and appears in a great variety of applications in different areas, such as control theory, statistics, quantum computation, geometric theorem proving, game theory, combinatorial optimization, power engineering, just to mention a few. Although problems in all these applications seek to efficiently optimize over the set of nonnegative polynomials, this task is unfortunately computationally hard. Indeed, simply testing whether a given polynomial of degree four is nonnegative is an NP-hard problem [98]. The recent years witnessed a growing interest in the topic because it was realized that one can write sufficient conditions for nonnegativity of a polynomial based on the concept of “sum of squares polynomials”, a subset of nonnegative polynomials. A polynomial p : Rⁿ → R with real coefficients is a sum of squares (sos) if there exist

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polynomials q₁(x), . . . , q_m(x) such that

p(x) =

m

X

i=1

q_i²(x),

i.e., if it can be written as the sum of squares of some other polynomials. Clearly, every sos polynomial is nonnegative. One of the reasons why the topic has been receiving renewed attention from the optimization community in recent years is the following observation:

while optimizing over nonnegative polynomials is NP-hard, optimization over the set of sos polynomials is computationally tractable and can be done via semidefinite programming, which can be solved to arbitrary accuracy in polynomial time (e.g., by interior point methods). More precisely, this fundamental link between sos polynomials and semidefinite programming is captured in the following statement [36, 103]: a polynomial p(x) in n variables and of degree 2d is sos if and only if there exists a positive semidefinite matrix Q such that

p(x) = z(x)^TQz(x),

where z(x) is the vector of monomials of degree up to d, i.e., z(x) = (1, x1, . . . , x_n, x₁x₂, . . . , x^d_n)^T. This result immediately leads to a semidefinite programming-based method for checking whether a polynomial is sos, and in fact, more interestingly, for optimizing a linear function over the intersection of the set of sos polynomials with an affine subspace. With this computational appeal, sos polynomials have impacted both discrete and continuous optimization over the last two decades (see, e.g., [25, 75]).

The fact that sos polynomials provide tractable sufficient conditions for computationally hard problems involving nonnegative polynomials revived an old fundamental question: when do nonnegative polynomials can be represented as sums of squares? This question was answered in 1888 by Hilbert [79], who gave a complete characterization of the degrees and dimensions in which all nonnegative polynomials can be written as sums of squares. In particular, Hilbert proved that there exist nonnegative polynomials that are not

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sos, though through a nonconstructive proof of existence. The first explicit examples of such polynomials appeared only about 80 years later and were produced by Motzkin [96]

and Robinson [116]. Although many other examples have appeared in the literature over the years (see, e.g., [115, 33, 34, 35]), the study of the gap between nonnegative and sos polynomials is still an active area of research to this day. In particular, constructing explicit examples of nonnegative polynomials that are not sos in relatively low dimensions and degrees still seems to be a nontrivial task. In one of the chapters of this thesis, we provide several infinite families of nonnegative polynomials that are not sos through various families of graphs and graph-theoretic operations.

Although the question of when nonnegative polynomials are also sos was answered by Hilbert for general polynomials, answering this question when additional structure is required on the polynomials certainly continues to have a practical interest. It might be the case that in some degrees and dimensions for which general nonnegative polynomials are not necessarily sos, nonnegative polynomials with certain additional structures admit a sum of squares representation. It is thus increasingly relevant to study the relationship between nonnegative and sos polynomials under additional structure. One of the chapters of this thesis investigates this relationship for polynomials under a structure that arises naturally in light of Hilbert’s original result as well as some other related questions and applications involving polynomials of that particular structure.

The second half of this thesis focuses on some problems from structural graph theory.

Graphs are a convenient way to represent different kinds of objects, and many real-world problems can be modeled as optimization problems on graphs. Although these problems are most of the time computationally intractable, significant progress has been made in recent years toward designing tools that address this intractability when certain restrictions are required on the input. In this sense, the study of the structural properties of restricted graph classes becomes crucial. One way of obtaining a restricted graph class, that recently draws great interest from graph theorists, is by forbidding induced subgraphs. Given two

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graphs G and H, the graph H is said to be an induced subgraph of G if V (H) ⊆ V (G), and two vertices of H are adjacent if and only if they are adjacent in G. Equivalently, H is an induced subgraph of G if H can be obtained from G by repeatedly deleting a vertex and all edges incident with it. For a (possibly infinite) family of graphs H, a graph G is called H-free if no member of H is isomorphic to an induced subgraph of G. As it turns out, several interesting families of graphs can be characterized as being H-free for some family H. The class of perfect graphs is surely one of the most notable ones.

A graph G is said to be perfect if for every induced subgraph H of G, the clique number of H equals the chromatic number of H. We remind the reader that the clique number of a graph G, denoted by ω(G), is the size of a maximum set of pairwise adjacent vertices in G, and that the chromatic number of G, denoted by χ(G), is the minimum number of colors needed to color the vertices of G such that adjacent vertices receive different colors. Perfect graphs were introduced by Claude Berge in 1960, who conjectured that being perfect is equivalent to being H-free for a certain infinite family H. More precisely, he conjectured that a graph G is perfect if and only if no induced subgraph of G is a cycle of odd length at least five, or the complement of one. Indeed, while every graph G clearly satisfies ω(G) ≤ χ(G), it is easy to see that the equality does not hold for cycles of odd length at least five or their complements. This conjecture, now known as the strong perfect graph theorem, was proved in 2002 by Chudnovsky, Robertson, Seymour and Thomas [41].

The class of perfect graphs is undoubtedly one of the most interesting families of graphs as it connects several mathematical fields in completely astonishing ways. These fields include graph theory, information theory, combinatorial optimization, polyhedral and convex geometry, semidefinite programming, and several others. Let us briefly review the connection of perfect graphs to information theory and semidefinite programming.

If we go back in time and review Berge’s motivation to introduce perfect graphs, we would encounter a problem from information theory, namely finding the “Shannon capacity” of a graph. This parameter originates from a problem of Shannon who was interested

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in finding the maximum rate at which information can be transmitted through a noisy com- munication channel, where the noise of the channel can be encoded by a graph. Although the Shannon capacity is a notoriously difficult parameter to compute and is not known for many small graphs (e.g., for an odd cycle of length seven), it is well known that for a graph G this parameter lies between ω(G) and χ(G). It follows that if ω(G) = χ(G) for a graph G, then the Shannon capacity of G is equal to this value. This is in particular the case for perfect graphs. Therefore, it is natural to ask what are the minimal graphs that do not satisfy ω(G) = χ(G), and this question led Berge to introduce perfect graphs and his famous conjecture.

Perfect graphs are also of particular interest due to the fact that several problems that are intractable to solve in general graphs can be solved efficiently when restricted to the class of perfect graphs. For example, the maximum clique problem, i.e., the problem of finding a clique of maximum cardinality in a given graph, is strongly NP-hard for general graphs [62], whereas it can be solved in polynomial time in the class of perfect graphs via semidefinite programming. This is due to the so-called theta number of a graph, which was introduced by Lovász [92] and which can be formulated as the optimal value of a semidefinite program. Grötschel, Lovász and Schrijver [71] showed that the theta number of the complement of a graph G is sandwiched between ω(G) and χ(G). Since the theta number can be computed with arbitrary precision in polynomial time via a semidefinite program, one of the consequences of this result is that for a perfect graph G, the clique number of G can be computed in polynomial time. Surprisingly, there is no known polynomial-time algorithm to find the clique number of a perfect graph without the use of semidefinite optimization. The existence of a polynomial-time algorithm of a purely combinatorial nature is still an open problem.

The first part of this thesis establishes a connection between perfect graphs and sos polynomials through a strengthening of the Lovász’s theta number. The succeeding chapter focuses on the relationship between nonnegative and sos polynomials, together with

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some accompanying questions and applications, under certain additional structures. Later chapters turn to problems related to forbidden induced subgraphs for classes of graphs that are closely related to perfect graphs.

1.1 Outline of This Thesis

In this section, we describe the contents and contributions of each part of this thesis more precisely.

In Chapter 2, we present an algebraic characterization of perfect graphs by introducing and studying the notion of “sos-perfectness”. For a graph G = (V, E) with clique number ω(G), we define a quartic polynomial p_G(x) in the variables x = (x₁, . . . , x|V (G)|)^T by

p_G(x) = −2 ω(G)

X

ij∈E(G)

x²_ix²_j + (ω(G) − 1)





|V (G)|

X

i=1

x²_i





2

.

We say that a graph G is sos-perfect if pH(x) is sos for every induced subgraph H of G.

The main result of this chapter is the characterization of perfect graphs by sos polynomials:

a graph is perfect if and only if it is sos-perfect. As a by-product, we obtain several infinite families of nonnegative polynomials that are not sos by considering different families of graphs (e.g., odd cycles, Paley graphs, Erd˝os-Rényi random graphs) and graph-theoretic operations (e.g., graph join, graph strong product). Using this connection between perfect graphs and sos polynomials, we also revisit some known structural results about perfect graphs and translate them into statements concerning sos polynomials.

In Chapter 3, we study separable plus quadratic (SPQ) polynomials. These are polynomials that can be written as p(x) = s(x) + q(x), where s(x) is a separable polynomial, i.e., s(x) = Pn

i=1s_i(x_i) for some univariate polynomials s_i : R → R, and q(x) is a quadratic polynomial. As both nonnegative univariate polynomials and nonnegative quadratic polynomials are sos, it is natural to wonder what would happen to the sum of a univariate and

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a quadratic polynomial. Would it also be the case that any nonnegative polynomial with such a structure admits a sum of squares representation? We answer this and other related questions in a more general setting where the ‘univariate’ polynomial is replaced by a

‘separable’ polynomial. This is a natural structure to consider in light of Hilbert’s original result, and it is also a structure of interest in various applications.

We investigate polynomials of this structure from different aspects. We first study whether nonnegative SPQ polynomials are sos. We establish that the answer to this question is positive for univariate plus quadratic polynomials and for convex SPQ polynomials, but in general negative for SPQ polynomials. We provide a complete characterization based on the degree and the number of variables of the SPQ polynomial. We further extend our study to understanding nonnegativity of convex SPQ polynomials. Using a decomposition result that we present for convex SPQ polynomials, we show that convex SPQ polynomial optimization problems can be solved by “small” semidefinite programs. We also prove that testing nonnegativity of SPQ polynomials is NP-hard when the degree is at least four. Finally, we present possible application areas where SPQ polynomials appear, such as sparsity of solutions to linear programs, polynomial regression, and a generalization of Newton’s method.

In the last two chapters of this thesis, we turn to problems related to graphs with forbidden induced subgraphs. Closely related to the class of perfect graphs is the class of even-hole-free graphs. These are H-free graphs where H is the family of all cycles of even length, i.e., if we denote a chordless cycle on k vertices by C_k, these are precisely (C₄, C₆, C₈, . . . )-free graphs. This class is structurally similar to the class of perfect graphs. Recall that by the strong perfect graph theorem, perfect graphs are precisely (C5, C7, C7, C9, C9, . . . )-free graphs. Observe also that in an even-hole-free graph by forbidding C4, we also forbid all Ck for k ≥ 6. Hence, an even-hole-free graph is (C₄, C₆, C₆, C₇, C₈, C₈, . . . )-free. It is also worth noting that the tools developed in the

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structural study of even-hole-free graphs were substantially made use of in the proof of the strong perfect graph theorem.

Although the class of even-hole-free graphs was extensively studied (see [120] for a survey), the complexity of the maximum independent set problem, i.e., the problem of finding a set of pairwise non-adjacent vertices of maximum cardinality, remains unknown in this class. In Chapter 4, we present a step forward in this direction by showing that there is a polynomial-time algorithm to solve the maximum independent set problem for the class of (pyramid, even hole)-free graphs. In fact, all our results hold for a slightly larger class of graphs, the class of (theta, pyramid, prism, turtle)-free graphs. Our result is based on a decomposition theorem and on bounding the number of “minimal separators” in a graph.

We show that graphs that do not contain a theta, pyramid, prism, or turtle as an induced subgraph have polynomially many minimal separators. This result is the best possible in the sense that there are graphs with exponentially many minimal separators if only three of the four induced subgraphs are forbidden.

In Chapter 5, we consider the class of strongly perfect graphs. A clique in a graph is a set of pairwise adjacent vertices, and an independent set is a set of pairwise non- adjacent vertices. A maximal clique is a clique that is not a subset of a larger clique.

A graph G is strongly perfect if every induced subgraph H of G has an independent set that intersects every maximal clique of H. Strongly perfect graphs form a subclass of perfect graphs. This may not be obvious at first sight but it can be easily deduced by the following characterization of perfect graphs: a graph G is perfect if and only if every induced subgraph H of G has an independent set that intersects every maximum clique of H. Since every maximum clique is also a maximal clique, it follows that every strongly perfect graph is perfect.

Can the class of strongly perfect graphs be characterized as being H-free for some family H? Unfortunately, the characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Let us say that a graph G is minimal non-strongly-perfect

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if G is not strongly perfect but every proper induced subgraph of G is. Although some results concerning the structure of minimal non-strongly-perfect graphs have been presented in [12, 80, 101], the characterization of strongly perfect graphs in general remains open. A conjecture in this direction was presented in [111] in 1990 stating that a graph is strongly perfect if and only if it does not contain a list of certain graphs as an induced subgraph. Although it was thought for a long time that this is a complete list of minimal non-strongly-perfect graphs, we disprove this conjecture in Chapter 5 by providing several new infinite families of minimal non-strongly-perfect graphs.

A graph is claw-free if no vertex in the graph has three pairwise non-adjacent neigh- bors. Although the characterization of strongly perfect graphs by a set of forbidden induced subgraphs remains open, the characterization of claw-free strongly perfect graphs was conjectured by Ravindra [111] in 1990, and was proved by Wang [121] in 2006. In Chapter 5, we also present a new shorter and quite different proof of this characterization.

Finally, we remark that each chapter is written to be completely self-contained for the convenience of the reader.

1.2 Related Publications

The material presented in this thesis is based on the following papers.

Chapter 2. A.A. Ahmadi, C. Dibek, “A sum of squares characterization of perfect graphs”. Under review. Available at arXiv:2110.08950.

Chapter 3. A.A. Ahmadi, C. Dibek, G. Hall, “Sums of separable and quadratic polynomials”. Under review. Available at arXiv:2105.04766.

Chapter 4. T. Abrishami, M. Chudnovsky, C. Dibek, S. Thomassé, N. Trotignon, K.

Vuškovi´c, “Graphs with polynomially many minimal separators”, Journal of Combinato- rial Theory (B), 152 (2022), 248–280.

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Chapter 5.

- Section 5.2. M. Chudnovsky, C. Dibek, P. Seymour, “New examples of minimal non-strongly-perfect graphs”, Discrete Mathematics, 344(5) (2021), 112334.

- Section 5.3. M. Chudnovsky, C. Dibek, “Strongly perfect claw-free graphs - A short proof”, Journal of Graph Theory, 97(3) (2021), 359–381.

In addition to these papers, the following papers were written during the graduate studies of the author at Princeton University but are not included in this thesis in order to provide a coherent story.

• T. Abrishami, M. Chudnovsky, C. Dibek, P. Rz ˛a˙zewski, “Polynomial-time algorithm for Maximum Independent Set in bounded-degree graphs with no long induced claws”, ACM-SIAM Symposium on Discrete Algorithms (SODA22). Accepted for publication. Available at arXiv:2107.05434.

• T. Abrishami, M. Chudnovsky, C. Dibek, S. Hajebi, P. Rz ˛a˙zewski, S. Spirkl, K. Vuškovi´c, “Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree”. Under review. Available at arXiv:2108.01162.

• T. Abrishami, M. Chudnovsky, C. Dibek, K. Vuškovi´c, “Submodular functions and perfect graphs”. Under review. Available at arXiv:2110.00108.

• M. Andrews, C. Dibek, K. Palyutina, “Evolution of Q-values for Deep Q-Learning in stable baselines”. Under review. Available at arXiv:2004.11766.

• M. Chudnovsky, C. Dibek, S. Spirkl, “The structure of apple-free graphs”. In prepa- ration.

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Chapter 2 A Sum of Squares Characterization of Perfect Graphs

In this chapter, we present an algebraic characterization of perfect graphs. We introduce and study a notion that builds a bridge between perfect graphs and sum of squares polynomials. More precisely, by bringing together a number of interesting results in graph theory and conic optimization, we show that a graph is perfect if and only if certain nonnegative polynomials associated with the graph are sums of squares. As a byproduct, we obtain several infinite families of nonnegative polynomials that are not sums of squares through various graph-theoretic constructions. We also examine certain subsets of sum of squares polynomials which admit a linear or second-order cone representation [8], or a more restricted semidefinite representation. We study the bounds that optimization over these subsets produces on the clique number of a graph, and characterize the graphs for which these bounds are tight for all induced subgraphs. Finally, in the hope of building more connections between structural graph theory and real algebraic geometry in the future, we reformulate some well-known results from the theory of perfect graphs as statements about sum of squares proofs of nonnegativity of certain polynomials. The material presented in this chapter is based on the work in [5].

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2.1 Introduction

A graph is perfect if for any of its induced subgraphs, the chromatic number equals the cardinality of a largest clique. Perfect graphs, introduced by Berge in 1960, have elegant the- oretical properties and curious connections with linear, integer, and semidefinite programming. For instance, perfect graphs appear in the study of exactness of linear programming relaxations of integer programs. As an example, for a matrix A ∈ {0, 1}^m×n, all vertices of the polytope {x ∈ Rⁿ: Ax ≤ 1, x ≥ 0} are integral if and only if the undominated rows of A are the incidence vectors of the maximal cliques of a perfect graph [40, 45]. Moreover, several combinatorial problems that are NP-hard on general graphs can be solved efficiently on perfect graphs using semidefinite programming [71]. Examples include the maximum independent set and the minimum clique cover problems. More generally, perfect graphs have been the subject of much research in recent decades due to the fact that they are at the crossroad of several mathematical disciplines, including graph theory, information theory, combinatorial optimization, polyhedral and convex geometry, and semidefinite programming [45, 52, 61, 71, 73, 92, 68].

The second notion of interest to this chapter is that of sum of squares polynomials. A polynomial is a sum of squares (sos) if it can be written as a sum of squares of some other polynomials. There has been a growing interest in sos polynomials recently due to the fact that they provide semidefinite programming-based sufficient conditions for problems involving nonnegative polynomials. It is well known that several important problems in applied and computational mathematics can be formulated as optimization problems over the set of nonnegative polynomials. Although these problems are generally intractable to solve exactly, they can be efficiently approximated by replacing nonnegativity constraints with sum of squares requirements. By connecting ideas from real algebraic geometry and semidefinite programming, sum of squares polynomials have significantly impacted both discrete and continuous optimization over the last two decades; see, e.g., [85, 104, 25, 87,

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In this work, we introduce and study the notion of sos-perfectness, a notion that brings together perfect graphs and sos polynomials. For a graph G = (V, E) with clique number ω(G), we define the following quartic (homogeneous) polynomial in the variables x = (x₁, . . . , x_{|V (G)|})^T:

(2.1) p_G(x) := −2 ω(G)

X

ij∈E(G)

x²_ix²_j + (ω(G) − 1)





|V (G)|

X

i=1

x²_i





2

.

It turns out that for every graph G, the polynomial p_G(x) is nonnegative by construction.

We say that a graph G is sos-perfect if p_H(x) is sos for every induced subgraph H of G. In this chapter, we prove the following theorem.

Theorem 2.1.1. A graph is perfect if and only if it is sos-perfect.

The remainder of this chapter is organized as follows. In Section 5.3.1, we recall some definitions and results related to perfect graphs and sos polynomials. In Section 2.3, we prove Theorem 4.1.2. Our proof brings together a number of interesting results in graph theory and conic optimization. In Section 2.4, we focus on the connection between imperfect graphs and nonnegative polynomials that are not sos. In Sections 2.4.1 and 2.4.2, we provide several infinite families of nonnegative polynomials that are not sos through various graph-theoretic constructions. In Section 2.4.3, by building on previous results on the probable values of certain parameters associated with Erd˝os-Rényi random graphs G_n,p, we show that for a fixed parameter p and for large enough n, the polynomial p_G_n,p(x) is nonnegative but not sos with high probability. In Section 2.4.4, we provide an explicit hyperplane that separates a given non-sos polynomial p_G(x) from the set of sos polynomials. In Section 2.4.5, we show that an example of a convex nonnegative polynomial that is not sos cannot arise from our graph-theoretic constructions. (The construction of such a polynomial was an open problem until recently [117]). In Section 2.5, we examine certain subsets of sos polynomials which admit a linear or second-order cone representation [8], or

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a more restricted semidefinite representation. We study the bounds that optimization over these subsets produces on the clique number of a graph, and characterize the graphs for which these bounds are tight for all induced subgraphs. Finally, in Section 2.6, we reformulate a number of results from the theory of perfect graphs as statements about sum of squares proofs of nonnegativity of certain polynomials. Our hope is that our reformulations will lead to more connections between structural graph theory and real algebraic geometry, and ideally to an algebraic proof of the “strong perfect graph theorem”.

2.2 Preliminaries

A (multivariate) polynomial p(x) in variables x := (x₁, . . . , x_n)^T is a function from Rⁿto R that is a finite linear combination of monomials:

p(x) =X

α

c_αx^α = X

α1,...,αn

c_α₁_,...,α_nx^α₁¹. . . x^α_nⁿ,

where the sum is over n-tuples of nonnegative integers αi. The degree of a monomial x^αis equal to α1+ · · · + α_n. The degree of a polynomial p(x) is defined to be the highest degree of its monomials. A form (or a homogeneous polynomial) is a polynomial where all the monomials have the same degree.

We denote the set of real symmetric n × n matrices by S_n. A matrix M ∈ S_nis positive semidefinite(psd) if x^TM x ≥ 0 for all x ∈ Rⁿ, and nonnegative if the entries of M are all nonnegative. We write M 0 if M is psd, and M ≥ 0 if M is nonnegative. We denote the set of n × n psd (resp. nonnegative) matrices by S_n⁺(resp. Nn). The trace of M is denoted by Tr(M ).

All graphs in this chapter are undirected, finite, and simple (i.e., have no loops or par- allel edges). Throughout the chapter, G = (V, E) denotes a graph with vertex set V (G) and edge set E(G). The complement of a graph G, denoted by G, is the graph with vertex

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G. The matrices A and A respectively denote the adjacency matrices of G and G. The matrices I and J respectively denote the identity matrix and the all-ones matrix. Observe that A + A + I = J .

A graph H is an induced subgraph of a graph G if V (H) ⊆ V (G) and any two vertices of H are adjacent if and only if they are adjacent in G. We say that G contains a graph H if G has an induced subgraph isomorphic to H, and that G is H-free if it does not contain H.

For an integer k ≥ 4, a hole (of length k) is a graph isomorphic to the chordless k-vertex cycle C_k, and an antihole (of length k) is a graph isomorphic to C_k. A hole (or an antihole) is odd if its length is odd.

A clique in a graph is a set of pairwise adjacent vertices, and an independent set is a set of pairwise non-adjacent vertices. The clique number of a graph G, denoted by ω(G), is the size of a maximum clique in G, and the independence number of G, denoted by α(G), is the size of a maximum independent set in G. The chromatic number of a graph G, denoted by χ(G), is the smallest number of independent sets of G with union V (G). Every graph G clearly satisfies χ(G) ≥ ω(G). The inequality, however, may be strict. For instance, if G is an odd hole, it is easy to see that ω(G) = 2 and χ(G) = 3. Similarly, if G is an odd antihole of length 2k + 1 for some k ≥ 2, then ω(G) = k and χ(G) = k + 1.

2.2.1 Perfect graphs

A graph G is perfect if every induced subgraph H of G satisfies χ(H) = ω(H). Berge introduced perfect graphs and made two conjectures [17]. The first, proved by Lovász [91]

and now known as the weak perfect graph theorem, states that a graph is perfect if and only if its complement is perfect. Berge’s second conjecture characterizes minimal imperfect graphs. A graph G is minimal imperfect if G is not perfect but every proper induced subgraph of G is perfect. Berge observed that odd holes and odd antiholes are minimal imperfect graphs, and conjectured that they are, in fact, the only minimal imperfect graphs.

This conjecture, now known as the strong perfect graph theorem, was proved by Chud-

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novsky, Robertson, Seymour, and Thomas [41]: A graph is perfect if and only if it does not contain an odd hole or an odd antihole. Indeed, since odd holes and odd antiholes satisfy χ(G) = ω(G) + 1, perfect graphs do not contain odd holes or odd antiholes. The proof of the converse direction is long and relies on structural graph theory.

The theta number of a graph G, introduced by Lovász [92] and denoted by ϑ(G), is given as the optimal value of the following semidefinite program:

(2.2)

ϑ(G) := max

X∈Sn

Tr(J X)

s.t. X_ij = 0 if ij ∈ E Tr(X) = 1

X 0.

Grötschel, Lovász, and Schrijver [71] showed that the theta number of the complement of a graph is sandwiched between the clique number and the chromatic number of the graph, that is, for any graph G, we have ω(G) ≤ ϑ(G) ≤ χ(G). Since ϑ(G) can be computed with arbitrary precision in polynomial time via the semidefinite program (2.2), one of the consequences of this result is that for a perfect graph G, the clique number of G can be computed in polynomial time.

A strengthening of the theta number was introduced by McEliece et al. [95] and Schri- jver [118], where an entry-wise nonnegativity constraint on the matrix X is added to (2.2):

(2.3)

ϑ⁰(G) := max

X∈Sn

Tr(J X)

s.t. X_ij = 0 if ij ∈ E Tr(X) = 1

X 0 X ≥ 0.

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Schrijver [118] observed that ϑ⁰(G), too, is an upper bound on ω(G), that is, for any graph G, we have

(2.4) ω(G) ≤ ϑ⁰(G) ≤ ϑ(G) ≤ χ(G).

2.2.2 Sum of squares polynomials

A polynomial p : Rⁿ → R with real coefficients is nonnegative if p(x) ≥ 0 for all x ∈ Rⁿ and a sum of squares (sos) if there exist polynomials q₁(x), . . . , q_m(x) such that p(x) =Pm

i=1q_i²(x). While every sos polynomial is clearly nonnegative, Hilbert showed in 1888 that there exist nonnegative polynomials that are not sos [79]. His proof was not con- structive and did not lead to an explicit example of a nonnegative polynomial that is not sos.

The first examples of such polynomials were found by Motzkin [96] and Robinson [116]

nearly eighty years after Hilbert’s proof.

From a complexity standpoint, testing nonnegativity of polynomials of any fixed degree 2d ≥ 4 is NP-hard [98]. By contrast, checking whether a polynomial is sos can be done by solving a semidefinite program. Indeed, a polynomial p(x) in n variables and of degree 2d is sos if and only if there exists a psd matrix Q such that p(x) = z(x)^TQz(x), where z(x) is the vector of monomials of degree up to d, i.e., z(x) = (1, x₁, . . . , x_n, x₁x₂, . . . , x^d_n)^T (see, e.g., [36, 103]). In fact, this statement leads to a semidefinite programming-based approach for optimizing a linear function over the intersection of the set of sos polynomials with an affine subspace. This observation has enabled wide-ranging applications, see, e.g., [75].

2.3 A Sum of Squares Characterization of Perfect Graphs

In this section, we prove Theorem 4.1.2 by establishing a few intermediary lemmas. We begin by stating some relevant results from prior literature.

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A matrix M ∈ S_n is copositive if x^TM x ≥ 0 for all x ≥ 0 (i.e., for all vectors x in the nonnegative orthant). We denote the set of n × n copositive matrices by Cn. It is not difficult to observe that minimization of a quadratic function over the standard simplex

∆ := {x ∈ Rⁿ : Pn

i=1x_i = 1, x ≥ 0} is equivalent to optimization of a linear function over Cn(see, e.g., [84, 27]):

(2.5)

min

x∈∆ x^TQx = max

k∈R k

s.t. Q − kJ ∈ C_n.

As shown by Motzkin and Straus [97], the problem on the left can be related to the clique number of a graph as follows:

(2.6) 1

ω(G) = min

x∈∆ x^T(I + A)x.

Here, A denotes the adjacency matrix of the complement graph G. It follows from (2.5) and (2.6) that

(2.7)

ω(G) = min

k∈R k

s.t. k(I + A) − J ∈ Cn.

It is easy to see that a matrix M belongs to C_nif and only if the quartic form

(2.8) p_M(x) :=

n

X

i,j=1

M_ijx²_ix²_j

is nonnegative. Let Kndenote the set of matrices M ∈ Snsuch that pM(x) is sos. Clearly, K_n⊆ C_n. Hence, a tractable upper bound on ω(G) can be obtained by replacing Cnin (2.7) with Kn. Using in part a result from [103, Section 5] (see also [33, Lemma 3.5]), De Klerk and Pasechnik [84] showed that the resulting upper bound coincides with the parameter

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ϑ⁰(G) defined in (2.3):

(2.9)

ϑ⁰(G) = min

k∈R k

s.t. k(I + A) − J ∈ K_n.

The following lemma sheds light on the construction of the polynomial p_G(x) in (2.1), and a corollary of it will be used in the proof of Theorem 4.1.2. Let us consider a more general family of quartic forms by replacing the constant ω(G) in pG(x) with an arbitrary scalar k ∈ R:

(2.10) p_G,k(x) := −2k

X

ij∈E(G)

x²_ix²_j + (k − 1)





|V (G)|

X

i=1

x²_i





2

.

Lemma 2.3.1. For any graph G,

(a) the polynomialp_G,k(x) is nonnegative if and only if k ≥ ω(G).

(b) the polynomialp_G,k(x) is sos if and only if k ≥ ϑ⁰(G).

Proof. Let G = (V, E) be a graph with |V (G)| = n and with adjacency matrix A. We first claim that the polynomial pG,k(x) is nonnegative if and only if k(I + A) − J ∈ Cn. Observe that since I + A = J − A, we have k(I + A) − J = −kA + (k − 1)J . Therefore,

k(I + A) − J ∈ C_n ⇐⇒

n

X

i,j=1

h− kA + (k − 1)Ji

ij

x²_ix²_j ≥ 0 ∀x ∈ Rⁿ

⇐⇒ −k

n

X

i,j=1

Aijx²_ix²_j + (k − 1)

n

X

i,j=1

Jijx²_ix²_j ≥ 0 ∀x ∈ Rⁿ

⇐⇒ −2k

X

ij∈E(G)

x²_ix²_j + (k − 1)





n

X

i=1

x²_i





2

≥ 0 ∀x ∈ Rⁿ

⇐⇒ p_G,k(x) ≥ 0 ∀x ∈ Rⁿ.

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Similarly, k(I + A) − J ∈ K_nif and only if p_G,k(x) is sos. Hence, by (2.7) and (2.9), we obtain the following formulations of ω(G) and ϑ⁰(G):

ω(G) = min

k∈R k

s.t. p_G,k(x) is nonnegative,

ϑ⁰(G) = min

k∈R k

s.t. p_G,k(x) is sos.

Therefore, if p_G,k(x) is nonnegative, then k ≥ ω(G). Similarly, if p_G,k(x) is sos, then k ≥ ϑ⁰(G). Observe also that if p_G,k(x) is nonnegative (resp. sos) for some k ∈ R, then pG,k⁰(x) is nonnegative (resp. sos) for every k⁰ ∈ R with k⁰ ≥ k. This is simply because

k⁰(I + A) − J − k(I + A) − J = (k⁰− k)(I + A)

is a nonnegative matrix, thus belongs to Cn (resp. Kn). It follows that pG,k(x) is nonnegative if and only if k ≥ ω(G), and that pG,k(x) is sos if and only if k ≥ ϑ⁰(G).

Corollary 2.3.2. For any graph G,

(a) the polynomialp_G(x) is nonnegative.

(b) the polynomialp_G(x) is sos if and only if ω(G) = ϑ⁰(G).

Proof. Part (a) follows from Lemma 2.3.1(a) since p_G(x) = p_G,ω(G)(x), and part (b) follows from Lemma 2.3.1(b) and (2.4).

Remark 2.3.3. In the statement of Corollary 2.3.2(b), one cannot replace the quantity ϑ⁰(G) with the theta number ϑ(G). For example, let G be the graph on 64 vertices corresponding to the vectors in{0, 1}⁶, with two vertices adjacent if and only if the Hamming distance between the corresponding vectors is at least 4. We haveω(G) = ϑ⁰(G) = 4 and ϑ(G) = 16/3; see [118]. Therefore, p_G(x) is sos by Corollary 2.3.2(b), but ω(G) 6= ϑ(G).

Remark 2.3.4. Corollary 2.3.2(b) characterizes the graphs G for which the polynomial

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G to be perfect. In fact, even the condition ω(G) = χ(G) is not enough for G to be perfect since perfectness requires this condition to hold for every induced subgraph.

As an example, let G be the graph with vertex set {v1, v2, v3, v4, v5, h} and edge set {v₁v₂, v₂v₃, v₃v₄, v₄v₅, v₅v₁, hv₁, hv₂}, i.e., G is the graph C₅ with an additional vertexh adjacent only tov₁ andv₂. Then, it is easy to verify thatω(G) = χ(G) = 3. However, G is not perfect as it contains the graph C₅, for which we have 2 = ω(C₅) < χ(C₅) = 3.

Observe also that this graph G is an example of an imperfect graph for which p_G(x) is sos. This observation justifies our definition of sos-perfectness which takes induced subgraphs into consideration. Indeed, if G is a perfect graph, then, by definition, every induced subgraph of G is perfect. However, being sos is not a “hereditary” property in the sense that it is possible for the polynomial p_G(x) to be sos and for G to have an induced subgraphH with p_H(x) not sos. The graph above with H = C₅provides one such example.

Although the condition “ω(G) = ϑ⁰(G)” is not sufficient for a graph G to be perfect, we prove next that the condition “ω(H) = ϑ⁰(H) for every induced subgraph H of G” is.

We follow a proof technique of Lovász [93] which makes use of a binary matrix associated with the maximum cliques of a graph. Let G be a graph with n vertices and m maximum cliques. Then, the max-clique matrix C of G is an m × n matrix with Cij = 1 if the i^th maximum clique contains the j^th vertex, and Cij = 0 otherwise. As an example, the max-clique matrix of the graph C₅ is given in Figure 2.1.

v₁

v₂

v₃ v₄ v₅

C =

v₁ v₂ v₃ v₄ v₅













1 1 0 0 0 clique 1

0 1 1 0 0 clique 2

0 0 1 1 0 clique 3

0 0 0 1 1 clique 4

1 0 0 0 1 clique 5

Figure 2.1: The graph C₅ and its max-clique matrix C

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Lemma 2.3.5. If G is a minimal imperfect graph, then ω(G) < ϑ⁰(G).

Proof. We follow a proof of Lovász’s (see Lemma 7.9 in [93, Section 3]). Let G = (V, E) be a minimal imperfect graph with |V (G)| = n and with clique number ω. Let C be the max-clique matrix of G. By a result of Padberg [102] (see also [63] for a different proof), G has n maximum cliques, every vertex of G is in exactly ω maximum cliques, and the matrix C is non-singular.

Let λ₁ be the smallest eigenvalue of C^TC. Observe that the diagonal entries of C^TC are all ω. It is then not difficult to see that λ₁ ∈ (0, ω). Indeed, since C^TC is psd and since Tr(C^TC) = nω, we have λ1 ∈ [0, ω]. Also, we have λ1 6= 0 since C is non-singular, and λ₁ 6= ω as otherwise all eigenvalues of C^TC would equal ω, in which case C^TC = ωI.

This is a contradiction since a minimal imperfect graph has at least one edge.

Now recall that

(2.11)

ϑ⁰(G) = max

X∈Sn

Tr(J X)

s.t. X_ij = 0 if ij /∈ E(G) Tr(X) = 1

X 0 X ≥ 0.

Consider the matrix X = _n(ω−λ¹

1)(C^TC − λ₁I). It is straightforward to check that X is a feasible solution to (2.11), and that

ϑ⁰(G) ≥ Tr(J X) = ω²− λ₁ ω − λ1

> ω.

Corollary 2.3.6. A graph G is perfect if and only if ω(H) = ϑ⁰(H) for every induced subgraphH of G.

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Proof. If G is perfect, then by definition, ω(H) = χ(H) for every induced subgraph H of G. Hence, by (2.4), we have ω(H) = ϑ⁰(H) for every induced subgraph H of G.

If G is not perfect, then it contains a minimal imperfect graph H?. By Lemma 2.3.5, ω(H_?) < ϑ⁰(H_?).

We are now ready to present the proof of Theorem 4.1.2, which we restate here for ease of reference.

Theorem 1.1. A graph is perfect if and only if it is sos-perfect.

Proof. By Corollary 2.3.6, a graph G is perfect if and only if ω(H) = ϑ⁰(H) for every induced subgraph H of G, which by Corollary 2.3.2 holds if and only if p_H(x) is sos for every induced subgraph H of G.

Remark 2.3.7. It is known that if G is an odd hole or an odd antihole, then ω(G) < ϑ⁰(G) (see, e.g., Proposition 15 and Proposition 19 in [106]).¹ Hence,assuming the strong perfect graph theorem, one can bypass Lemma 2.3.5 in the proof of Theorem 1.1. Indeed, if a graph G is not perfect, then by the strong perfect graph theorem, it contains either an odd hole or an odd antihole, call itH. Since ω(H) < ϑ⁰(H), by Corollary 2.3.2(b), the polynomial pH(x) is not sos. Therefore, G is not sos-perfect. However, we purposefully want to avoid the use of the highly-nontrivial strong perfect graph theorem in the proof of Theorem 1.1.

Indeed, our hope is that Theorem 1.1 could lead to an algebraic proof of the strong perfect graph theorem in the future (see Section 2.6.1).

2.4 Nonnegative Polynomials That Are Not Sums of Squares

As mentioned in Section 2.2.2, Hilbert proved the existence of nonnegative polynomials that are not sums of squares in [79], while the first examples of such polynomials were

1The fact that for an odd hole or an odd antihole G, the inequality ω(G) < ϑ⁰(G) holds also follows immediately from Lemma 2.3.5 since odd holes and odd antiholes are clearly minimal imperfect graphs.

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constructed by Motzkin [96] and Robinson [116] many years later. Many other examples have appeared in the literature over the years; see, e.g., [115, 33, 34, 35]. Understanding the distinction between nonnegative polynomials and sos polynomials is an active area of research. In relatively low degrees and dimensions, constructing examples of nonnegative polynomials that are not sos seems to be a nontrivial task.

In Section 2.4.1, we provide several infinite families of nonnegative polynomials that are not sos through various families of imperfect graphs. In Section 2.4.2, we describe certain operations on graphs that allow us to generate more nonnegative polynomials that are not sos starting from existing ones. In Section 2.4.3, by appealing to the literature on random graph theory, we show that for a fixed parameter p and for large enough n, the polynomial pGn,p(x) associated with the Erd˝os-Rényi random graph G_n,p is nonnegative but not sos with high probability. In Section 2.4.4, we provide an explicit hyperplane that separates a given non-sos polynomial p_G(x) from the set of sos polynomials. Finally, in Section 2.4.5, we show that an example of a convex nonnegative polynomial that is not sos cannot arise from our graph-theoretic constructions. The construction of such a polynomial was an open problem until recently [117].

2.4.1 From imperfect graphs to nonnegative polynomials that are not sos

Odd holes and odd antiholes

Recall that odd holes and odd antiholes are minimal imperfect graphs, i.e., they are not perfect but their proper induced subgraphs are all perfect. Hence, by Lemma 2.3.5 and Corollary 2.3.2, for every odd hole and odd antihole G, the polynomial pG(x) is a nonnegative polynomial that is not sos. This yields an infinite family of degree-4 polynomials that are nonnegative but not sos. As an example, consider the smallest minimal imperfect graph

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C₅and the corresponding polynomial p_C₅(x):

p_C₅(x) = −4(x²₁x²₂+ x²₂x²₃+ x²₃x²₄+ x²₄x²₅+ x²₁x²₅) + (x²₁+ x²₂ + x²₃+ x²₄+ x²₅)².

Similarly, the polynomials p_C₇(x), p_C₉(x), . . . , p_C₇(x), p_C₉(x), . . . are all nonnegative but not sos. We recall that for m ≥ 2, we have ω(C_2m+1) = 2 and ω(C_2m+1) = m, and therefore it is immediate to explicitly write down the polynomial pG(x) when G is an odd hole or an odd antihole.

Powers of cycles and their complements

The conclusion of Lemma 2.3.5 holds for a more general class of graphs than minimal imperfect graphs. A graph G is called partitionable (or an (α, ω)-graph) if there exist integers α ≥ 2 and ω ≥ 2 such that |V (G)| = αω + 1, and for every vertex v, there is a partition of V (G) \ {v} into α many cliques of size ω and ω many independent sets of size α. Using only elementary linear algebra, the authors in [22] show that every partitionable graph G with n vertices has n maximum cliques, every vertex of G is in exactly ω(G) maximum cliques, and the max-clique matrix C of G is non-singular. Notice that these were the only properties of minimal imperfect graphs that were used to prove Lemma 2.3.5. Hence, the same proof implies that if G is a partitionable graph, then ω(G) < ϑ⁰(G). Consequently, by Corollary 2.3.2, partitionable graphs provide an infinite family of polynomials that are nonnegative but not sos.

It can be shown that a minimal imperfect graph G is partitionable with α = α(G) and ω = ω(G). There are, however, many other partitionable graphs (see, e.g., [48]). The simplest examples are “powers” of cycles. The k^thpowerof a graph G, denoted by G^k, is a graph with the same vertex set as G, and with two vertices adjacent if and only if their distance² in G is at most k. For every integer α ≥ 2 and ω ≥ 2, the graph C_αω+1^ω−1 is a partitionable graph; see [48]. Notice that if ω = 2, then the graphs C_αω+1^ω−1 = C_2α+1 are

2The distance between two vertices in a graph is the number of edges in a shortest path between them.

PERFECT GRAPHS SUMS OF SQUARES AND CEMIL DIBEK A DISSERTATION PRESENTED TO THE FACULTY RECOMMENDED FOR ACCEPTANCE

P ERFECT G RAPHS

AND

S UMS OF S QUARES

Abstract

Acknowledgements

Contents

List of Tables

List of Figures

Chapter 1

Introduction

1.1 Outline of This Thesis

X

X

1.2 Related Publications

Chapter 2

A Sum of Squares Characterization of Perfect Graphs

2.1 Introduction

X

X

2.2 Preliminaries

2.2.1 Perfect graphs

2.2.2 Sum of squares polynomials

2.3 A Sum of Squares Characterization of Perfect Graphs

X

X

X

X

2.4 Nonnegative Polynomials That Are Not Sums of Squares

2.4.1 From imperfect graphs to nonnegative polynomials that are not sos

P ^ERFECT G ^RAPHS

S ^{UMS OF} S ^QUARES