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Sublinear Extensions of Polygons

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Sublinear Extensions of Polygons SUBLINEAR EXTENSIONS OF POLYGONS YAROSLAV SHITOV Abstract. Every convex polygon with n vertices is a linear projection of a higher-dimensional polytope with at most 147 n2=3 facets. We revisit the following appealing question, which survived a thorough discussion in recent literature without receiving any definitive progress towards its resolution. Question 1. What is the smallest number, pc(n), such that every convex polygon with n vertices is a linear projection of some polytope with at most pc(n) facets? Apart from being a natural question to ask in polyhedral combinatorics, Ques- tion 1 is being studied from the point of view of modern optimization theory and linear algebra. Our approach is purely geometric, but still we need to recall several concepts of optimization theory to explain the relevance of Question 1. 1. Introduction Let P be a polytope, that is, the convex hull of a finite collection of points in a d-dimensional Euclidean space. An extended formulation of P is a pair (Q; π) consisting of a polytope Q ⊂ Rm and a linear projection π : Rm ! Rd such that π(Q) = P . The number of facets of such a polytope Q is called the size of the formulation (Q; π), and the extension complexity of P is the smallest possible size of any extended formulation of P . This quantity, further denoted by xc(P ), equals the smallest possible number of inequalities in a representation of any polytope that can be sent to P by a linear projection [17, 60]. As we see, Question 1 is devoted to the worst-case extension complexities of two-dimensional polytopes, so we decided to use the notation `pc' as an abbreviation of `polygon complexity.' Extended formulations became a prominent technique in optimization after a 1991 paper by Yannakakis [60], who discussed a possibility of building fast al- gorithms for hard combinatorial problems by constructing smaller extended for- mulations to linear programs corresponding to these problems. Several intrigu- ing questions of this nature were solved quite recently, which include the lack of arXiv:1412.0728v2 [math.CO] 29 Feb 2020 polynomial-size extended formulations for the polytope associated to the traveling salesman problem [18]. Another notable result [49] gives an exponential lower bound for the extension complexity of the matching polytope, which is in contrast to the existence of a polynomial-time algorithm solving the matching problem [20]. Other results on the topic include strong lower bounds for the complexities of the poly- topes associated to different exact and approximate optimization problems such as max-cut [9], independent set [24], knapsack [46] and many others [2]. The extension complexity is also being discussed for polytopes not necessarily arising from particu- lar combinatorial problems, including those with few vertices and facets [28, 29, 43], 2000 Mathematics Subject Classification. 52B05, 52B12, 15A23. Key words and phrases. Convex polytope, extended formulation, nonnegative matrices. 1 2 YAROSLAV SHITOV the correlation polytope [35], the permutahedron [25], 0-1 polytopes [8, 48], two- level polytopes [1], orbitopes [15], Cartesian products and hypersimplices [31] and others. These also include a prominent example of polygons [5, 19, 50, 55], which this paper is devoted to and which have `prototypical importance' to applications, according to Braun and Pokutta [6]. Also, the study of extended formulations has been developed to reflect the case of semidefinite programming [16, 27, 30], and it may require tools of linear algebra [60], information theory [7], communication complexity [14], quantum learning [38], tropical mathematics [51] and other fields; an interested reader is referred to survey papers [12, 34] for further details. Yannakakis [60] developed a linear algebraic approach to extended formulations based on nonnegative matrices, that is, matrices with nonnegative real entries. The nonnegative rank of such a matrix M is the smallest integer k for which M can be written as a sum of k nonnegative rank-one matrices. Now consider a polytope P , with v vertices and f facets, defined as a subset of Rd by the conditions f"(x) = α"; g'(x) > β'; where the letter " indexes a finite set of linear equations, the ' runs over the facets d of P , and the mappings f" and g' are linear functionals R ! R.A slack matrix of P can be defined by the formula Sν' = g'(ν) − β' with rows and columns of S labeled by the vertices and facets of P , respectively. It is not hard to see that the rank of S equals dim P +1; Yannakakis [60] proves that the nonnegative rank of S equals the extension complexity of P . We note in passing that the nonnegative rank is an important concept of linear algebra in its own right [21, 57] and for theoretical studies in demography, quantum mechanics [11], and statistics [37, 42]. Also, this notion is relevant for several real-life challenges like signal processing [23], text mining [10], and image processing [39]. Theorem 2. (See [60].) The extension complexity of a polytope P equals the non- negative rank of any slack matrix of P . Corollary 3. For n > 3, the value pc(n) equals the largest nonnegative rank of a nonnegative matrix with at most n rows and conventional rank three. Proof. This follows from Theorem 2 by standard techniques. In particular, we can realize pc(n) as the nonnegative rank of an n × n matrix of rank three by a direct application of this result to a slack matrix of a convex n-gon with extension com- plexity pc(n). The fact that the nonnegative rank of any n×m nonnegative matrix A with rank three cannot exceed pc(n) also follows from Theorem 2, as outlined in the proof of Theorem 3.1 in [50]. To give a sketch of the proof, we consider such a matrix A, and we denote by ∆ ⊂ Rn the simplex consisting of nonnegative vectors whose coordinates sum to 1. Since ∆ has n facets, the intersection of ∆ with the column space of A is a polygon P with k 6 n vertices. Denoting by S the matrix of the column coordinate vectors of the vertices of P , we have A = SB with B nonnegative. Since S is a slack matrix of P , it has nonnegative rank at most pc(n), and so does A. 2. Polygons on a plane Question 1 takes an important place in the study of extended formulations. A detailed account of this problem was carried out by the authors of [19], who adopted SUBLINEAR EXTENSIONS OF POLYGONS 3 p the dimension counting method and gave an Ω( n) lower bound for the extension complexity of a generic convex n-gon. More presicely, they obtained the inequalities p (2.1) 2n 6 pc(n) 6 n; in which the upper bound is trivial. Question 1 and its analogues were studied in at least a dozen of papers [3, 22, 27, 30, 33, 40, 43, 44, 45, 50, 51, 53, 54, 56] and mentioned in several dozens of other works, including a number of textbooks, surveys and highly cited research papers [7, 12, 17, 41, 47, 48, 58]. This question was discussed at open problem sessions of computer science conferences [4, 36] and online in popular media [13, 26], but it has seenp no progress on either bound except for improving the constant factors in front of n and n. Concerning the asymptotic behavior of pc(n), the authors of the initial paper [19] and most other experts seemed to expect that the upper bound in (2.1) is closer to the actual value. Braun and Pokutta [7] point out the importance of the following question and mention it alongside a list of major problems in the theory of extended formulations. Question 4. Do we have pc(n) > "n for some fixed " > 0? This question appeared in online media such as the Open Problem Garden [13] and MathOverflow [26]. An affirmative answer to Question 4 was conjectured in [3] and claimed to be proven in [40]. However, Hrubeˇs[33] showed that the argument of [40] is flawed and formulated a problem equivalent to Question 4; a further version of this question but specified to " = 0:5 appears in [56]. The authors of [19] expected an affirmative answer to another version obtained by replacing "n with Ω(~ n) in the formulation of Question 4, which would mean that the upper bound in (2.1) is tight up to logarithmic factors. Contrary to these expectations, we show that the lower bound in (2.1) is closer to the actual value of pc(n) in the asymptotical sense. 2=3 Theorem 5. We have pc(n) 6 147 n for all n > 3. We close the preliminary part of our paper with a more detailed description of the progress achieved on Questions 1 and 4 before the work we present. As far as we know, Beasley and Laffey [3] were the first to consider this question; they followed the linear algebraic approach in terms of Corollary 3. They proved that a special family of nonnegative n×n rank-three matrices, called Euclidean distance matrices, have nonnegative rank at least log n, and conjectured that the maximal value of this rank is n. In our notation, their result stated that pc(n) > log n, and the conjecture was pc(n) = n. A subsequent paper [40] claimed to prove this conjecture, but later works [22, 33] pointed out a flaw in their proof. The authors of [22] saved a part of the argument in [40] and showed that the so-called restricted nonnegative rank of Euclidean distance matrices equals n. They reiterated the question as to whether the equality pc(n) = n holds for general n and proved it for n not exceeding five.
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