Nonconstructive Argument Research Articles

A Convex Form That Is Not a Sum of Squares

Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist convex forms that are not sums of squares via a nonconstructive argument. We provide an explicit example of a convex form of degree 4 in 272 variables that is not a sum of squares. The form is related to the Cauchy-Schwarz inequality over the octonions. The proof uses symmetry reduction together with the fact (due to Blekherman) that forms of even degree that are near-constant on the unit sphere are convex. Using this same connection, we obtain improved bounds on the approximation quality achieved by the basic sum-of-squares relaxation for optimizing quaternary quartic forms on the sphere. Funding: This work was supported by the Australian Research Council [Grant DE210101056].

Preprocessing for Outerplanar Vertex Deletion: An Elementary Kernel of Quartic Size

In the {varvec{mathcal {F}}}-Minor-Free Deletion problem one is given an undirected graph {varvec{G}}, an integer {varvec{k}}, and the task is to determine whether there exists a vertex set {varvec{S}} of size at most {varvec{k}}, so that {varvec{G}}-{varvec{S}} contains no graph from the finite family {varvec{mathcal {F}}} as a minor. It is known that whenever {varvec{mathcal {F}}} contains at least one planar graph, then {varvec{mathcal {F}}}-Minor-Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size {varvec{k}}^{{varvec{mathcal {O}}}{} {textbf {(1)}}} [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most {varvec{k}} vertices from a graph to make it outerplanar. This is a special case of {varvec{mathcal {F}}}-Minor-Free Deletion for the family {varvec{mathcal {F}}} = {{varvec{K}}_{{textbf {4}}}, {varvec{K}}_{{{textbf {2,3}}}}}. The class of outerplanar graphs is arguably the simplest class of graphs for which no explicit kernelization size bounds are known. By exploiting the combinatorial properties of outerplanar graphs we present elementary reduction rules decreasing the size of a graph. This yields a constructive kernel with {varvec{mathcal {O}}}({varvec{k}}^{textbf {4}}) vertices and edges. As a corollary, we derive that any minor-minimal obstruction to having an outerplanar deletion set of size {varvec{k}} has {varvec{mathcal {O}}}({varvec{k}}^{textbf {4}}) vertices and edges.

About the complexity of two-stage stochastic IPs

We consider so called 2-stage stochastic integer programs (IPs) and their generalized form, so called multi-stage stochastic IPs. A 2-stage stochastic IP is an integer program of the form max { c^T x mid {mathcal {A}} x = b, ,l le x le u,, x in {mathbb {Z}}^{s + nt} } where the constraint matrix {mathcal {A}} in {mathbb {Z}}^{r n times s +nt} consists roughly of n repetitions of a matrix A in {mathbb {Z}}^{r times s} on the vertical line and n repetitions of a matrix B in {mathbb {Z}}^{r times t} on the diagonal. In this paper we improve upon an algorithmic result by Hemmecke and Schultz from 2003 [Hemmecke and Schultz, Math. Prog. 2003] to solve 2-stage stochastic IPs. The algorithm is based on the Graver augmentation framework where our main contribution is to give an explicit doubly exponential bound on the size of the augmenting steps. The previous bound for the size of the augmenting steps relied on non-constructive finiteness arguments from commutative algebra and therefore only an implicit bound was known that depends on parameters r, s, t and Delta , where Delta is the largest entry of the constraint matrix. Our new improved bound however is obtained by a novel theorem which argues about intersections of paths in a vector space. As a result of our new bound we obtain an algorithm to solve 2-stage stochastic IPs in time f(r,s,Delta ) cdot mathrm {poly}(n,t), where f is a doubly exponential function. To complement our result, we also prove a doubly exponential lower bound for the size of the augmenting steps.

A sufficiently complicated noded Schottky group of rank three

In 1974, Marden proved the existence of non-classical Schottky groups by a theoretical and non-constructive argument. Explicit examples are only known in rank two; the first one by Yamamoto in 1991 and later by Williams in 2009. In 2006, Maskit and the author provided a theoretical method to construct non-classical Schottky groups in any rank. The method assumes the knowledge of certain algebraic limits of Schottky groups, called sufficiently complicated noded Schottky groups. The aim of this paper is to provide explicitly a sufficiently complicated noded Schottky group of rank three and explain how to use it to construct explicit non-classical Schottky groups.

Polytopal Bier Spheres and Kantorovich–Rubinstein Polytopes of Weighted Cycles

The problem of deciding if a given triangulation of a sphere can be realized as the boundary sphere of a simplicial, convex polytope is known as the ‘Simplicial Steinitz problem’. It is known by an indirect and non-constructive argument that a vast majority of Bier spheres are non-polytopal. Contrary to that, we demonstrate that the Bier spheres associated to threshold simplicial complexes are all polytopal. Moreover, we show that all Bier spheres are starshaped. We also establish a connection between Bier spheres and Kantorovich–Rubinstein polytopes by showing that the boundary sphere of the KR-polytope associated to a polygonal linkage (weighted cycle) is isomorphic to the Bier sphere of the associated simplicial complex of “short sets”.

On Multiparty Communication with Large versus Unbounded Error

The unbounded-error communication complexity of a Boolean function $F$ is the limit of the $\epsilon$-error randomized complexity of $F$ as $\epsilon\to1/2.$ Communication complexity with weakly unbounded error is defined similarly but with an additive penalty term that depends on $1/2-\epsilon$. Explicit functions are known whose two-party communication complexity with unbounded error is logarithmic compared to their complexity with weakly unbounded error. Chattopadhyay and Mande (ECCC TR16-095, Theory of Computing 2018) recently generalized this exponential separation to the number-on-the-forehead multiparty model. We show how to derive such an exponential separation from known two-party work, achieving a quantitative improvement along the way. We present several proofs here, some as short as half a page. In more detail, we construct a $k$-party communication problem $F\colon(\{0,1\}^{n})^{k}\to\{0,1\}$ that has complexity $O(\log n)$ with unbounded error and $\Omega(\sqrt n\,/\,4^{k})$ with weakly unbounded error, reproducing the bounds of Chattopadhyay and Mande. In addition, we prove a quadratically stronger separation of $O(\log n)$ versus $\Omega(n\,/\,4^k)$ using a nonconstructive argument. A preliminary version of this paper appeared in ECCC, Report TR16-138, 2016.

Belief Propagation Guided Decimation Fails on Random Formulas

Let Φ be a uniformly distributed random k -SAT formula with n variables and m clauses. Nonconstructive arguments show that Φ is satisfiable for clause/variable ratios m / n ⩽ r k − SAT ∼ 2 k ln 2 with high probability. Yet no efficient algorithm is known to find a satisfying assignment beyond m / n ∼ 2 k ln ( k )/ k with a nonvanishing probability. On the basis of deep but nonrigorous statistical mechanics ideas, a message passing algorithm called Belief Propagation Guided Decimation has been put forward (Mézard, Parisi, Zecchina: Science 2002; Braunstein, Mézard, Zecchina: Random Struc. Algorithm 2005). Experiments suggested that the algorithm might succeed for densities very close to r k − SAT for k = 3, 4, 5 (Kroc, Sabharwal, Selman: SAC 2009). Furnishing the first rigorous analysis of this algorithm on a nontrivial input distribution, in the present article we show that Belief Propagation Guided Decimation fails to solve random k -SAT formulas already for m / n = O (2 k / k ), almost a factor of k below the satisfiability threshold r k − SAT . Indeed, the proof refutes a key hypothesis on which Belief Propagation Guided Decimation hinges for such m / n .

Uniform observability estimates for linear waves

In this article, we give a completely constructive proof of the observability/controllability of the wave equation on a compact manifold under optimal geometric conditions. This contrasts with the original proof of Bardos–Lebeau–Rauch [C. Bardos, G. Lebeau and J. Rauch, SIAM J. Control Optim. 30 (1992) 1024–1065], which contains two non-constructive arguments. Our method is based on the Dehman-Lebeau [B. Dehman and G. Lebeau, SIAM J. Control Optim. 48 (2009) 521–550] Egorov approach to treat the high-frequencies, and the optimal unique continuation stability result of the authors [C. Laurent and M. Leautaud. Preprint arXiv:1506.04254 (2015)] for the low-frequencies. As an application, we first give estimates of the blowup of the observability constant when the time tends to the limit geometric control time (for wave equations with possibly lower order terms). Second, we provide (on manifolds with or without boundary) with an explicit dependence of the observability constant with respect to the addition of a bounded potential to the equation.

Lower bounds on complexity of Lyapunov functions for switched linear systems

We show that for any positive integer d, there are families of switched linear systems–in fixed dimension and defined by two matrices only–that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤d, or (ii) a polytopic Lyapunov function with ≤d facets, or (iii) a piecewise quadratic Lyapunov function with ≤d pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates.Several constructive and non-constructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and non-algebraicity of the joint spectral radius. In particular, we show that existence of an extremal piecewise algebraic Lyapunov function implies the finiteness property of the optimal product, generalizing a result of Lagarias and Wang. As a corollary, we prove that the finiteness property holds for sets of matrices with an extremal Lyapunov function belonging to some of the most popular function classes in controls.

A navigation system for tree space

The reconstruction of evolutionary trees from data sets on overlapping sets of species is a central problem in phylogenetics. Provided that the tree reconstructed for each subset of species is rooted and that these trees fit together consistently, the space of all parent trees that 'display' these trees was recently shown to satisfy the following strong property: there exists a path from any one parent tree to any other parent tree by a sequence of local rearrangements (nearest neighbour interchanges) so that each intermediate tree also lies in this same tree space. However, the proof of this result uses a non-constructive argument. In this paper we describe a specific, polynomial-time procedure for navigating from any given parent tree to another while remaining in this tree space. The results are of particular relevance to the recent study of 'phylogenetic terraces'.

From Small Space to Small Width in Resolution

In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of a Conjunctive Normal Form (CNF) formula is always an upper bound on the width needed to refute the formula. Their proof is beautiful but uses a nonconstructive argument based on Ehrenfeucht-Fraïssé games. We give an alternative, more explicit, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a “black-box” technique for proving space lower bounds via a “static” complexity measure that works against any resolution refutation—previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.

Systematic Error-Correcting Codes for Rank Modulation

The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. In this paper, we explore [n, k, d] systematic error-correcting codes for rank modulation. Such codes have length n, k information symbols, and minimum distance d. Systematic codes have the benefits of enabling efficient information retrieval in conjunction with memory-scrubbing schemes. We study systematic codes for rank modulation under Kendall's T-metric as well as under the l ∞ -metric. In Kendall's T-metric, we present [k + 2, k, 3] systematic codes for correcting a single error, which have optimal rates, unless systematic perfect codes exist. We also study the design of multierror-correcting codes, and provide a construction of [k + t + 1, k, 2t + 1] systematic codes, for large-enough k. We use nonconstructive arguments to show that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes. Finally, in the l ∞ -metric, we construct two [n, k, d] systematic multierror-correcting codes, the first for the case of d = 0(1) and the second for d = Θ(n). In the latter case, the codes have the same asymptotic rate as the best codes currently known in this metric.

Revisiting Zariski Main Theorem from a constructive point of view

This paper deals with the Peskine version of Zariski Main Theorem published in 1965 and discusses some applications. It is written in the style of Bishop's constructive mathematics. Being constructive, each proof in this paper can be interpreted as an algorithm for constructing explicitly the conclusion from the hypothesis. The main non-constructive argument in the proof of Peskine is the use of minimal prime ideals. Essentially we substitute this point by two dynamical arguments; one about gcd's, using subresultants, and another using our notion of strong transcendence. In particular we obtain algorithmic versions for the Multivariate Hensel Lemma and the structure theorem of quasi-finite algebras.

On Complexity of Lyapunov Functions for Switched Linear Systems

We show that for any positive integer d, there are families of switched linear systems—in fixed dimension and defined by two matrices only—that are stable under arbitrary switching but do not admit (i) a polynomial Lyapunov function of degree ≤ d, or (ii) a polytopic Lyapunov function with ≤ d facets, or (iii) a piecewise quadratic Lyapunov function with ≤ d pieces. This implies that there cannot be an upper bound on the size of the linear and semidefinite programs that search for such stability certificates. Several constructive and non-constructive arguments are presented which connect our problem to known (and rather classical) results in the literature regarding the finiteness conjecture, undecidability, and non-algebraicity of the joint spectral radius. In particular, we show that existence of a sum of squares Lyapunov function implies the finiteness property of the optimal product.

Approximating fixed points of α-nonexpansive mappings in uniformly convex Banach spaces and "Equation missing"spaces

AbstractAn existence theorem for a fixed point of anα-nonexpansive mapping of a nonempty bounded, closed and convex subset of a uniformly convex Banach space has been recently established by Aoyama and Kohsaka with a non-constructive argument. In this paper, we show that appropriate Ishikawa iterate algorithms ensure weak and strong convergence to a fixed point of such a mapping. Our theorems are also extended toCAT(0)spaces.AMS Subject Classification:54E40, 54H25, 47H10, 37C25.

Amount of nonconstructivity in deterministic finite automata

When D. Hilbert used nonconstructive methods in his famous paper on invariants (1888), P. Gordan tried to prevent the publication of this paper considering these methods as non-mathematical. L.E.J. Brouwer in the early twentieth century initiated intuitionist movement in mathematics. His slogan was “nonconstructive arguments have no value for mathematics”. However, P. Erdös got many exciting results in discrete mathematics by nonconstructive methods. It is widely believed that these results either cannot be proved by constructive methods or the proofs would have been prohibitively complicated. The author (Freivalds, 2008) [10] showed that nonconstructive methods in coding theory are related to the notion of Kolmogorov complexity. We study the problem of the quantitative characterization of the amount of nonconstructiveness in nonconstructive arguments. We limit ourselves to computation by deterministic finite automata. The notion of nonconstructive computation by finite automata is introduced. Upper and lower bounds of nonconstructivity are proved.

Random Constraint Satisfaction Problems

Random instances of constraint satisfaction problems such as k-SAT provide challenging benchmarks. If there are m constraints over n variables there is typically a large range of densities r=m/n where solutions are known to exist with probability close to one due to non-constructive arguments. However, no algorithms are known to find solutions efficiently with a non-vanishing probability at even much lower densities. This fact appears to be related to a phase transition in the set of all solutions. The goal of this extended abstract is to provide a perspective on this phenomenon, and on the computational challenge that it poses.

Easily refutable subformulas of large random 3CNF formulas.

A simple nonconstructive argument shows that most 3CNF formulas with cn clauses (where c is a large enough constant) are not satisfiable. It is an open question whether there is an efficient refutation algorithm that for most formulas with cn clauses proves that they are not satisfiable. We present a polynomial time algorithm that for most 3CNF formulas with cn 3/2 clauses (where c is a large enough constant) finds a subformula with O(c 2 n) clauses and then proves that this subformula is not satisfiable (and hence that the original formula is not satisfiable). Previously, it was only known how to efficiently certify the unsatisfiability of random 3CNF formulas with at least poly(log(n)) · n 3/2 clauses. Our algorithm is simple enough to run in practice. We present some preliminary experimental results.

On Heilbronn?s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]2, there must be three points which form a triangle of area at most O(1/n2). This conjecture was disproved by a nonconstructive argument of Komlos, Pintz and Szemeredi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]2 where all triangles have area at least Ω(log n/n2). Considering a generalization of this problem to dimensions d≥3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least Ω(1/nd). We improve on this lower bound by a logarithmic factor Θ(log n).

Thue-like Sequences and Rainbow Arithmetic Progressions

A sequence $u=u_{1}u_{2}...u_{n}$ is said to be nonrepetitive if no two adjacent blocks of $u$ are exactly the same. For instance, the sequence $a{\bf bcbc}ba$ contains a repetition $bcbc$, while $abcacbabcbac$ is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set $\{a,b,c\}$. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this paper we consider a stronger property defined as follows. Let $k\geq 2$ be a fixed integer and let $C$ denote a set of colors (or symbols). A coloring $f:{\bf N}\rightarrow C$ of positive integers is said to be $k$-nonrepetitive if for every $r\geq 1$ each segment of $kr$ consecutive numbers contains a $k$-term rainbow arithmetic progression of difference $r$. In particular, among any $k$ consecutive blocks of the sequence $f=f(1)f(2)f(3)...$ no two are identical. By an application of the Lovász Local Lemma we show that the minimum number of colors in a $k$-nonrepetitive coloring is at most $2^{-1}e^{k(2k-1)/(k-1)^{2}}k^{2}(k-1)+1$. Clearly at least $k+1$ colors are needed but whether $O(k)$ suffices remains open. This and other types of nonrepetitiveness can be studied on other structures like graphs, lattices, Euclidean spaces, etc., as well. Unlike for the classical Thue sequences, in most of these situations non-constructive arguments seem to be unavoidable. A few of a range of open problems appearing in this area are presented at the end of the paper.