Polynomials Of Bounded Degree Research Articles

Upper tails via high moments and entropic stability

Suppose that X is a bounded-degree polynomial with nonnegative coefficients on the p-biased discrete hypercube. Our main result gives sharp estimates on the logarithmic upper tail probability of X whenever an associated extremal problem satisfies a certain entropic stability property. We apply this result to solve two long-standing open problems in probabilistic combinatorics: the upper tail problem for the number of arithmetic progressions of a fixed length in the p-random subset of the integers and the upper tail problem for the number of cliques of a fixed size in the random graph Gn,p. We also make significant progress on the upper tail problem for the number of copies of a fixed regular graph H in Gn,p. To accommodate readers who are interested in learning the basic method, we include a short, self-contained solution to the upper tail problem for the number of triangles in Gn,p for all p=p(n) satisfying n−1logn≪p≪1.

Existence and Efficiency of Equilibria for Cost-Sharing in Generalized Weighted Congestion Games

This work studies the impact of cost-sharing methods on the existence and efficiency of (pure) Nash equilibria in weighted congestion games. We also study generalized weighted congestion games , where each player may control multiple commodities. Our results are fairly general; we only require that our cost-sharing method and our set of cost functions satisfy certain natural conditions. For general weighted congestion games, we study the existence of pure Nash equilibria in the induced games, and we exhibit a separation from the standard single-commodity per player model by proving that the Shapley value is the only cost-sharing method that guarantees existence of pure Nash equilibria. With respect to efficiency, we present general tight bounds on the price of anarchy, which are robust and apply to general equilibrium concepts. Our analysis provides a tight bound on the price of anarchy, which depends only on the used cost-sharing method and the set of allowable cost functions. Interestingly, the same bound applies to weighted congestion games and generalized weighted congestion games. We then turn to the price of stability and prove an upper bound for the Shapley value cost-sharing method, which holds for general sets of cost functions and which is tight in special cases of interest, such as bounded degree polynomials. Also for bounded degree polynomials, we provide a somewhat surprising result, showing that a slight deviation from the Shapley value has a huge impact on the price of stability. In fact, for this case, the price of stability becomes as bad as the price of anarchy. Again, our bounds on the price of stability are independent on whether players are single or multi-commodity.

Sparsification of SAT and CSP Problems via Tractable Extensions

Unlike polynomial kernelization in general, for which many non-trivial results and methods exist, only few non-trival algorithms are known for polynomial-time sparsification. Furthermore, excepting problems on restricted inputs (such as graph problems on planar graphs), most such results rely upon encoding the instance as a system of bounded-degree polynomial equations. In particular, for satisfiability (SAT) problems with a fixed constraint language Γ, every previously known result is captured by this approach; for several such problems, this is known to be tight. In this work, we investigate the limits of this approach—in particular, does it really cover all cases of non-trivial polynomial-time sparsification? We generalize the method using tools from the algebraic approach to constraint satisfaction problems (CSP). Every constraint that can be modelled via a system of linear equations, over some finite field F, also admits a finite domain extension to a tractable CSP with a Maltsev polymorphism; using known algorithms for Maltsev languages, we can show that every problem of the latter type admits a “basis” of O ( n ) constraints, which implies a linear sparsification for the original problem. This generalization appears to be strict; other special cases include constraints modelled via group equations over some finite group G . For sparsifications of polynomial but super-linear size, we consider two extensions of this. Most directly, we can capture systems of bounded-degree polynomial equations in a “lift-and-project” manner, by finding Maltsev extensions for constraints over c -tuples of variables, for a basis with O ( n c ) constraints. Additionally, we may use extensions with k -edge polymorphisms instead of requiring a Maltsev polymorphism. We also investigate characterizations of when such extensions exist. We give an infinite sequence of partial polymorphisms φ 1 , φ 2 , …which characterizes whether a language Γ has a Maltsev extension (of possibly infinite domain). In the complementary direction of proving lower bounds on kernelizability, we prove that for any language not preserved by φ 1 , the corresponding SAT problem does not admit a kernel of size O ( n 2−ε ) for any ε > 0 unless the polynomial hierarchy collapses.

On the Bias of Reed--Muller Codes over Odd Prime Fields

We study the bias of random bounded-degree polynomials over odd prime fields and show that, with probability exponentially close to 1, $n$-variate polynomials of degree $d$ over $\mathbb{F}_p$ have...

Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials

Hilbert famously showed that polynomials in n n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n n ) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert’s results are not uniform in n n : unsurprisingly, polynomials in n n variables will exhibit greater complexity as n n increases. However, an array of recent work has shown that in a certain regime—namely, that where the number of polynomials and their degrees are fixed—the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman’s conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman’s conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables.

The method of shifted partial derivatives cannot separate the permanent from the determinant

The method of shifted partial derivatives introduced A. Gupta et al. [Approaching the chasm at depth four, IEEE Comp. Soc., 2013, pp. 65-73] and N. Kayal [An exponential lower bound for the sum of powers of bounded degree polynomials, ECCC 19, 2010, p. 81], was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent ℓ n − m p e r m m \ell ^{n-m}\mathrm {perm}_m cannot be realized inside the G L n 2 GL_{n^2} -orbit closure of the determinant d e t n \mathrm {det}_n when n > 2 m 2 + 2 m n>2m^2+2m . Our proof relies on several simple degenerations of the determinant polynomial, Macaulay’s theorem, which gives a lower bound on the growth of an ideal, and a lower bound estimate from [Approaching the chasm at depth four, IEEE Comp. Soc., 2013, pp. 65-73] regarding the shifted partial derivatives of the determinant.

Kinetic Spanners in ℝd

We present a new (1+ε)-spanner for sets of n points in ℝ d . Our spanner has size O(n/ε d−1) and maximum degree O(log d n). The main advantage of our spanner is that it can be maintained efficiently as the points move: Assuming that the trajectories of the points can be described by bounded-degree polynomials, the number of topological changes to the spanner is O(n 2/ε d−1), and using a supporting data structure of size O(nlog d n), we can handle events in time O(log d+1 n). Moreover, the spanner can be updated in time O(log n) if the flight plan of a point changes. This is the first kinetic spanner for points in ℝ d whose performance does not depend on the spread of the point set.

Exact Price of Anarchy for Polynomial Congestion Games

We show exact values for the worst-case price of anarchy in weighted and unweighted (atomic unsplittable) congestion games, provided that all cost functions are bounded-degree polynomials with nonnegative coefficients. The given values also hold for weighted and unweighted network congestion games.

Kinetic Facility Location

We present a deterministic kinetic data structure for the facility location problem that maintains a subset of the moving points as facilities such that, at any point of time, the accumulated cost for the whole point set is at most a constant factor larger than the optimal cost. In our scenario, each point can change its status between client and facility and moves continuously along a known trajectory in a d-dimensional Euclidean space, where d is a constant. Our kinetic data structure requires $\mathcal{O}(n(\log^{d}(n)+\log (nR)))$space in total, where $R:=\frac{\max_{p_{i}\in\mathcal{P}}{f_{i}}\cdot\max_{p_{i}\in\mathcal{P}}{d_{i}}}{\min_{p_{i}\in\mathcal {P}}{f_{i}}\cdot\min_{p_{i}\in\mathcal{P}}{d_{i}}}$, ℘={p 1,p 2,…,p n } is the set of given points, and f i , d i are the maintenance cost and the demand of a point p i , respectively. In case that each trajectory can be described by a bounded degree polynomial, we process $\mathcal{O}(n^{2}\log^{2}(nR))$events, each requiring $\mathcal{O}(\log^{d+1}(n)\cdot\log(nR))$time and $\mathcal {O}(\log(nR))$status changes.

Learning functions represented as multiplicity automata

We study the learnability of multiplicity automata in Angluin's exact learning model , and we investigate its applications. Our starting point is a known theorem from automata theory relating the number of states in a minimal multiplicity automaton for a function to the rank of its Hankel matrix. With this theorem in hand, we present a new simple algorithm for learning multiplicity automata with improved time and query complexity, and we prove the learnability of various concept classes. These include (among others): -The class of disjoint DNF, and more generally satisfy- O (1) DNF. -The class of polynomials over finite fields. -The class of bounded-degree polynomials over infinite fields. -The class of XOR of terms. -Certain classes of boxes in high dimensions. In addition, we obtain the best query complexity for several classes known to be learnable by other methods such as decision trees and polynomials over GF(2). While multiplicity automata are shown to be useful to prove the learnability of some subclasses of DNF formulae and various other classes, we study the limitations of this method. We prove that this method cannot be used to resolve the learnability of some other open problems such as the learnability of general DNF formulas or even k -term DNF for k = ω(log n ) or satisfy- s DNF formulas for s = ω(1). These results are proven by exhibiting functions in the above classes that require multiplicity automata with super-polynomial number of states.

Polycyclic-by-finite groups admit a bounded-degree polynomial structure

For a polycyclic-by-finite group $\Gamma$ , of Hirsch length $h$ , an affine (resp. polynomial) structure is a representation of $\Gamma$ into ${\rm Aff}({\Bbb R}^{h})$ (resp. ${\rm P}({\Bbb R}^h)$ , the group of polynomial diffeomorphisms) letting $\Gamma$ act properly discontinuously on ${\Bbb R}^{h}$ . Recently it was shown by counter-examples that there exist groups $\Gamma$ (even nilpotent ones) which do not admit an affine structure, thus giving a negative answer to a long-standing question of John Milnor. We prove that every polycyclic-by-finite group $\Gamma$ admits a polynomial structure, which moreover appears to be of a special (“simple”) type (called ”canonical”) and, as a consequence of this, consists entirely of polynomials of a bounded degree. The construction of this polynomial structure is a special case of an iterated Seifert Fiber Space construction, which can be achieved here because of a very strong and surprising cohomology vanishing theorem.

On range searching with semialgebraic sets

LetP be a set ofn points in ?d (whered is a small fixed positive integer), and let Γ be a collection of subsets of ?d, each of which is defined by a constant number of bounded degree polynomial inequalities. We consider the following Γ-range searching problem: GivenP, build a data structure for efficient answering of queries of the form, Given a ??Γ, count (or report) the points ofP lying in ?. Generalizing the simplex range searching techniques, we give a solution with nearly linear space and preprocessing time and withO(n1?1/b+?) query time, whered≤b≤2d?3 and ?>0 is an arbitrarily small constant. The acutal value ofb is related to the problem of partitioning arrangements of algebraic surfaces into cells with a constant description complexity. We present some of the applications of Γ-range searching problem, including improved ray shooting among triangles in ?3.

Lower bounds for algebraic decision trees

A topological method is given for obtaining lower bounds for the height of algebraic decision trees. The method is applied to the knapsack problem where an Ω(n 2) bound is obtained for trees with bounded-degree polynomial tests, thus extending the Dobkin-Lipton result for linear trees. Applications to the convex hull problem and the distinct element problem are also indicated. Some open problems are discussed.