The independence polynomial of a graph G is the generating polynomial corresponding to its independent sets of different sizes. More formally, if a_k(G) denotes the number of independent sets of G of size k then I(G,z) := ∑_k (-1)^k a_k(G) z^k. The study of evaluating I(G,z) has several deep connections to problems in combinatorics, complexity theory and statistical physics. Consequently, the roots of the independence polynomial have been studied in detail. In particular, many works have provided regions in the complex plane that are devoid of any roots of the polynomial. One of the first such results showed a lower bound on the absolute value of the smallest root β(G) of the polynomial. Furthermore, when G is connected, Goldwurm and Santini established that β(G) is a simple real root of I(G,z) smaller than one. An alternative proof was given by Csikvári. Both proofs do not provide a gap from β(G) to the smallest absolute value amongst all the other roots of I(G,z). In this paper, we quantify this gap.

Root separation bounds play an important role as a complexity measure in understanding the behaviour of various algorithms in computational algebra, e.g., root isolation algorithms. A classic result in the univariate setting is the Davenport-Mahler-Mignotte (DMM) bound. One way to state the bound is to consider a directed acyclic graph (V, E) on a subset of roots of a degree d polynomial f (z) ∈ C[z], where the edges point from a root of smaller absolute value to one of larger absolute, and the in-degrees of all vertices is at most one. Then the DMM bound is an amortized lower bound on the following product: Π(α, β)∈E |α - β|. However, the lower bound involves the discriminant of the polynomial f, and becomes trivial if the polynomial is not square-free. This was resolved by Eigenwillig, 2008, by using a suitable subdiscriminant instead of the discriminant. Escorcielo-Perrucci, 2016, further dropped the in-degree constraint on the graph by using the theory of finite differences. Emiris et al., 2019, have generalized their result to handle the case where the exponent of the term |α - β| in the product is at most the multiplicity of either of the roots. In this paper, we generalize these results by allowing arbitrary positive integer weights on the edges of the graph, i.e., for a weight function w : E → Z<0, we derive an amortized lower bound on Π(α, β)∈E |α - β|w(α, β). Such a product occurs in the complexity estimates of some recent algorithms for root clustering (e.g., Becker et al., 2016), where the weights are usually some function of the multiplicity of the roots. Because of its amortized nature, our bound is arguably better than the bounds obtained by manipulating existing results to accommodate the weights.

In this paper, we focus on lower bounds for data structures supporting orthogonal range querying on m points in n-dimensions in the semigroup model. Such a data structure usually maintains a family of "canonical subsets" of the given set of points and on a range query, it outputs a disjoint union of the appropriate subsets. Fredman showed that in order to prove lower bounds in the semigroup model, it suffices to prove a lower bound on a certain combinatorial tradeoff between two parameters: (a) the total sizes of the canonical subsets, and (b) the total number of canonical subsets required to cover all query ranges. In particular, he showed that the arithmetic mean of these two parameters is Omega(m log^n m). We strengthen this tradeoff by showing that the geometric mean of the same two parameters is Omega(m log^n m). Our second result is an alternate proof of Fredman's tradeoff in the one dimensional setting. The problem of answering range queries using canonical subsets can be formulated as factoring a specific boolean matrix as a product of two boolean matrices, one representing the canonical sets and the other capturing the appropriate disjoint unions of the former to output all possible range queries. In this formulation, we can ask what is an optimal data structure, i.e., a data structure that minimizes the sum of the two parameters mentioned above, and how does the balanced binary search tree compare with this optimal data structure in the two parameters? The problem of finding an optimal data structure is a non-linear optimization problem. In one dimension, Fredman's result implies that the minimum value of the objective function is Omega(m log m), which means that at least one of the parameters has to be Omega(m log m). We show that both the parameters in an optimal solution have to be Omega(m log m). This implies that balanced binary search trees are near optimal data structures for range querying in one dimension. We derive intermediate results on factoring matrices, not necessarily boolean, while trying to minimize the norms of the factors, that may be of independent interest.

A multivariate polynomial F(x1,x2,...,xn) is said to be absolutely positive from a real number B if F and all its partial derivatives are non-negative for x1,x2,...,xn ≥ B. One of the well known bounds on absolute positiveness in the literature is due to Hong. His bound is dependent on the first maximum of a certain sequence of radicals defined using the absolute value of the coefficients of the polynomial. In the univariate setting, a bound due to Lagrange considers the first and the second maximum in the same radical sequence and is shown by Collins to be better than Hong's bound. In the 1930's, Westerfield had proposed a bound that consides every value in the same radical sequence and improves on Lagrange's bound. In this paper, we provide a generalization of Westerfield's bound to the multivariate setting. As a specialization of this bound, we also derive a generalization of Lagrange's bound, which is a strict improvement upon Hong's bound. Finally, we give an algorithm to compute this improved bound. The running time of this algorithm matches the running time of the best known algorithm to compute Hong's bound.

Let F(z) be an arbitrary complex polynomial. We introduce the {local root clustering problem}, to compute a set of natural epsilon-clusters of roots of F(z) in some box region B0 in the complex plane. This may be viewed as an extension of the classical root isolation problem. Our contribution is two-fold: we provide an efficient certified subdivision algorithm for this problem, and we provide a bit-complexity analysis based on the local geometry of the root clusters.

Isolating real roots of a square-free polynomial in a given interval is a fundamental problem. Subdivision based algorithms are a standard approach to solve this problem. E.g., Sturm's method,or various algorithms based on the Descartes's rule of signs. For isolating all the real roots of a degree n polynomial with root separation σ, the subdivision tree size of most of these algorithms is bounded by O(log 1/σ) (assume σ < 1). Recently Sagraloff (2012) and Sagraloff-Mehlhorn (2013) have developed algorithms that combine subdivision with Newton iteration to reduce the size of the subdivision tree to O(n (log (nlog 1/σ))). Their algorithms and analysis crucially depend on the terminating predicates. We describe a subroutine that improves the running time of any subdivision algorithm for real root isolation. The subdivision tree size of our algorithm using predicates based on the Descartes's rule of signs is bounded by O(nlog n). Our analysis differs in two key aspects from earlier approaches. First, we use the general technique of continuous amortization from Burr-Krahmer-Yap (2009), and hence the analysis extends to other predicates; and second, we use the geometry of clusters of roots instead of root bounds.

The problem of isolating all real roots of a square-free integer polynomial f(X) inside any given interval I0 is a fundamental problem. EVAL is a simple and practical exact numerical algorithm for this problem: it recursively bisects I0, and any sub-interval I ⊆ I0, until a certain numerical predicate C0(I) V C1(I) holds on each I. We prove that the size of the recursion tree is

In this paper, we provide polynomial bounds on the worst case bit-complexity of two formulations of the continued fraction algorithm. In particular, for a square-free integer polynomial of degree $n$ with coefficients of bit-length $L$, we show that the bit-complexity of Akritas' formulation is $\wt{O}(n^8L^3)$, and the bit-complexity of a formulation by Akritas and Strzebo\'nski is $\wt{O}(n^7L^2)$; here $\wt{O}$ indicates that we are omitting logarithmic factors. The analyses use a bound by Hong to compute the floor of the smallest positive root of a polynomial, which is a crucial step in the continued fraction algorithm. We also propose a modification of the latter formulation that achieves a bit-complexity of $\wt{O}(n^5L^2)$.

We give a unified ("basis free") framework for the Descartes method for real root isolation of square-free real polynomials. This framework encompasses the usual Descartes' rule of sign method for polynomials in the power basis as well as its analog in the Bernstein basis. We then give a new bound on the size of the recursion tree in the Descartes method for polynomials with real coefficients. Applied to polynomials A(X) = Εni=0 aiXi with integer coefficients |ai| < 2L, this yields a bound of O(n(L + logn)) on the size of recursion trees. We show that this bound is tight for L = Ω(logn), and we use it to derive the best known bit complexity bound for the integer case.