We study the problem of deciding whether an n-variate polynomial, presented as an arithmetic circuit G, contains a degree k square-free term with an odd coefficient. We show that if G can be evaluated over the integers modulo 2 k+1 in time t and space s, the problem can be decided with constant probability in O((kn + t)2 k ) time and O(kn + s) space. Based on this, we present new and faster algorithms for two well studied problems: (i) an O * (2 mk ) algorithm for the m-set k-packing problem and (ii) an O * (2 3k/2 ) algorithm for the simple k-path problem, or an O * (2 k ) algorithm if the graph has an induced k-subgraph with an odd number of Hamiltonian paths. Our algorithms use poly(n) random bits, comparing to the 2 O(k) random bits required in prior algorithms, while having similar low space requirements.

We present an algorithm that on input a graph G with n vertices and m + n − 1 edges and a value k, produces an incremental sparsifierĜ with n − 1 + m/k edges, such that the condition number of G withĜ is bounded above byÕ(k log 2 n), with probability 1 − p. The algorithm runs in timẽ O((m log n + n log 2 n) log(1/p)). 1As a result, we obtain an algorithm that on input an n × n symmetric diagonally dominant matrix A with m + n − 1 non-zero entries and a vector b, computes a vectorx satisfying ||x − AThe solver is based on a recursive application of the incremental sparsifier that produces a hierarchy of graphs which is then used to construct a recursive preconditioned Chebyshev iteration.

We present an improved algorithm for solving symmetrically diagonally dominant linear systems. On input of an n × n symmetric diagonally dominant matrix A with m non-zero entries and a vector b such that Ax = b for some (unknown) vectorx, our algorithm computes a vector x such that ||x −x||A < ||x||A 1 in timẽThe solver utilizes in a standard way a 'preconditioning' chain of progressively sparser graphs. To claim the faster running time we make a two-fold improvement in the algorithm for constructing the chain. The new chain exploits previously unknown properties of the graph sparsification algorithm given in [Koutis,Miller,Peng, FOCS 2010], allowing for stronger preconditioning properties. We also present an algorithm of independent interest that constructs nearly-tight low-stretch spanning trees in timeÕ(m log n), a factor of O(log n) faster than the algorithm in [Abraham,Bartal,Neiman, FOCS 2008]. This speedup directly reflects on the construction time of the preconditioning chain.

The algebraic framework introduced in [Koutis, Proc. of the 35 th ICALP 2008] reduces several combinatorial problems in parameterized complexity to the problem of detecting multilinear degree-k monomials in polynomials presented as circuits. The best known (randomized) algorithm for this problem requires only O * (2 k ) time and oracle access to an arithmetic circuit, i.e. the ability to evaluate the circuit on elements from a suitable group algebra. This algorithm has been used to obtain the best known algorithms for several parameterized problems. In this paper we use communication complexity to show that the O * (2 k ) algorithm is essentially optimal within this evaluation oracle framework. On the positive side, we give new applications of the method: finding a copy of a given tree on k nodes, a spanning tree with at least k leaves, a minimum set of nodes that dominate at least t nodes, and an m-dimensional k-matching. In each case we achieve a faster algorithm than what was known. We also apply the algebraic method to problems in exact counting. Among other results, we show that a combination of dynamic programming and a variation of the algebraic method can break the trivial upper bounds for exact parameterized counting in fairly general settings.

Linear systems and eigen-calculations on symmetric diagonally dominant matrices (SDDs) occur ubiquitously in computer vision, computer graphics, and machine learning. In the past decade a multitude of specialized solvers have been developed to tackle restricted instances of SDD systems for a diverse collection of problems including segmentation, gradient inpainting and total variation. In this paper we explain and apply the support theory of graphs, a set of of techniques developed by the computer science theory community, to construct SDD solvers with provable properties. To demonstrate the power of these techniques, we describe an efficient multigrid-like solver which is based on support theory principles. The solver tackles problems in fairly general and arbitrarily weighted topologies not supported by prior solvers. It achieves state of the art empirical results while providing robust guarantees on the speed of convergence. The method is evaluated on a variety of vision applications.