k-SUM Problem Research Articles

Counting vanishing matrix-vector products

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces. Let v∈Qd be a rational vector, (T1,T2…,Tm) a list of d×d rational matrices, S∈Qh×d a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices Ti1,Ti2,…,Tik from the list such that STik⋯Ti1v=0∈Qh.In this paper, we show that this problem is #W[2]-hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for d>3 is #W[2]-hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show W[1]/W[2]-hardness. This is in contrast to the parameterized k-sum problem, which is only W[1]-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.

Polyomino: A 3D-SRAM-Centric Accelerator for Randomly Pruned Matrix Multiplication With Simple Reordering Algorithm and Efficient Compression Format in 180-nm CMOS

We have developed a sparse matrix reordering algorithm with a novel 3D-SRAM-centric Polyomino accelerator that enables efficient processing of the reordered matrix for parameter compression. By reordering randomly pruned, irregularly structured sparse matrices into regularly structured matrices, both the compression ratio of the data and the efficiency of the hardware processing increase. The reordering algorithm can be implemented simply by attributing it to the widely known k-sum problem. We also developed a compression format for storing the reordered matrices and show that the reordered regular structure can reduce the amount of required memory by 63% compared with the conventional method. The proposed Polyomino accelerator can efficiently process reordered matrices by using a 3D stacked SRAM, which is an external memory with random accessibility and low latency. The measurement results using a test chip fabricated in a 180-nm CMOS process demonstrate that the proposed accelerator can achieve high area-efficiency and high energy-efficiency and scales well with the pruning rate.

A Nearly Quadratic Bound for Point-Location in Hyperplane Arrangements, in the Linear Decision Tree Model

We consider the point location problem in an arrangement of n arbitrary hyperplanes in any dimension d, in the linear decision tree model, in which we only count linear comparisons involving the query point, and all other operations do not explicitly access the query and are for free. We mainly consider the simpler variant (which arises in many applications) where we only want to determine whether the query point lies on some input hyperplane. We present an algorithm that performs a point location query with $$O(d^2\log n)$$ linear comparisons, improving the previous best result by about a factor of d. Our approach is a variant of Meiser’s technique for point location (Inf Comput 106(2):286–303, 1993) (see also Cardinal et al. in: Proceedings of the 24th European symposium on algorithms, 2016), and its improved performance is due to the use of vertical decompositions in an arrangement of hyperplanes in high dimensions, rather than bottom-vertex triangulation used in the earlier approaches. The properties of such a decomposition, both combinatorial and algorithmic (in the standard real RAM model), are developed in a companion paper (Ezra et al. arXiv:1712.02913 , 2017), and are adapted here (in simplified form) for the linear decision tree model. Several applications of our algorithm are presented, such as the k-SUM problem and the Knapsack and SubsetSum problems. However, these applications have been superseded by the more recent result of Kane et al. (in: Proceedings of the 50th ACM symposium on theory of computing, 2018), obtained after the original submission (and acceptance) of the conference version of our paper (Ezra and Sharir in: Proceedings of the 33rd international symposium on computational geometry, 2017). This result only applies to ‘low-complexity’ hyperplanes (for which the $$\ell _1$$ -norm of their coefficient vector is a small integer), which arise in the aforementioned applications. Still, our algorithm has currently the best performance for arbitrary hyperplanes.

On the Power of Non-adaptive Learning Graphs

We introduce a notion of the quantum query complexity of a certificate structure. This is a formalization of a well-known observation that many quantum query algorithms only require the knowledge of the position of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays and generalizes the quantum query lower bound for the k-sum problem derived recently by Belovs and ?palek (Proceeding of 4th ACM ITCS, 323---328, 2013). Finally, we use these results to show that the learning graph for the triangle problem by Lee et al. (Proceeding of 24th ACM-SIAM SODA, 1486---1502, 2013) is almost optimal in the above settings. This also gives a quantum query lower bound for the triangle sum problem.