Computer Science

In 2016, Karney proposed an exact sampling algorithm for the standard normal distribution. In this paper, we study the computational complexity of this algorithm under the random deviate model. Specifically, Karney's algorithm requires the access to an infinite sequence of independently and uniformly random deviates over the range (0,1). We give an estimate of the expected number of uniform deviates used by this algorithm until outputting a sample value, and present an improved algorithm with lower uniform deviate consumption. The experimental results also shows that our improved algorithm has better performance than Karney's algorithm.

We show that the CNF satisfiability problem can be solved O ∗ ( 1.2226 m ) time, where m is the number of clauses in the formula, improving the known upper bounds O ∗ ( 1.234 m ) given by Yamamoto 15 years ago and O ∗ ( 1.239 m ) given by Hirsch 22 years ago. By using an amortized technique and careful case analysis, we successfully avoid the bottlenecks in previous algorithms and get the improvement.

An SPQR-tree is a data structure that efficiently represents all planar embeddings of a biconnected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints. We develop an SPQR-tree-like data structure that represents all level-planar embeddings of a biconnected level graph with a single source, called the LP-tree, and give a simple algorithm to compute it in linear time. Moreover, we show that LP-trees can be used to adapt three constrained planarity algorithms to the level-planar case by using them as a drop-in replacement for SPQR-trees.

We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09] for low-rank matrices [Wossnig et al., Physical Review Letters'18], when the input matrix A is stored in a data structure applicable for QRAM-based state preparation. Namely, given an A∈ C m×n with minimum singular value σ and which supports certain efficient ℓ 2 -norm importance sampling queries, along with a b∈ C m , we can output a description of an x∈ C n such that ∥x− A + b∥≤ε∥ A + b∥ in O ~ ( ∥A ∥ 6 F ∥A ∥ 2 σ 8 ε 4 ) time, improving on previous "quantum-inspired" algorithms in this line of research by a factor of ∥A ∥ 14 σ 14 ε 2 [Chia et al., STOC'20]. The algorithm is stochastic gradient descent, and the analysis bears similarities to those of optimization algorithms for regression in the usual setting [Gupta and Sidford, NeurIPS'18]. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired setting, for comparison against future quantum computers.

The weighted maximal bisection problem is, given an edge weighted graph, to find a bipartition of the vertex set into two sets such that their cardinality differs by at most one and the sum of the weight of the edges between vertices that are not in the same set is maximized. This problem is known to be NP-hard, even when a tree decomposition of width t and O(n) nodes is given as an advice as part of the input, where n is the number of vertices of the input graph. But, given such an advice, the problem is decidable in FPT time in n parametrized by t . In particular Jansen et al. presented an algorithm with running time O( 2 t n 3 ) . Hanaka, Kobayashi, and Sone enhanced the analysis of the complexity to O( 2 t (nt ) 2 ) . By slightly modifying the approach, we improve the running time to O( 2 t n 2 ) in the RAM model, which is asymptotically optimal in n under the hardness of MinConv. We proof that this is also asymptotically optimal in its dependence on t assuming SETH by showing for a slightly easier problem (maximal cut) that there is no O( 2 ϵt polyn) algorithm for any ε<1 under SETH. This was already claimed by Hanaka, Kobayashi, and Sone but without a correct proof. We also present a hardness result (no O( 2 t n 2?��?) algorithm for any ε>0 ) for a broad family of subclasses of the weighted maximal bisection problem that are characterized only by the dependence of t from n , more precisely, all instances with t=f(n) for an arbitrary but fixed f(n)?�o(logn) . This holds even when only considering planar graphs. Moreover we present a detailed description of the implementation details and assumptions that are necessary to achieve the optimal running time.

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this conjecture, and also studied its implied fractional versions. We establish the fractional version of the Aharoni-Zerbib conjecture up to lower order terms. Specifically, we give a factor t/2+O( tlogt − − − − − √ ) approximation based on LP rounding for an algorithmic version of the hypergraph Turán problem (AHTP). The objective in AHTP is to pick the smallest collection of (t−1) -sized subsets of vertices of an input t -uniform hypergraph such that every hyperedge contains one of these subsets. Aharoni and Zerbib also posed whether Tuza's conjecture and its hypergraph versions could follow from non-trivial duality gaps between vertex covers and matchings on hypergraphs that exclude certain sub-hypergraphs, for instance, a "tent" structure that cannot occur in the incidence of triangles and edges. We give a strong negative answer to this question, by exhibiting tent-free hypergraphs, and indeed F -free hypergraphs for any finite family F of excluded sub-hypergraphs, whose vertex covers must include almost all the vertices. The algorithmic questions arising in the above study can be phrased as instances of vertex cover on simple hypergraphs, whose hyperedges can pairwise share at most one vertex. We prove that the trivial factor t approximation for vertex cover is hard to improve for simple t -uniform hypergraphs. However, for set cover on simple n -vertex hypergraphs, the greedy algorithm achieves a factor (lnn)/2 , better than the optimal lnn factor for general hypergraphs.

Anonymous social networks present a number of new and challenging problems for existing Social Network Analysis techniques. Traditionally, existing methods for analysing graph structure, such as community detection, required global knowledge of the graph structure. That implies that a centralised entity must be given access to the edge list of each node in the graph. This is impossible for anonymous social networks and other settings where privacy is valued by its participants. In addition, using their graph structure inputs for learning tasks defeats the purpose of anonymity. In this work, we hypothesise that one can re-purpose the use of the HyperANF a.k.a HyperBall algorithm -- intended for approximate diameter estimation -- to the task of privacy-preserving community detection for friend recommending systems that learn from an anonymous representation of the social network graph structure with limited privacy impact. This is possible because the core data structure maintained by HyperBall is a HyperLogLog with a counter of the number of reachable neighbours from a given node. Exchanging this data structure in future decentralised learning deployments gives away no information about the neighbours of the node and therefore does preserve the privacy of the graph structure.

The ridesharing problem is that given a set of trips, each trip consists of an individual, a vehicle of the individual and some requirements, select a subset of trips and use the vehicles of selected trips to deliver all individuals to their destinations satisfying the requirements. Requirements of trips are specified by parameters including source, destination, vehicle capacity, preferred paths of a driver, detour distance and number of stops a driver is willing to make, and time constraints. We analyze the relations between the time complexity and parameters for two optimization problems: minimizing the number of selected vehicles and minimizing total travel distance of the vehicles. We consider the following conditions: (1) all trips have the same source or same destination, (2) no detour is allowed, (3) each participant has one preferred path, (4) no limit on the number of stops, and (5) all trips have the same departure and same arrival time. It is known that both minimization problems are NP-hard if one of Conditions (1), (2) and (3) is not satisfied. We prove that both problems are NP-hard and further show that it is NP-hard to approximate both problems within a constant factor if Conditions (4) or (5) is not satisfied. We give K+2 2 -approximation algorithms for minimizing the number of selected vehicles when condition (4) is not satisfied, where K is the largest capacity of all vehicles.

We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density α c (?) and provide (i) for α< α c (?) randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most αn in n -vertex graphs of maximum degree ? ; and (ii) a proof that unless NP=RP, no such algorithms exist for α> α c (?) . The critical density is the occupancy fraction of hard core model on the clique K ?+1 at the uniqueness threshold on the infinite ? -regular tree, giving α c (?)??e 1+e 1 ? as ??��? .

Given a matrix A∈ R n×n , we consider the problem of maximizing x T Ax subject to the constraint x∈{−1,1 } n . This problem, called MaxQP by Charikar and Wirth [FOCS'04], generalizes MaxCut and has natural applications in data clustering and in the study of disordered magnetic phases of matter. Charikar and Wirth showed that the problem admits an Ω(1/lgn) approximation via semidefinite programming, and Alon, Makarychev, Makarychev, and Naor [STOC'05] showed that the same approach yields an Ω(1) approximation when A corresponds to a graph of bounded chromatic number. Both these results rely on solving the semidefinite relaxation of MaxQP, whose currently best running time is O ~ ( n 1.5 ⋅min{N, n 1.5 }) , where N is the number of nonzero entries in A and O ~ ignores polylogarithmic factors. In this sequel, we abandon the semidefinite approach and design purely combinatorial approximation algorithms for special cases of MaxQP where A is sparse (i.e., has O(n) nonzero entries). Our algorithms are superior to the semidefinite approach in terms of running time, yet are still competitive in terms of their approximation guarantees. More specifically, we show that: - MaxQP admits a (1/2Δ) -approximation in O(nlgn) time, where Δ is the maximum degree of the corresponding graph. - UnitMaxQP, where A∈{−1,0,1 } n×n , admits a (1/2d) -approximation in O(n) time when the corresponding graph is d -degenerate, and a (1/3δ) -approximation in O( n 1.5 ) time when the corresponding graph has δn edges. - MaxQP admits a (1−ε) -approximation in O(n) time when the corresponding graph and each of its minors have bounded local treewidth. - UnitMaxQP admits a (1−ε) -approximation in O( n 2 ) time when the corresponding graph is H -minor free.