Approximate Set Union Via Approximate Randomization

We develop an randomized approximation algorithm for the size of set union problem \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert, which given a list of sets $A_1,...,A_m$ with approximate set size $m_i$ for $A_i$ with $m_i\in \left((1-\beta_L)|A_i|, (1+\beta_R)|A_i|\right)$, and biased random generators with Prob(x=\randomElm(A_i))\in \left[{1-\alpha_L\over |A_i|},{1+\alpha_R\over |A_i|}\right] for each input set $A_i$ and element $x\in A_i,$ where $i=1, 2, ..., m$. The approximation ratio for \arrowvert A_1\cup A_2\cup...\cup A_m\arrowvert is in the range $[(1-\epsilon)(1-\alpha_L)(1-\beta_L), (1+\epsilon)(1+\alpha_R)(1+\beta_R)]$ for any $\epsilon\in (0,1)$, where $\alpha_L, \alpha_R, \beta_L,\beta_R\in (0,1)$. The complexity of the algorithm is measured by both time complexity, and round complexity. The algorithm is allowed to make multiple membership queries and get random elements from the input sets in one round. Our algorithm makes adaptive accesses to input sets with multiple rounds. Our algorithm gives an approximation scheme with O(\setCount\cdot(\log \setCount)^{O(1)}) running time and $O(\log m)$ rounds, where $m$ is the number of sets. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. Our algorithm gives a flexible tradeoff with time complexity O\left(\setCount^{1+\xi}\right) and round complexity $O\left({1\over \xi}\right)$ for any $\xi\in(0,1)$

Approximate Set Union via Approximate Randomization

We develop an randomized approximation algorithm for the size of set union problem |A1 U A2 U...UAm|, which given a list of sets A1,...,Am with approximate set size m i for Ai with mi ∈ ((1–βL)|A i|,(1+βR)|Ai|), and biased random generators with Prob(x = RandomElement(Ai)) ∈ [1–a L/Ai, 1 +aR/Ai] for each input set Ai and element x ∈ Ai, where i = 1,2,...,m. The approximation |Ai | |Ai | ratio for |A1 U A2 U...UAm| is in the range [(1–ϵ)(1–aL)(1–βL),(1+ϵ)(1+β R)(1+βR)] for any ϵ ∈ (0,1), where α L,αR,βL,βR ∈ (0,1). The complexity of the algorithm is measured by both time complexity, and round complexity. The algorithm is allowed to make multiple membership queries and get random elements from the input sets in one round. Our algorithm makes adaptive accesses to input sets with multiple rounds. Our algorithm gives an approximation scheme with O(m˙(logm) O(1)) running time and O(logm) rounds, where m is the number of sets. Our algorithm can handle input sets that can generate random elements with bias, and its approximation ratio depends on the bias. Our algorithm gives a flexible tradeoff with time complexity O m1 and round complexity O 1 for any (0, 1). We prove that our algorithm runs sublinear in time under certain condition that each element in A 1 U A2 U ... U Am belong to ma for any fixed a \u3e 0. A O r(r + l|)3l3d4 running time dynamic programming algorithm is proposed to deal with an interesting problem in number theory area that is to count the number of lattice points in a d—dimensional ball Bd( r,p,d) of radius r with center at p ∈ D(λ,d,l), where D(λ, d,l) = {(x1,˙˙˙ , xd) : (x1,˙˙˙ ,xd) with xk = ik + jkλ for an integer jk ∈ [–l, l], and another arbitrary integer ik for k = 1,2,...,d.} We prove that it is #P-hard to count the number of lattice points in a set of balls, and we also show that there is no polynomial time algorithm to approximate the number of lattice points in the intersection of n-dimenisonal k-degree balls unless P=NP

Approximate Randomization of Quantum States With Fewer Bits of Key

Randomization of quantum states is the quantum analogue of the classical one-time pad. We present an improved, efficient construction of an approximately randomizing map that uses O(d/epsilon^2) Pauli operators to map any d-dimensional state to a state that is within trace distance epsilon of the completely mixed state. Our bound is a log d factor smaller than that of Hayden, Leung, Shor, and Winter (2004), and Ambainis and Smith (2004). Then, we show that a random sequence of essentially the same number of unitary operators, chosen from an appropriate set, with high probability form an approximately randomizing map for d-dimensional states. Finally, we discuss the optimality of these schemes via connections to different notions of pseudorandomness, and give a new lower bound for small epsilon.Comment: 18 pages, Quantum Computing Back Action, IIT Kanpur, March 2006, volume 864 of AIP Conference Proceedings, pages 18--36. Springer, New Yor

Approximating the Held-Karp Bound for Metric TSP in Nearly Linear Time

We give a nearly linear time randomized approximation scheme for the Held-Karp bound [Held and Karp, 1970] for metric TSP. Formally, given an undirected edge-weighted graph $G$ on $m$ edges and $\epsilon > 0$, the algorithm outputs in $O(m \log^4n /\epsilon^2)$ time, with high probability, a $(1+\epsilon)$-approximation to the Held-Karp bound on the metric TSP instance induced by the shortest path metric on $G$. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the $O(m^2 \log^2(m)/\epsilon^2)$ running time achieved previously by Garg and Khandekar. The LP solution can be used to obtain a fast randomized $\big(\frac{3}{2} + \epsilon\big)$-approximation for metric TSP which improves upon the running time of previous implementations of Christofides' algorithm

Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems, such as maximum matching and minimum vertex cover, on large datasets. For massive inputs, several different computational models have been introduced, including the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In each model, algorithms are analyzed in terms of resources such as space used or rounds of communication needed, in addition to the more traditional approximation ratio. In this paper, we give a single unified approach that yields better approximation algorithms for matching and vertex cover in all these models. The highlights include: * The first one pass, significantly-better-than-2-approximation for matching in random arrival streams that uses subquadratic space, namely a $(1.5+\epsilon)$-approximation streaming algorithm that uses $O(n^{1.5})$ space for constant $\epsilon > 0$. * The first 2-round, better-than-2-approximation for matching in the MPC model that uses subquadratic space per machine, namely a $(1.5+\epsilon)$-approximation algorithm with $O(\sqrt{mn} + n)$ memory per machine for constant $\epsilon > 0$. By building on our unified approach, we further develop parallel algorithms in the MPC model that give a $(1 + \epsilon)$-approximation to matching and an $O(1)$-approximation to vertex cover in only $O(\log\log{n})$ MPC rounds and $O(n/poly\log{(n)})$ memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj~et.al. [STOC 2018]