인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

Symposium on Theoretical Aspects of Computer Science

We prove exponential lower bounds on the size of homogeneous depth 4 arithmetic circuits computing an explicit polynomial in $VP$. Our results hold for the iterated matrix multiplication polynomial---in particular we show that any homogeneous depth 4 circuit computing the (1,1) entry in the product of $n$ generic matrices of dimension $n^{O(1)}$ must have size $n^{\Omega(\sqrt{n})}$. Our results strengthen previous works in two significant ways: (1) Our lower bounds hold for a polynomial in $VP$. Prior to our work, Kayal et al. [Proceedings of FOCS, 2014, pp. 61--70] proved an exponential lower bound for homogeneous depth 4 circuits (over fields of characteristic zero) computing a poly in $VNP$. The best known lower bounds for a depth 4 homogeneous circuit computing a poly in $VP$ was the bound of $n^{\Omega(\log n)}$ by Kayal et al. Our exponential lower bounds also give the first exponential separation between general arithmetic circuits and homogeneous depth 4 arithmetic circuits. In particular they im...

On the power of homogeneous depth 4 arithmetic circuits.

The multi-depot vehicle routing problem is a well-known non-deterministic polynomial-time hard combinatorial optimization problem, which is crucial for transportation and logistics systems. We proposed a novel fitness-scaling adaptive genetic algorithm with local search FISAGALS. The fitness-scaling technique converts the raw fitness value to a new value that is suitable for selection. The adaptive rates strategy changes the crossover and mutation probabilities depending on the fitness value. The local search mechanism exploits the problem space in a more efficient way. The experiments employed 33 benchmark problems. Results showed the proposed FISAGALS is superior to the standard genetic algorithm, simulated annealing, tabu search, and particle swarm optimization in terms of success instances and computation time. Furthermore, FISAGALS performs better than parallel iterated tabu search PITS and fuzzy logic guided genetic algorithm FLGA, and marginally worse than ILS-RVND-SP in terms of the maximum gap. It performs faster than PITS and ILS-RVND-SP a combination of iterated local search framework [ILS], a variable neighborhood descent with random neighborhood ordering [RVND] and the the set partitioning [SP] model and slower than FLGA. In summary, FISAGALS is a competitive method with state-of-the-art algorithms.

Fitness-scaling adaptive genetic algorithm with local search for solving the Multiple Depot Vehicle Routing Problem

Traveling salesman problem TSP and its quasi problem Quasi-TSP are typical problems in path optimization, and ant colony optimization ACO algorithm is considered as an effective way to solve TSP. However, when the problems come to high dimensions, the classic algorithm works with low efficiency and accuracy, and usually cannot obtain an ideal solution. To overcome the shortcoming of the classic algorithm, this paper proposes an improved ant colony optimization I-ACO algorithm which combines swarm intelligence with local search to improve the efficiency and accuracy of the algorithm. Experiments are carried out to verify the availability and analyze the performance of I-ACO algorithm, which cites a Quasi-TSP based on a practical problem in a tourist area. The results illustrate the higher accuracy and efficiency of the I-ACO algorithm to solve Quasi-TSP, comparing with greedy algorithm, simulated annealing, classic ant colony algorithm and particle swarm optimization algorithm, and prove that the I-ACO algorithm is a positive effective way to tackle Quasi-TSP.

An improved ant colony optimization I-ACO method for the quasi travelling salesman problem Quasi-TSP

Average-case approximation ratio of the 2-opt algorithm for the TSP

We study the size blow-up that is necessary to convert an algebraic circuit of product-depth δ+1 to one of product-depth δ in the multilinear setting. We show that for every positive δ = δ(n) = o(log n/log log n), there is an explicit multilinear polynomial P^(δ) on n variables that can be computed by a multilinear formula of product-depth δ+1 and size O(n), but not by any multilinear circuit of product-depth δ and size less than exp(n^Ω(1/δ)). This result is tight up to the constant implicit in the double exponent for all δ = o(log n/log log n). This strengthens a result of Raz and Yehudayoff (Computational Complexity 2009) who prove a quasipolynomial separation for constant-depth multilinear circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an exponential separation in the case δ = 1. Our separating examples may be viewed as algebraic analogues of variants of the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan (STOC 2016), who used them to prove lower bounds for constant-depth Boolean circuits.

A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant's model. We are given a fixed graph $H$ and want to find all graphs, from some graph class, homomorphic to this $H$. These graphs will be encoded by a family of polynomials. 
We give dichotomies for the polynomials for cycles, cliques, trees, outerplanar graphs, planar graphs and graphs of bounded genus.

Dichotomy Theorems for Homomorphism Polynomials of Graph Classes

We construct a hitting set generator for sparse multivariate polynomials over the reals The seed length of our generator is O(log^2 (mn/epsilon)) where m is the number of monomials, n is number of variables, and 1 - epsilon is the hitting probability The generator can be evaluated in time polynomial in log m, n, and log 1/epsilon This is the first hitting set generator whose seed length is independent of the degree of the polynomial The seed length of the best generator so far by Klivans and Spielman [STOC 2001] depends logarithmically on the degree

From this, we get a randomized algorithm for testing sparse black box polynomial identities over the reals using O(log^2 (mn/epsilon)) random bits with running time polynomial in log m, n, and log(1/epsilon)

We also design a deterministic test with running time ~O(m^3 n^3) Here, the ~O-notation suppresses polylogarithmic factors The previously best deterministic test by Lipton and Vishnoi [SODA 2003] has a running time that depends polynomially on log delta, where $delta$ is the degree of the black box polynomial

https://drops.dagstuhl.de/opus/volltexte/2011/3043/pdf/51.pdf/

Randomness Efficient Testing of Sparse Black Box Identities of Unbounded Degree over the Reals

In this thesis we will try to answer the question why specific polynomials have no small suspected arithmetic circuits. We will look at this general problem in three different ways. First, we study non-commutative computation. Here we show matching upper and lower bounds for the non-commutative permanent for various restricted graph classes. Our main result gives algebraic branching program upper and lower bounds for graphs with connected component size 6 as well as a #P hardness result. We introduce a measure that characterizes this complexity on these instances. Secondly, we introduce a new framework for arithmetic circuits, similar to fixed parameter tractability in the boolean setting. This framework shows that specific polynomials based on graph problems have the expected complexity as in the counting FPT case. We introduce classes BVW[t] which are close to the boolean setting but hardly use the power of arithmetic circuits. We then introduce VW[t] with modified problems to remedy this situation. Thirdly, we study polynomials defined by graph homomorphisms and show various dichotomy theorems. This shows that even restrictions on the graphs can already give us hard instances. Finally, we stray from our main continuous thread and handle simple heuristics for metric graphs. Instead of studying specific metrics we look at a randomized process giving us shortest path metric instances.

/pdf/why-are-certain-polynomials-hard-a-look-at-non-commutative-409hnmaare.pdf

Christian Engels

Papers

Average-case approximation ratio of the 2-opt algorithm for the TSP

A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear Circuits

Dichotomy Theorems for Homomorphism Polynomials of Graph Classes

Randomness Efficient Testing of Sparse Black Box Identities of Unbounded Degree over the Reals

Why are certain polynomials hard? : A look at non-commutative, parameterized and homomorphism polynomials