Deterministic Polynomial Time Algorithm Research Articles

Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (Comput Geom 2(3):169---186, 1992) and the optimal but randomized algorithm of Ramos (Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG'99, 1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, $$({\le }k)$$(≤k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, $$\varepsilon $$?-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (Discrete Comput Geom 6(1):385---406, 1991) and Chazelle (Discrete Comput Geom 9(1):145---158, 1993).

Roots of Unity in Orders

We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of roots of unity can be solved in polynomial time. As an auxiliary result, we solve the discrete logarithm problem for certain unit groups in finite rings. Our techniques, which are taken from commutative algebra, may have further potential in the context of cryptology and computer algebra.

Multicasting in Linear Deterministic Relay Network by Matrix Completion

We provide a deterministic polynomial time algorithm for multicasting in a linear deterministic relay network and a wireless communication framework proposed by Avestimehr et al. The running time of our algorithm is faster than existing ones and matches the current best complexity of unicast computations for each sink. Our approach is based on the polylinking flow model of Goemans et al. and the mixed matrix completion technique of Harvey et al.

AND-Compression of NP-Complete Problems: Streamlined Proof and Minor Observations

Drucker (2012) proved the following result: Unless the unlikely complexity-theoretic collapse coNP is in NP/poly occurs, there is no AND-compression for SAT. The result has implications for the compressibility and kernelizability of a whole range of NP-complete parameterized problems. We present a streamlined proof of Drucker's theorem. An AND-compression is a deterministic polynomial-time algorithm that maps a set of SAT-instances $x_1,\dots,x_t$ to a single SAT-instance of size poly(max $|x_i|$) such that is if and only if all are The in the name stems from the fact that the predicate $y$ is satisfiable can be written as the AND of all predicates $x_i$ is satisfiable. Drucker's result complements the result by Bodlaender et al. (2009) and Fortnow and Santhanam (2010), who proved the analogous statement for OR-compressions, and Drucker's proof not only subsumes that result but also extends it to randomized compression algorithms that are allowed to have a certain probability of failure. Drucker (2012) presented two proofs: The first uses information theory and the minimax theorem from game theory, and the second is an elementary, iterative proof that is not as general. In our proof, we realize the iterative structure as a generalization of the arguments of Ko (1983) for P-selective sets, which use the fact that tournaments have dominating sets of logarithmic size. We generalize this fact to hypergraph tournaments. Our proof achieves the full generality of Drucker's theorem, avoids the minimax theorem, and restricts the use of information theory to a single, intuitive lemma about the average noise sensitivity of compressive maps. To prove this lemma, we use the same information-theoretic inequalities as Drucker.

Reassembling Trees for the Traveling Salesman

Many recent approximation algorithms for different variants of the traveling salesman problem (asymmetric TSP, graph TSP, $s$-$t$-path TSP) exploit the well-known fact that a solution of the natural linear programming relaxation can be written as convex combination of spanning trees. The main argument then is that randomly sampling a tree from such a distribution and then completing the tree to a tour at minimum cost yields a better approximation guarantee than simply taking a minimum cost spanning tree (as in Christofides' algorithm). We argue that an additional step can help: reassembling the spanning trees before sampling. Exchanging two edges in a pair of spanning trees can improve their properties under certain conditions. We demonstrate the usefulness for the metric $s$-$t$-path TSP by devising a deterministic polynomial-time algorithm that improves on Sebo's previously best approximation ratio of $\frac{8}{5}$.

A Dynamic Programming Algorithm for Leveraging Probabilistic Detection of Energy Theft in Smart Home

In the modern smart home and community, smart meters have been massively deployed for the replacement of traditional analog meters. Although it significantly reduces the cost of data collection as the meter readings are wireless transmitted, a smart meter is not tamper-resistant. As a consequence, the smart grid infrastructure is under threat of energy theft, by means of attacking a smart meter so that it undercounts the electricity usage. Deployment of feeder remote terminal unit (FRTU) helps narrow the search zone of energy theft in smart home and community. However, due to budgetary limit, utility companies can only afford to insert the minimum number of FRTUs. This imposes a signifcant challenge to deploy the minimum number of FRTUs while each smart meter is still effectively monitored. To the best of our knowledge, the only work addressing this problem is [1] , which uses stochastic optimization methods. Their algorithm is not very practical as it cannot handle large distribution networks because of the scalability issue. Due to the inherent heuristic and non-deterministic nature, there is no guarantee on the solution quality as well. Thus, the high performance energy theft detection is still needed for this energy theft problem. In order to resolve this challenge, we propose a new dynamic programming algorithm that inserts the minimum number of FRTUs satisfying the detection rate constraint. It evaluates every candidate solution in a bottom-up fashion using an innovative pruning technique. As a deterministic polynomial time algorithm, it is able to handle large distribution networks. In contrast to [1] which can only handle small system, our technique can perform FRTU insertion for a large scale power system. Our experimental results demonstrate that the average number of FRTUs required is only 26% of the number of smart meters in the community. Compared with the previous work, the number of FRTUs is reduced by 18.8% while the solution quality in terms of anomaly coverage index metric is still improved.

Deterministic root finding in finite fields

Finding roots of polynomials with coefficients in a finite field is a special instance of the polynomial factorization problem. The most well known algorithm for factoring and root-finding is the Cantor-Zassenhaus algorithm. It is a Las Vegas algorithm with running time polynomial in both the polynomial degree and the field size. Its running time is quasi-optimal for the special case of root-finding. No deterministic polynomial time algorithm for these problems is known in general, however several deterministic algorithms are known for special instances, most notably when the characteristic of the finite field is small. The goal of this poster is to review the best deterministic algorithms for root-finding, in a systematic way. We present, in a unified way, four algorithms: • Berlekamp's Trace Algorithm [2] (BTA), • Moenk's Subgroup Refinement Method [7] (SRM), • Menezes, van Oorschot and Vanstone's Affine Refinement Method [5, 10] (ARM), and • Petit's Successive Resultants Algorithm [8] (SRA). It is the first time that these algorithms are presented together in a comprehensive way, and that they are rigorously analysed, implemented and compared to each other. In doing so, we obtain several new results: • We significantly improve the complexity ARM, matching that of BTA and SRA. • We highlight a profound duality relationship between ARM and SRA. • We show how to combine ARM with SRM to obtain a new algorithm, which always performs better, and of which ARM and SRM are special instances. The new algorithm considerably extends the range of finite fields for which deterministic polynomial time root-finding is possible. • We present several practical and asymptotic improvements to special instances of the algorithms. Part of these results were submitted in response to the call for papers of ISSAC '15, but were rejected. This poster corrects some minor imperfections, improves the asymptotic complexities of some algorithms, and presents a new algorithm not previously known.

Solving Closest Vector Instances Using an Approximate Shortest Independent Vectors Oracle

Given an n-dimensional lattice L and some target vector, this paper studies the algorithms for approximate closest vector problem (CVPγ) by using an approximate shortest independent vectors problem oracle (SIVPγ). More precisely, if the distance between the target vector and the lattice is no larger than c \( {\scriptscriptstyle \frac{c}{\gamma n}}{\uplambda}_1\left(\mathrm{L}\right) \) for arbitrary large but finite constant c > 0, we give randomized and deterministic polynomial time algorithms to find a closest vector, while previous reductions were only known for \( {\scriptscriptstyle \frac{c}{2\gamma n}}{\uplambda}_1\left(\mathrm{L}\right) \). Moreover, if the distance between the target vector and the lattice is larger than some quantity with respect to λn(L), using SIVPγ oracle and Babai’s nearest plane algorithm, we can solve \( \mathrm{CVP}\upgamma \sqrt{n} \) in deterministic polynomial time. Specially, if the approximate factor γ ϵ (1, 2) in the SIVPγ oracle, we obtain a better reduction factor for CVP.

Analysis of a certain polycyclic-group-based cryptosystem

Abstract We investigate security properties of the Anshel–Anshel–Goldfeld commutator key-establishment protocol [Math. Res. Lett. 6 (1999), 287–291] used with certain polycyclic groups described by Eick and Kahrobaei [http://arxiv.org/abs/math.GR/0411077]. We show that despite low success of the length based attack shown by Garber, Kahrobaei and Lam [J. Math. Crypt. 9 (2015), 33–43] the protocol can be broken by a deterministic polynomial-time algorithm.

Using shape grammars and extremal optimization in 3D IC layout design

Computer-aided 3D IC layout design task is computationally difficult and no deterministic polynomial time algorithms are able to solve it. The elaborated intelligent framework architecture for visual kind of design uses a simple shape grammar to generate topologically feasible solutions which are further optimized to meet design requirements. The search process is driven by an intelligent derivation controller that employs design specific knowledge given in a form of goals and constraints. The paper focuses on the post-generation 3D IC wirelength optimization stage. The original partitioning heuristics implemented by the means of the extremal optimization is applied to the MCNC set of benchmark circuits. The achieved results confirm the effectiveness of the proposed approach while applied to sparse circuits.

The Parameterized Complexity of Geometric Graph Isomorphism

We study the parameterized complexity of Geometric Graph Isomorphism (Known as the Point Set Congruence problem in computational geometry): given two sets of n points A and B with rational coordinates in k-dimensional euclidean space, with k as the fixed parameter, the problem is to decide if there is a bijection $$\pi :A \rightarrow B$$ź:AźB such that for all $$x,y \in A$$x,yźA, $$\Vert x-y\Vert = \Vert \pi (x)-\pi (y)\Vert $$źx-yź=źź(x)-ź(y)ź, where $$\Vert \cdot \Vert $$ź·ź is the euclidean norm. Our main result is the following: We give a $$O^*(k^{O(k)})$$Oź(kO(k)) time (The $$O^*(\cdot )$$Oź(·) notation here, as usual, suppresses polynomial factors) FPT algorithm for Geometric Isomorphism. This is substantially faster than the previous best time bound of $$O^*(2^{O(k^4)})$$Oź(2O(k4)) for the problem (Evdokimov and Ponomarenko in Pure Appl Algebra 117---118:253---276, 1997). In fact, we show the stronger result that even canonical forms for finite point sets with rational coordinates can also be computed in $$O^*(k^{O(k)})$$Oź(kO(k)) time. We also briefly discuss the isomorphism problem for other $$l_p$$lp metrics. Specifically, we describe a deterministic polynomial-time algorithm for finite point sets in $$\mathbb {Q}^2$$Q2.

An Improved Deterministic #SAT Algorithm for Small de Morgan Formulas

We give a deterministic #SAT algorithm for de Morgan formulas of size up to $$n^{2.63}$$ , which runs in time $$2^{n-n^{{\varOmega }(1)}}$$ . This improves upon the deterministic #SAT algorithm of Chen et al. (Proceedings of the twenty-ninth annual IEEE conference on computational complexity, 2014), which has similar running time but works only for formulas of size less than $$n^{2.5}$$ . Our new algorithm is based on the shrinkage of de Morgan formulas under random restrictions, shown by Paterson and Zwick (Random Struct Algorithms 4(2):135–150, 1993). We prove a concentrated and constructive version of their shrinkage result. Namely, we give a deterministic polynomial-time algorithm that selects variables in a given de Morgan formula so that, with high probability over the random assignments to the chosen variables, the original formula shrinks in size, when simplified using a given deterministic polynomial-time formula-simplification algorithm.

Determining cyclicity of finite modules

We present a deterministic polynomial-time algorithm that determines whether a finite module over a finite commutative ring is cyclic, and if it is, outputs a generator.

Deterministic polynomial identity tests for multilinear bounded-read formulae

We present a polynomial-time deterministic algorithm for testing whether constant-read multilinear arithmetic formulae are identically zero. In such a formula, each variable occurs only a constant number of times, and each subformula computes a multilinear polynomial. Our algorithm runs in time $${s^{O(1)}\cdot n^{k^{O(k)}}}$$sO(1)·nkO(k) , where s denotes the size of the formula, n denotes the number of variables, and k bounds the number of occurrences of each variable. Before our work, no subexponential-time deterministic algorithm was known for this class of formulae. We also present a deterministic algorithm that works in a blackbox fashion and runs in time $${n^{k^{O(k)} + O(k \log n)}}$$nkO(k)+O(klogn) in general, and time $${n^{k^{O(k^2)} + O(kD)}}$$nkO(k2)+O(kD) for depth D. Finally, we extend our results and allow the inputs to be replaced with sparse polynomials. Our results encompass recent deterministic identity tests for sums of a constant number of read-once formulae and for multilinear depth-four formulae.

Close values of shifted modular inversions and the decisional modular inversion hidden number problem

We give deterministic polynomialtime algorithms for two different decision version the modularinversion hidden number problemintroduced by D. Boneh, S. Halevi and N. A. Howgrave-Grahamin 2001. For example, for one of our algorithms we need to be given about $1/2$ of thebits of each inversion, while for the computationalversion the best known algorithm requires about $2/3$ of the bitsand is probabilistic.

Approximate tradeoffs on weighted labeled matroids

We consider problems where a solution is evaluated with a couple. Each coordinate of this couple represents the utility of an agent. Due to the possible conflicts, it is unlikely that one feasible solution is optimal for both agents. Then a natural aim is to find a tradeoff. We investigate tradeoff solutions with worst case guarantees for the agents. The focus is on discrete problems having a matroid structure and the utility of an agent is modeled with a function which is either additive or weighted labeled. We provide polynomial-time deterministic algorithms which achieve several guarantees and we prove that some guarantees are not possible to reach.

The remote set problem on lattices

We initiate studying the Remote Set Problem ( $${\mathsf{RSP}}$$ RSP ) on lattices, which given a lattice asks to find a set of points containing a point which is far from the lattice. We show a polynomial-time deterministic algorithm that on rank n lattice $${\mathcal{L}}$$ L outputs a set of points, at least one of which is $${\sqrt{\log n / n} \cdot \rho(\mathcal{L})}$$ log n / n · ? ( L ) -far from $${\mathcal{L}}$$ L , where $${\rho(\mathcal{L})}$$ ? ( L ) stands for the covering radius of $${\mathcal{L}}$$ L (i.e., the maximum possible distance of a point in space from $${\mathcal{L}}$$ L ). As an application, we show that the covering radius problem with approximation factor $${\sqrt{n / \log n}}$$ n / log n lies in the complexity class $${\mathsf{NP}}$$ NP , improving a result of Guruswami et al. (Comput Complex 14(2): 90---121, 2005) by a factor of $${\sqrt{\log n}}$$ log n . Our results apply to any $${\ell_p}$$ l p norm for $${2 \leq p \leq \infty}$$ 2 ≤ p ≤ ? with the same approximation factors (except a loss of $${\sqrt{\log \log n}}$$ log log n for $${p = \infty}$$ p = ? ). In addition, we show that the output of our algorithm for $${\mathsf{RSP}}$$ RSP contains a point whose $${\ell_2}$$ l 2 distance from $${\mathcal{L}}$$ L is at least $${(\log n/n)^{1/p} \cdot \rho^{(p)}(\mathcal{L})}$$ ( log n / n ) 1 / p · ? ( p ) ( L ) , where $${\rho^{(p)}(\mathcal{L})}$$ ? ( p ) ( L ) is the covering radius of $${\mathcal{L}}$$ L measured with respect to the $${\ell_p}$$ l p norm. The proof technique involves a theorem on balancing vectors due to Banaszczyk (Random Struct Algorithms 12(4):351---360, 1998) and the "six standard deviations" theorem of Spencer (Trans Am Math Soc 289(2):679---706, 1985).

Boolean matrices with prescribed row/column sums and stable homogeneous polynomials: Combinatorial and algorithmic applications

We prove a new efficiently computable lower bound on the coefficients of stable homogeneous polynomials and present its algorithmic and combinatorial applications. Our main application is the first poly-time deterministic algorithm which approximates the partition functions associated with boolean matrices with prescribed row and column sums within simply exponential multiplicative factor. This new algorithm is a particular instance of new polynomial time deterministic algorithms related to the multiple partial differentiation of polynomials given by evaluation oracles.

Black box groups isomorphic to [formula omitted

A deterministic polynomial-time algorithm constructs an isomorphism between PGL(2,2e) and a black box group to which it is isomorphic.

Fast Approximation Algorithm for Restricted Euclidean Bottleneck Steiner Tree Problem

Bottleneck Steiner tree problem asks to find a Steiner tree for n terminals with at most k Steiner points such that the length of the longest edge in the tree is minimized. The problem has applications in the design of wireless communication networks. In this paper we study a restricted version of the bottleneck Steiner tree problem in the Euclidean plane which requires that only degree-2 Steiner points are possibly adjacent in the optimal solution. We first show that the problem is NP-hard and cannot be approximated within unless P=NP, and provide a fast polynomial time deterministic approximation algorithm with performance ratio .