scholarly journals Combinatorial Nullstellensatz Modulo Prime Powers and the Parity Argument

10.37236/4124 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
László Varga
Keyword(s):  
Complexity Class ◽  
Combinatorial Nullstellensatz ◽  
Search Problem ◽  
Search Problems ◽  
Computational Search

We present new generalizations of Olson's theorem and of a consequence of Alon's Combinatorial Nullstellensatz. These enable us to extend some of their combinatorial applications with conditions modulo primes to conditions modulo prime powers. We analyze computational search problems corresponding to these kinds of combinatorial questions and we prove that the problem of finding degree-constrained subgraphs modulo $2^d$ such as $2^d$-divisible subgraphs and the search problem corresponding to the Combinatorial Nullstellensatz over $\mathbb{F}_2$ belong to the complexity class Polynomial Parity Argument (PPA).

THE THOMPSON–HIGMAN MONOIDS Mk,i: THE ${\mathcal J}$-ORDER, THE ${\mathcal D}$-RELATION, AND THEIR COMPLEXITY

2011 ◽  
Vol 21 (01n02) ◽  
pp. 1-34 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Complexity Class ◽  
Maximal Subgroups ◽  
Complexity Classes ◽  
Search Problem ◽  
Image Size ◽  
Search Problems ◽  
Generating Set ◽  
Counting Complexity ◽  
Slight Generalization ◽  
Np Complete

The Thompson–Higman groups Gk,i have a natural generalization to monoids, called Mk,i, and inverse monoids, called Invk,i. We study some structural features of Mk,i and Invk,i and investigate the computational complexity of related decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity. The maximal subgroups of Mk,1 and Invk,1 are isomorphic to the groups Gk,j (1 ≤ j ≤ k - 1); so we rediscover all the Thompson–Higman groups within Mk,1. Deciding the Green relations [Formula: see text] and [Formula: see text] of Mk,1, when the inputs are words over a finite generating set of Mk,1, is in P. When a circuit-like generating set is used for Mk,1 then deciding [Formula: see text] is coDP-complete (where DP is the complexity class consisting of differences of sets in NP). The multiplier search problem for [Formula: see text] is xNPsearch-complete, whereas the multiplier search problems of [Formula: see text] and [Formula: see text] are not in xNPsearch unless NP = coNP. The class of search problems xNPsearch is introduced as a slight generalization of NPsearch. Deciding [Formula: see text] for Mk,1 when the inputs are words over a circuit-like generating set, is ⊕k-1• NP -complete; for any h ≥ 2, ⊕h•NP is a modular counting complexity class, whose verification problems are in NP. Related problems for partial circuits are the image size problem (which is # • NP-complete), and the image size modulo h problem (which is ⊕h•NP-complete). For Invk,1 over a circuit-like generating set, deciding [Formula: see text] is ⊕k-1P-complete. It is interesting that the little known complexity classes coDP and ⊕k-1•NP play a central role in Mk,1.

An improved version of the AAG cryptographic protocol

2019 ◽  
Vol 11 (1) ◽  
pp. 35-41 ◽  
Author(s):  
Vitaliĭ Roman’kov
Keyword(s):  
Linear Span ◽  
Original Version ◽  
Search Problem ◽  
Cryptographic Protocol ◽  
Membership Problem ◽  
Search Problems ◽  
Hard Problems ◽  
Cryptographic Key ◽  
Conjugacy Search Problem ◽  
Conjugacy Search

AbstractAn improved version of the Anshel–Anshel–Goldfeld (AAG) algebraic cryptographic key-exchange scheme, that is in particular resistant against the Tsaban linear span cryptanalysis, is established. Unlike the original version, that is based on the intractability of the simultaneous conjugacy search problem for the platform group, the proposed version is based on harder simultaneous membership-conjugacy search problems, and the membership problem needs to be solved for a subset of the platform group that can be easily and efficiently built to be very complicated and without any good structure. A number of other hard problems need to be solved first before start solving the simultaneous membership-conjugacy search problem to obtain the exchanged key.

scholarly journals MICROSERVICES ARCHITECTURE FOR SOLVE SEARCH PROBLEMS IN OBJECT-BASED STORAGES

2020 ◽  
Vol 7 (3) ◽  
pp. 018-023
Author(s):  
P. V. Prokhorov ◽  
◽  
D. I. Gerasimenko ◽  
◽  
Keyword(s):  
Search Engines ◽  
Search Problem ◽  
Search Problems ◽  
Object Based ◽  
Object Storage ◽  
Microservice Architecture

This article discusses microservice architecture when solving the problem of searching object storage similar to Amazon S3. The main focus is on popular solutions and core services used in microservice architecture. Ready-made search engines are also considered: Elastissearch; Apache Solr, when solving a search problem in a developed application. Describes tools that facilitate development and debugging in a microservice architecture.

scholarly journals THE CONJUGACY PROBLEM IN AMALGAMATED PRODUCTS I: REGULAR ELEMENTS AND BLACK HOLES

2007 ◽  
Vol 17 (07) ◽  
pp. 1299-1333 ◽  
Author(s):  
ALEXANDRE V. BOROVIK ◽  
ALEXEI G. MYASNIKOV ◽  
VLADIMIR N. REMESLENNIKOV
Keyword(s):  
Black Holes ◽  
Time Complexity ◽  
Conjugacy Problem ◽  
Search Problem ◽  
Search Problems ◽  
Free Products ◽  
Regular Elements ◽  
Amalgamated Products ◽  
Conjugacy Search Problem ◽  
Conjugacy Search

We discuss the time complexity of the word and conjugacy search problems for free products G = A ⋆C B of groups A and B with amalgamation over a subgroup C. We stratify the set of elements of G with respect to the complexity of the word and conjugacy problems and show that for the generic stratum the conjugacy search problem is decidable under some reasonable assumptions about groups A, B, C.

scholarly journals Incremental Symmetry Breaking Constraints for Graph Search Problems

2020 ◽  
Vol 34 (02) ◽  
pp. 1536-1543
Author(s):  
Avraham Itzhakov ◽  
Michael Codish
Keyword(s):  
Problem Solving ◽  
Symmetry Breaking ◽  
Special Property ◽  
Graph Search ◽  
Search Problem ◽  
Search Problems ◽  
Canonical Order ◽  
Parallel Graph ◽  
Canonical Graph

This paper introduces incremental symmetry breaking constraints for graph search problems which are complete and compact. We show that these constraints can be computed incrementally: A symmetry breaking constraint for order n graphs can be extended to one for order n + 1 graphs. Moreover, these constraints induce a special property on their canonical solutions: An order n canonical graph contains a canonical subgraph on the first k vertices for every 1 ≤ k ≤ n. This facilitates a “generate and extend” paradigm for parallel graph search problem solving: To solve a graph search problem φ on order n graphs, first generate the canonical graphs of some order k < n. Then, compute canonical solutions for φ by extending, in parallel, each canonical order k graph together with suitable symmetry breaking constraints. The contribution is that the proposed symmetry breaking constraints enable us to extend the order k canonical graphs to order n canonical solutions. We demonstrate our approach through its application on two hard graph search problems.

scholarly journals Systemic Evolutionary Chemical Space Exploration For Drug Discovery

2021 ◽  
Author(s):  
Chong Lu ◽  
Shien Liu ◽  
Weihua Shi ◽  
Jun Yu ◽  
Zhou Zhou ◽  
...  
Keyword(s):  
Small Molecules ◽  
De Novo ◽  
Chemical Space ◽  
Space Exploration ◽  
De Novo Design ◽  
Search Problem ◽  
Chemical Libraries ◽  
Major Task ◽  
Building Process ◽  
Computational Search

Chemical space exploration is a major task of the hit-finding process during the pursuit of novel chemical entities. Compared with other screening technologies, computational de novo design has become a popular approach to overcome the limitation of current chemical libraries. Here, we reported a de novo design platform named systemic evolutionary chemical space explorer (SECSE). The platform was conceptually inspired by fragment-based drug design, that miniaturized a “lego-building” process within the pocket of a certain target. The key of virtual hits generation was then turned into a computational search problem. To enhance search and optimization, human intelligence and deep learning were integrated. Application of SECSE against PHGDH, proved its potential in finding novel and diverse small molecules that are attractive starting points for further validation. This platform is open-sourced and the code is available at http://github.com/KeenThera/SECSE.

scholarly journals CONSISTENCY OF CIRCUIT EVALUATION, EXTENDED RESOLUTION AND TOTAL NP SEARCH PROBLEMS

2016 ◽  
Vol 4 ◽  
Author(s):  
JAN KRAJÍČEK
Keyword(s):  
Symbolic Logic ◽  
Search Problem ◽  
Bounded Arithmetic ◽  
Search Problems ◽  
Open Problems ◽  
Proof Systems ◽  
Arithmetic Theory ◽  
Propositional Proof Systems ◽  
Propositional Proof ◽  
Definition Of

We consider sets ${\it\Gamma}(n,s,k)$ of narrow clauses expressing that no definition of a size $s$ circuit with $n$ inputs is refutable in resolution R in $k$ steps. We show that every CNF with a short refutation in extended R, ER, can be easily reduced to an instance of ${\it\Gamma}(0,s,k)$ (with $s,k$ depending on the size of the ER-refutation) and, in particular, that ${\it\Gamma}(0,s,k)$ when interpreted as a relativized NP search problem is complete among all such problems provably total in bounded arithmetic theory $V_{1}^{1}$. We use the ideas of implicit proofs from Krajíček [J. Symbolic Logic, 69 (2) (2004), 387–397; J. Symbolic Logic, 70 (2) (2005), 619–630] to define from ${\it\Gamma}(0,s,k)$ a nonrelativized NP search problem $i{\it\Gamma}$ and we show that it is complete among all such problems provably total in bounded arithmetic theory $V_{2}^{1}$. The reductions are definable in theory $S_{2}^{1}$. We indicate how similar results can be proved for some other propositional proof systems and bounded arithmetic theories and how the construction can be used to define specific random unsatisfiable formulas, and we formulate two open problems about them.

Best Neighbor Heuristic Search for Finding Minimum Paths in Transportation Networks

1998 ◽  
Vol 1651 (1) ◽  
pp. 48-52 ◽  
Author(s):  
Jeffrey L. Adler
Keyword(s):  
Heuristic Search ◽  
Transportation Networks ◽  
Transportation Network ◽  
Search Problems ◽  
A Algorithm ◽  
Running Time ◽  
Search Effort ◽  
Wide Range ◽  
Path Search ◽  
Sparse Networks

For a wide range of transportation network path search problems, the A* heuristic significantly reduces both search effort and running time when compared to basic label-setting algorithms. The motivation for this research was to determine if additional savings could be attained by further experimenting with refinements to the A* approach. We propose a best neighbor heuristic improvement to the A* algorithm that yields additional benefits by significantly reducing the search effort on sparse networks. The level of reduction in running time improves as the average outdegree of the network decreases and the number of paths sought increases.

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