Theorem 7.56 SUBSET-SUM is NP Complete ANSHUMAN MOHANTY SUBSET-SUM - PowerPoint PPT Presentation

Aug 04, 2023 •232 likes •353 views

Theorem 7.56 SUBSET-SUM is NP Complete ANSHUMAN MOHANTY SUBSET-SUM Problem Consider a set of numbers S. and a target number t. We have to determine of the exists a subset of S such that the summation of the numbers present in the subset is

Theorem 7.56 SUBSET-SUM is NP Complete ANSHUMAN MOHANTY
SUBSET-SUM Problem  Consider a set of numbers S. and a target number t. We have to determine of the exists a subset of S such that the summation of the numbers present in the subset is equal to t.  The subset is considered a multi-set i.e. repetitions are allowed.  Thus we have, SUBSET-SUM = {  S,t  | S = {x 1 , x 2 , ... x k }, and for some {y 1 , y 2 , … y k }  S, we have  y i = t)
NP and NP Complete  NP is the class of languages that have polynomial time verifiers. i.e. we can verify if a given solution is true or not for a given problem in polynomial time.  A problem P is said to be NP Complete if the it belongs to NP and if there exists a problem in NP that is reducible to P.
SUBSET-SUM is NP
SUBSET-SUM is NP Complete  The idea of the proof is to reduce 3SAT NP Complete problem to SUBSET-SUM.  We create a 3 cnf formula  to construct an instance of SUBSET- SUM which contains a subcollection whose summation is target t.  Using variables and clauses we find structure to the SUBSET-SUM  Each variable x is represented by y and z which establishes the truth value of x in the subcollection.
Proof  Let x 1 to l be the variables and c 1 to k be the clauses.  We construct a table with l+k columns and 2(l+k) rows.  For every y, z in the i-th row, set the i-th column value as 1 and the rest be 0.  For every y, z in the j-th row, set y = 1 for the j-th column if x is true else set z as true. Everything else is to be set as 0.  For every g, h in the (l+j)-th row, set the j-th column as 1 and set everything else as 0.  Set the first l columns of final row as 1 and the remaining k columns as 3.
Subset exists if formula is satisfiable  Considering a sample Boolean expression, take y i if x is true else take z i if x is false.  If the number of true literals in c j is at most 2, take g j  If the number of true literals in c j is 1, take h j
Example
Conclusion  We can conclude that the reduction can take place in polynomial time.  The size of the table is finite, roughly (l+k) 2 and that each entry can be easily calculated for any   The time complexity is O(n 2 ).  Thus SUBSET-SUM is NP Complete after reducing 3SAT.
“ Thank You ”

Download Presentation

Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

31. Stokes Theorem Stokes theorem is to Greens theorem, for the work done, as the

31. Stokes Theorem Stokes theorem is to Greens theorem, for the work done, as the divergence theorem is to Greens theorem, for the flux. Both are 3D generalisations of 2D theorems. Theorem 31.1 (Stokes Theorem) . Let C be any closed

406 views • 3 slides

29. The divergence theorem Theorem 29.1 (Divergence Theorem; Gauss, Ostrogradsky) . Let S be a

29. The divergence theorem Theorem 29.1 (Divergence Theorem; Gauss, Ostrogradsky) . Let S be a closed surface bounding a solid D , oriented outwards. Let F be a vector field with continuous partial derivatives. Then F d

471 views • 3 slides

Kleenes Theorem 4-0 Kleenes Theorem Theorem For every language L (over

Griffith University 3130CIT Theory of Computation (Based on slides by Harald Sndergaard of The University of Melbourne) Kleenes Theorem 4-0 Kleenes Theorem Theorem For every language L (over a finite

422 views • 17 slides

PCP Theorem [PCP Theorem is] the most important result in complexity theory since Cooks

PCP Theorem [PCP Theorem is] the most important result in complexity theory since Cooks Theorem. Ingo Wegener, 2005 Computational Complexity, by Fu Yuxi PCP Theorem 1 / 119 S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof

1.53k views • 120 slides

On Theorem Proving for Program Checking Historical perspective and recent developments Maria

Outline Introduction Where is theorem proving in program checking Inside theorem proving Big and little engines together: a new theorem proving style Decision procedures with speculative inferences Current and future challenges On Theorem

944 views • 69 slides

The Replacement Theorem Theorem (Theorem 1.10) Let V be a vector space and suppose G and L are

The Replacement Theorem Theorem (Theorem 1.10) Let V be a vector space and suppose G and L are finite subsets of V such that V = Span ( G ) , |G| = n , L is linearly independent, and |L| = m . Then m n and there is a set H G , such that

468 views • 36 slides

1 What is a Theorem Prover? What is a Theorem? Theorem: A formalizable statement which is

Prerequisites Course 2D1453, 2006-07 Undergraduate logic and discrete maths CS literacy Advanced Formal Methods Some functional programming experience useful The theorem prover Isabelle is programming in SML Some SML

644 views • 4 slides

22. Greens Theorem Theorem 22.1 (Greens Theorem) . If C is a positively oriented closed curve

22. Greens Theorem Theorem 22.1 (Greens Theorem) . If C is a positively oriented closed curve enclosing a region R then curl F d r = F d A. C R The circle in the centre of the integral sign is simply to emphasize

355 views • 4 slides

Generalized Intermediate Value Theorem Intermediate Value Theorem Theorem Intermediate Value

Generalized Intermediate Value Theorem Intermediate Value Theorem Theorem Intermediate Value Theorem Suppose f is continuous on [ a , b ] and let N be any number between f ( a ) and f ( b ) , where f ( a ) = f ( b ) . Then there exists a

59 views • 4 slides

From Shang Gao Magic 15 If a 2 + b 2 = c 2 , then ( a , b , c ) is called to be a Background

Magic 15 Magic 15 Background Two Square Theorem Galois Finite Fields Magic 15: A STORY OVER 380 YEARS Proofs of 2-Square Theorem Three Square Theorem Four Square Theorem Universal Polynomials Magic 15 Proof of Fifteen Theorem 15

1.31k views • 97 slides

Ch04. Maximum Theorem, Implicit Function Theorem and Envelope Theorem Ping Yu Faculty of

Ch04. Maximum Theorem, Implicit Function Theorem and Envelope Theorem Ping Yu Faculty of Business and Economics The University of Hong Kong Ping Yu (HKU) MIFE 1 / 27 The Maximum Theorem 1 The Implicit Function Theorem 2 The Envelope

599 views • 28 slides

Visual theorem proving with the Incredible Proof Machine The idea Theorem Proving without

Interactive Theorem Proving, Nancy August 22, 2016 Joachim Breitner Karlsruhe Institute of Technology University of Pennsylvania, Philadelphia Visual theorem proving with the Incredible Proof Machine The idea Theorem Proving without Syntax

579 views • 43 slides

The Bellows Theorem (Proof) Giovanni Viglietta JAIST July 5, 2018 The Bellows Theorem

The Bellows Theorem (Proof) Giovanni Viglietta JAIST July 5, 2018 The Bellows Theorem (Proof) Bellows theorem: statement Theorem (Sabitov, 1996) The volume V of a polyhedron (of any genus) with edge lengths 1 , , e

1.22k views • 93 slides

E LLIPTIC CURVES II ( THE ASSOCIATIVITY ) The algebraic proof of associativity the ring of

Elliptic curves over F q Formulas for Addition Computer assited proof of associativity Proof of associativity via combinatorial incidence Geometry Bezout Theorem Cayley-Bacharah Theorem Pappus Theorem Pascals Theorem Associativity E

474 views • 21 slides

The endpoint multilinear Kakeya theorem via the BorsukUlam/LusternikSchnirelmann theorem

. . . . . . . . . . . . . . . . The endpoint multilinear Kakeya theorem via the BorsukUlam/LusternikSchnirelmann theorem After L. Guth, A. Carbery, I. Valdimarsson Pavel Zorin-Kranich University of Bonn Kopp, October 2017

327 views • 20 slides

Green's Theorem is a special case of Stoke's 1 Some examples for Stoke's Theorem 2 3 4 5 6

Green's Theorem is a special case of Stoke's 1 Some examples for Stoke's Theorem 2 3 4 5 6 7 8 9 Divergence Theorem 10 11 12 13 Some commonlyused formulas resulting from the Divergence Theorem 14 15 16 17 18 Extension of the

854 views • 31 slides

Preliminaries and Problem Formulation Simplification 1 Roomba stores the complete map;

Preliminaries and Problem Formulation Simplification 1 Roomba stores the complete map; Computation is done off-line Simplification 2 Roomba does not switch between cleaning and traveling modes within one trip Algorithm Design Key idea:

324 views • 6 slides

Computational Complexity of the GPAC Amaury Pouly Joint work with Olivier Bournez and Daniel

Computational Complexity of the GPAC Amaury Pouly Joint work with Olivier Bournez and Daniel Graa April 10, 2014 Pouly, Bournez, Graa Computational Complexity of the GPAC April 10, 2014 / 17 Outline Introduction 1 GPAC

1.13k views • 84 slides

Complexity of Circuit Satisfiability Ramamohan Paturi University of California, San Diego

Complexity of Circuit Satisfiability Ramamohan Paturi University of California, San Diego jointly with Pavel Pudl ak, Czech Academy of Sciences November 9, 2009 Paturi/Pudl ak Complexity of Circuit Satisfiability Overview Exact

1.16k views • 73 slides

PHASE CORRECTION FOR DYNAMIC MEASUREMENTS Timothy Muyimbwa, Dr. Tony Schmitz Background

PHASE CORRECTION FOR DYNAMIC MEASUREMENTS Timothy Muyimbwa, Dr. Tony Schmitz Background Monitoring vibrations is an important aspect of industrial processes, such as machining operations, in the aerospace, automotive, and medical fields

330 views • 11 slides

On the temporal robustness of uniprocessor real-time systems scheduled with FP and EDF Laurent

On the temporal robustness of uniprocessor real-time systems scheduled with FP and EDF Laurent George Contact: lgeorge@ieee.org Univ. of Paris 12 / ECE ETR'07 September 3-7, Nantes Outline 1. Introduction 2. Classical FC 3. Sensitivity of

311 views • 27 slides

Equations over sets of natural numbers. Artur Je z University of Wroclaw December 13, 2007

Equations over sets of natural numbers. Artur Je z University of Wroclaw December 13, 2007 Artur Je z ( University of Wroclaw ) Equations over sets of natural numbers. December 13, 2007 1 / 27 Part I Conjunctive Grammars Artur Je

1.51k views • 140 slides

Probabilistic analysis on the rank of Macaulay matrices over finite fields Andrea Tenti Selmer

Probabilistic analysis on the rank of Macaulay matrices over finite fields Andrea Tenti Selmer Senter Univerity of Bergen Finse, May 08, 2018 Joint work with Igor Semaev I. Semaev, A.Tenti (UiB) Rank of Macaulay matrices on finite fields

260 views • 22 slides

Switch ON Ayush Jain (aj2672) Donovan Chan(dc3095) Shivam Choudhary(sc3973) An

Switch ON Ayush Jain (aj2672) Donovan Chan(dc3095) Shivam Choudhary(sc3973) An extremely efficient Hardware Switch!! Architecture Userspace generates packets. Input module sorts and places in RAM(s). Scheduler avoids

419 views • 19 slides