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COMP 3804 — Assignment
创建日期：2024-05-04 17:14:20
标签： COMP 3804
Assignment

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COMP 3804 — Assignment 

Assignment Policy:
• Your assignment must be submitted as one single PDF file through cuLearn.
Use the following format to name your file:
LastName StudentId a4.pdf
• Late assignments will not be accepted. I will not reply to emails of the
type “my internet connection broke down at 23:57” or “my scanner stopped
working at 23:58”, or “my dog ate my laptop charger”.
• You are encouraged to collaborate on assignments, but at the level of discussion only.
When writing your solutions, you must do so in your own words.
• Past experience has shown conclusively that those who do not put adequate effort into
the assignments do not learn the material and have a probability near 1 of doing poorly
on the exams.
• When writing your solutions, you must follow the guidelines below.
– You must justify your answers.
– The answers should be concise, clear and neat.
– When presenting proofs, every step should be justified.
Question 1: Write your name and student number.
Question 2: The (0, 1)-integer programming problem is defined as follows:
IntProg := {(A, c) : A is an integer m× n matrix for some integers m and n,
c is an integer vector of length m, and
∃x ∈ {0, 1}n such that Ax ≤ c (componentwise) }.
Prove that the language IntProg is in NP.
1
Question 3: The subset sum problem is defined as follows:
SubsetSum := {(a1, a2, . . . , am, b) : m, a1, a2, . . . , am, b are integers and
∃I ⊆ {1, 2, . . . ,m} such that ∑i∈I ai = b}.
Prove that SubsetSum ≤P IntProg, i.e., in polynomial time, SubsetSum can be reduced
to IntProg.
Question 4: The three-coloring problem is defined as follows:
3Color := {G : G is a graph whose vertices can be colored using three colors
such that any two adjacent vertices have distinct colors }.
The clique covering problem is defined as follows:
CliqueCover := {(G, k) : G = (V,E) is a graph whose vertex set V can be
partitioned into k subsets V1, V2, . . . , Vk such that
each subset Vi forms a clique in G }.
(4.1) Prove that the language 3Color is in NP.
(4.2) Prove that the language CliqueCover is in NP.
(4.3) Prove that 3Color ≤P CliqueCover, i.e., in polynomial time, 3Color can be
reduced to CliqueCover.
Question 5: You are given
• a sequence A = (A1, A2, . . . , Ak), where each Ai is a set consisting of three elements,
and
• a sequence B = (B1, B2, . . . , B`), where each Bj is a set consisting of two elements.
This pair (A,B) of sequences is called good if there exists a set T such that
• T contains at least one element of each set Ai, and
• T contains at most one element of each set Bj.
The Problem without a Name is defined as follows:
ProblemWithoutAName := {(A,B) : the pair (A,B) is good}.
Prove that the language ProblemWithoutAName is NP-complete.
Hint: You may remember from class that 3Sat is NP-complete.
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关键词一 关键词二 关键词三 关键词四 关键词五 和法规和的发更好的风格和 ECON 615 FINC 612 CSC148 COMP 639 ACCT 203 ACCT 211 PSYC 101 MTH2009 COMP52315 PSYCO 432 statistics MATH4063 11 xxx 1 MA1MSP MAST30028 MATH3888 CS341 EEE30002 ES2F5 MCD4160 CC0933 ECON 1101 MCHM3001 CHEM3118 COMMGMT 2511 COMP41280 STAT 631 24T1 IRE379 SENG365 ECON433 FIT3139 MATH4602 LING5621M FINS5516 ECON3208 MN2177 L2 SM EEET1415 Finance DS4023 COMP3004 7CCSMRTS K1S5B6 Stat230 CSCI4211 CSI2132 ECON30130 MAS6100 ENVS222 COM2107 INFO1112 STAT 4004 FIT2100 comp2022 COMP4691 Physics 307 Econ 659 probability CSCI316 fir29 CSCI203 COMP3160 COMP6714 INFO1113 Physics 7B Physics Photonics S203031 STAT2005 COMP90015 CSCI 2110 CSCC46 Information CSC477 COMP4650 Statistical COMP 346 COMP1017 ELEC340 MATH1007 Programming function STAT 244 CS 134 Java cse13s Math 108A FW20 ECON3389 CSE 130 Science 331 COMP9020 engineering COSC 445 CSE 5523 GAME2012 visualization MATH 235 Math 370 CSC343 COGS 108 EE524 COMP10200 STAT 416 3FK4 MSCI 311 ECON6113 ACC40700 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