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SWEN30006 Software Modelling and Design Project 1: Snakes and Ladders

创建日期：2024-06-19 16:39:27

所属分类：数学mathematics

标签： SWEN30006

Software Modelling and Design Project 1: Snakes and Ladders

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SWEN30006 Software Modelling and Design Project 1: Snakes and Ladders

Anu is making holiday travel plans. There are 6 cities that she is interested in visiting. For each
k 2 f1; : : : ; 6g, it would cost ck to incorporate city k into the itinerary but she would gain hk
units of happiness from visiting city k (both amounts are regardless of which other cities are on the
itinerary). Unfortunately, she has a limited budget of B, so she may not be able to visit all the cities.

There are also other considerations. Cities 3 and 5 are too similar to each other, so Anu prefers to
visit at most one of them. The same is true for cities 1, 5, and 6. Some cities can only be enjoyed in
the fullest if visited together with some other cities. Anu doesn’t want to visit cities 5 or 6, unless
at least one of cities 3 and 4 is also on the itinerary. Finally, she wants to visit at least one and at
most three among the cities 1; 2; 4; 5; 6.

Anu obviously wants to nd an itinerary (a subset of cities to visit) that will maximize her happiness,
given the constraints listed above.

(a) [10 Points] Formulate this problem as a binary integer linear program. Here, binary means
that every variable xk in your program should be in f0; 1g.

(b) [10 Points] Imagine a relaxation of the program from part (a), where you allow each variable
xk to take any non-negative real value (i.e., xk ≥ 0). This relaxation becomes a linear program.

Write the dual of this linear program.

Anu has begun her much awaited journey. While she planned a bit before leaving, she didn’t work
out all the details in advance. She now faces the problem of planning her transportation.

There are n buses running along a road (think of it as the real axis). Each bus i starts at location
si and goes to location ti (where ti > si). If she decides to use bus i, she can board at any point
x 2 [si; ti], drop o at any later point y 2 [x; ti], and must pay a xed cost ci regardless of where
she boards and drops o. Anu wants to get from point a to point b on the road (where b > a).

(a) [15 Points] Write a binary integer linear program that helps Anu achieve this goal while mini
mizing her total cost. Your program only needs to nd the minimum total cost. It is OK if it does
not nd the locations at which Anu should board and drop o buses. Your program must have a
nite (and ideally, a polynomial) number of constraints. Prove that your program is correct.

(b) [5 Points] Suppose each bus is operated by one of two companies. Let T1 and T2 be the indices
of buses operated by companies 1 and 2, respectively (T1 \ T2 = ; and T1 [ T2 = f1; : : : ; ng). For
some reason, Anu wants to make sure that she don’t spend more than 70% of the total cost on any
single company. Note that this constraint might raise the minimum total cost she needs to pay in
order to get from point a to point b. How would you add this constraint to the program in part (a)?

As Anu was thinking more about her dilemma of avoiding similar cities from Q1, she realizes that
she perhaps modeled the problem too simply. The actual problem is a bit more involved and graph
theoretic.

There is an undirected graph G = (V; E), where every node in V is a city and every edge in E is a
road connecting two cities. Anu wants to travel from source city s to destination city t. Anu has
made a list L containing pairs of similar cities: for every (u; v) 2 L, cities u and v are too similar
to each other, so Anu would prefer to avoid visiting them both. Anu is interested in solving the
following decision problem.

AvoidSimilar:

Input: Undirected graph G = (V; E), list L containing pairs of similar cities, source city s, desti
nation city t.

Question: Does there exist a path from s to t such that for every (u; v) 2 L, it visits at most one
of cities u and v?

(a) [10 Points] Prove that AvoidSimilar is NP-complete. For hardness, provide a reduction from
the Exact 3SAT problem we saw in class.

Exact 3SAT:

Input: An exact 3CNF formula ’ with n variables x1; : : : ; xn and m clauses C1; : : : ; Cm.

Question: Does ’ admit a satisfying assignment?

[Hint: The gadgets shown below may be useful. In the variable gadget on the left, going from node
a to node b requires making a binary choice, similar to setting a variable to true or false. In the
clause gadget on the right, going from node a to node b requires making a ternary choice, similar
to choosing a literal of the clause to set to true. Think about combining these gadgets.]

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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