CS3000: Algorithms & Data Jonathan Ullman Lecture 16: - PowerPoint PPT Presentation

Hello CS3000: Algorithms & Data Jonathan Ullman Lecture 16: Applications of Network Flow • March 23, 2020

Midterm II • Logistics: • Wednesday April 1 st • Administratered through Gradescope • Exam will be available from 2:50pm until _____ • You have 90 minutes from the time you start • Academic Honesty: • You will have to take an honor pledge • Do not discuss the exam with anyone for 24 hours • Please be a good citizen

Midterm II • Format: • Same as Midterm I, but condensed for Gradescope Minimum • Topics (graph algos): • Key graph definitions and concepts spanning µ Tree • Adjacency list / matrix representations • DFS + topological sort • Shortest paths: BFS, Dijkstra, Bellman-Ford • Network flow: concepts, Ford-Fulkerson, choosing paths • Does not include this week

Applications of Network Flow

Applications of Network Flow • Algorithms for maximum flow can be used to solve: • Bipartite Matching 2 • Disjoint Paths me • Survey Design • Matrix Rounding If can be problem a • Auction Design in polynomial time solved • Fair Division • Project Selection can be written as then it • Baseball Elimination application of max flow an • Airline Scheduling • …

Reduction • Definition: a reduction is an efficient algorithm that solves problem A using calls to function that solves problem B.

Mechanics of Reductions • What exactly is a problem ? a list of AGI numbers ACD • A set of legal inputs ! e.g • A set "($) of legal outputs for each $ ∈ ! A in soiled order e g • Example: integer maximum flow V E Ede 3 t where G s Inputs integer for every e C E is an e c f e 3 far flow G A Outputs maximum is an integer fer every et E f where e

Use an alg for Problem A Mechanics of Reductions solve Problem B to Input x for Input u for Problem B Problem A SolveA Output y in B(x) Output v in A(u) for Problem B for Problem A we only make Simplified Version solve A call to one

When is a Reduction Correct? Input x for Input u for Problem B Problem A SolveA Output y in B(x) Output v in A(u) for Problem B for Problem A Assume for valid For every valid input u for ProbA input Ala If valid output in is any solve A returns w in BH valid output VEA valid orbit u then y a is a T c Alu can be any

What is the Running Time? Input x for Input u for Problem B Problem A SolveA Output y in B(x) Output v in A(u) for Problem B for Problem A t Total Time Solve A the running t.me of depends on the input u on differ from that of x size of u may The

Max flow A Example: Minimum Cut B at min Input x for Input u for Problem B Problem A SolveA Output y in B(x) Output v in A(u) for Problem B for Problem A Ecce 3 x for G B E t V Inpol s 14 t Edel s G V E u for A Input H G max flow f u C Alu A m Outpol

Max flow A Example: Minimum Cut B at min Input x for Input u for Problem B Problem A SolveA Output y in B(x) Output v in A(u) for Problem B for Problem A Hr compte residual graph Of f Output ytB x Take in Gf Fund nodes reachable from s those nodes as the art Off

Max flow A Example: Minimum Cut B at min Input x for Input u for Problem B Problem A SolveA Output y in B(x) Output v in A(u) for Problem B for Problem A sff.IE Running time j.sndeocna Donothng 0Cn a edges in g 0 nodes in a 3OBFS.OFt.me

A sorting Example: Median median B Input x for Input u for Problem B Problem A SolveA Output y in B(x) Output v in A(u) for Problem B for Problem A An array A of length Input x for B n same array u for A The Input i S

Bipartite Matching DEE He u UEL Ef vice or versa • Input: bipartite graph ' = (*, ,) with * = - ∪ / • Output: a maximum cardinality matching • A matching 0 ⊆ , is a set of edges such that every node 2 is an endpoint of at most one edge in 0 • Cardinality = 0 R L Models any problem where one type of object is assigned to another type: • doctors to hospitals • jobs to processors • advertisements to websites red edges are a matching

ng MBM max bipartite matching Bipartite Matching max flow IMF integer forMBMJ stole Tgif hmg • There is a reduction that uses integer maximum s-t flow to solve maximum bipartite matching.

Step 1: Transform the Input Input G for Input G’ for MCBM MAXFLOW t e t.se t.II.IE

Step 1: Transform the Input Input G for Input G’ for MCBM MAXFLOW

Step 2: Receive the Output Matching will be all of the L Input G’ for R edges MAXFLOW that carry flow SolveA I Output f for MAXFLOW Ts I Red arrow means f(e)=1 Black means f(e) = 0

Step 3: Transform the Output Output M for Output f for MCBM MAXFLOW all edges from L to R M carry flow t

Reduction Recap • Step 1: Transform the Input • Given G = (L,R,E), produce G’ = (V,E,{c(e)},s,t) by... • ... orienting edges e from L to R • ... adding a node s with edges from s to every node in L • ... adding a node t with edges from every not in R to t • ... seting all capacities to 1 • Step 2: Receive the Output • Find an integer maximum s-t flow f in G’ • Step 3: Transform the Output • Given an integer s-t flow f(e)… • Let M be the set of edges e going from L to R that have f(e)=1

Correctness • Need to show: • (1) This algorithm returns a matching • (2) This matching is a maximum cardinality matching the endpoint of is node v Every Matching M 1 in most edge at Its 011 Exactly one edge out of can carry flow u

Correctness • Need to show: • (1) This algorithm returns a matching • (2) This matching is a maximum cardinality matching

Correctness • This algorithm returns a matching

Correctness • Claim: G has a matching of cardinality at least k if and only if G’ has an s-t flow of value at least k A matching of size kuaffineska flow of a E

Correctness • Claim: G has a matching of cardinality at least k if and only if G’ has an s-t flow of value at least k A flow of value k must edges carrying flow k have R from L to R L x is 7 matching of size k floor k

G has m edges nodes Running Time n • Need to analyze the time for: • (1) Producing G’ given G • (2) Finding a maximum flow in G’ • (3) Producing M given G’ edges 2 nodes has ntm G nt OCn in t.ve G Can write 0µF m is the 1 m n where n 12 ntm 1 flow a graph c th n in solve mat to time nm edges nodes m which have flow Scan the edges of G and see t.me 0 ntm

Summary integer Solving maximum s-t flow in a graph with n+2 nodes and m+n edges and c(e) = 1 in time T Solving maximum bipartite matching in a graph with n nodes and m edges in time T + O(m+n) • Can solve max bipartite matching in time O(nm) using Ford-Fulkerson • Improvement for maximum flow gives improvement for maximum bipartite matching

Hallis heoremo gewrymdeojf.IE'd G has a perfect matching o 0 iff does not 0 exist there a SEL of nodes set 0 EIS INDI that 8 such is the 0 PCs 00 set of all neighbors 80 in S of nodes