Lecture 32: Bipartite Matchings and Reductions

Overview

1. Reductions
2. Maximum Bipartite Matchings
3. Reduction to Maximum Flow

Big Picture, So Far

Efficient algorithms for many problems:

• sorting
• graph problems
  • shortest paths
  • Eulerian circuits
  • minimum spanning trees
• interval scheduling
• sequence alignment
• stable matching

All solved in $O(N^2)$ time ($N = $ size of instance)

A Question

What algorithmic problems cannot be solved efficiently?

Last Unit of the Semester

Algorithmic Reality

For many practical problems:

1. no efficient algorithm is known
2. no proof that there isn’t an efficient algorithm

What do we do in this situation?

Algorithm Life

Observation. Algorithm design is challenging.

Reductions

Properties of nice transformations:

1. transforming instances of $A$ to instances of $B$ can be done efficiently
2. solution to $B$ instance can be efficiently transformed back to solution for $A$ instance

$1 + 2 = $ reduction from problem $A$ to problem $B$

Practical Value of Reductions

If $A \leq_P B$, then:

• any efficient solution to $B$ gives an efficient solution to $A$
• an improved algorithm for $B$ may give an improved algorithm for $A$

New Challenge

• Given a problem $A$, solve it by reducing to another problem $B$ that you already have an algorithm for

Internship Assignments, Again

In a small world…

• Three students: $a, b, c$
• Three internships: $X, Y, Z$

Students/Internships have acceptability criteria (not preferences)

How to match students an internships?

Viewed as a Graph

Not Great Matching

Best Matching

Maximum Bipartite Matching (MBM)

Input:

• bipartite graph $G = (V, E)$
• $V$ partitioned into two disjoint sets, $S, T$
• All edges are pairs $(s, t)$ with $s \in S$, $t \in T$

Output:

• a matching $M = \{(s_1, t_1), (s_2, t_2), \ldots, (s_k, t_k)\}$
  • each $(s_i, t_i)$ is an edge in $G$
  • each $s \in S$, $t \in T$ appears in at most one pair
• $M$ is a maximum matching: there is no matching of size $\ell > k$

Simplest Interesting Example

Does greedily adding edges to form a matching always result in a maximum matching?

Greedy Doesn’t Work

Issue: choosing an edge greedily/prematurely may block other edges that could result in a larger matching

• does this sound familiar?

Adapting to Our New Lifestyle

Don’t design a new algorithm…

…instead make a reduction to…

Reduction to Max-Flow

How to turn MBM into a flow problem?

A Max-Flow Instance

Formalizing the Reduction

Input:

• Bipartite graph $G = (V, E)$ with $V = S \cup T$

Output:

• Directed graph $G’ = (V’, E’)$
• $V’ = V \cup \{s, t\}$
• For $E’$:
  • direct all edges $(s_i, t_j) \in E$ from $S$ to $T$
  • add edges $(s, s_i)$ for all $s_i \in S$
  • add edges $(t_j, t)$ for all $t_j \in T$
  • all capacities are $1$

I Start with Bipartite Graph

II Form Max Flow Instance

III Solve Max Flow

IV Form Matching from Flow

A Technicality

Assume all flows are integral flows:

• amount of flow across each edge is an integer

Note. Ford-Fulkerson always gives an integral flow.

Claim 1

If $G$ has a matching of size $k$, then $G’$ admits a flow $f$ of value $k$.

Claim 2

If $G’$ admits an integral flow of value $k$, then $G$ has a matching of size $k$.

Implications

1. Value of the maximum flow in $G’$ equals the size maximum matching in $G$.
2. Given a maximum (integral) flow in $G’$, we can find a maximum matching in $G$

Conclusions

• Found a reduction from Maximum Bipartite Matching to Max Flow
• Any (integral) Max Flow algorithm can be used to solve MBM
  • any improvement in Max Flow algorithms will yield a corresponding improvement in MBM solution
• We showed MBM $\leq_P$ Max Flow
  • MBM is “no harder” than Max Flow
  • Max Flow is “no easier” than MBM

Next Time

Another view of $\leq_P$:

• suppose $A$ is a “hard” problem
• we show $A \leq_P B$
• then we’ve established that $B$ is no easier than $A$
  • any efficient solution to $B$ would imply an efficient solution to $A$