I Finally Understood Ford–Fulkerson by Solving Splitwise’s “Simplify Debts”

For a long time, Ford–Fulkerson felt like one of those DSA algorithms that:

• shows up in textbooks


• appears in interviews


• but never really shows up in real products

I could recite:

“It finds the maximum flow in a graph”

But I never understood:

• what problem it truly solves


• why reverse edges exist


• why source and sink are even needed


• how this has anything to do with real apps

That changed when I tried to deeply understand Splitwise’s “Simplify Debts” feature.

This article is written for developers who:

• struggled with augmenting paths


• were confused by reverse edges


• didn’t get why “infinity” is used


• wondered why max-flow works for money problems

I’ll explain everything intuitively, step by step.

The real problem: what does “simplify debt” actually mean?

As far as I guess, Splitwise is not trying to preserve:

• who paid whom originally


• the sequence of transactions

Its only goal is:

Find the cleanest way to settle money so everyone ends up correct.

That’s it.

Step 1: Reduce everything to net balances (pure accounting)

Before algorithms, we do accounting.

For each person:

net_balance = money_received − money_paid

Example after reduction:

Interpretation:

• A must pay ₹50


• B must pay ₹30


• C must receive ₹40


• D must receive ₹40

Total owed = total received = ₹80

At this point, I was stuck:

Who should pay whom?

The key insight (this is the click)

_Debt simplification is routing money from people who owe to people who should receive, without creating or destroying money.

_

That is exactly what a flow network models.

Money is flow.

People are nodes.

Payments are edges.

Step 2: Why we need a Source and a Sink

This confused me the most at first.

Why a Source?

We want a single place that represents:

“All money that must be paid”

So we introduce a Source (S).

Example:

Meaning:

• A can send at most ₹50 into the system


• B can send at most ₹30

Why a Sink?

We also want a single place that represents:

“All money that must be received”

So we introduce a Sink (T).

If someone should receive money:

Example:

Meaning:

• C can absorb at most ₹40


• D can absorb at most ₹40

Critical mental model

_

• Source injects money.


• Sink absorbs money.


• Everyone else just routes it._

Without a sink:

• there is no definition of “maximum”


• flow could circulate forever


• the problem has no goal

Step 3: Connecting people (the “infinity” confusion)

Now we connect debtors to creditors:

Why is this “infinite”?

This does NOT mean infinite money.

It means:

Do not restrict who can pay whom.

The real limits already exist:

• A cannot pay more than ₹50 (from S → A)


• C cannot receive more than ₹40 (from C → T)

So “∞” just means:

Large enough to never be the bottleneck

In practice:

∞ = total owed (or any sufficiently large number)

Step 4: What Ford–Fulkerson actually solves

Ford–Fulkerson answers one question only:

Any path from S to T where money can still flow.

Example:

S → A → C → T

Translated to English:

“Take money from the debt pool → A pays C → C receives it”

Each augmenting path is a valid payment.

Step-by-step execution (no fear, no magic)

Augmenting Path 1

S → A → C → T

Bottleneck:

min(50, ∞, 40) = 40

Action:

• A pays C ₹40

Remaining:

• A owes ₹10


• C is fully settled

Augmenting Path 2

S → A → D → T

min(10, ∞, 40) = 10

Action:

• A pays D ₹10

Remaining:

• A is settled


• D still needs ₹30

Augmenting Path 3

S → B → D → T

Bottleneck:

min(30, ∞, 30) = 30

Action:

• B pays D ₹30

Step 5: Reading the final answer (this is crucial)

Ignore Source and Sink.

Look only at person → person flows:

That is the simplified debt list.

Why reverse edges are inevitable?

This was my biggest confusion.

Reverse edges mean:

You are allowed to undo a previous payment if a better routing exists.

They do not represent real payments.

They represent:

• refunds


• cancellations


• rerouting permissions

Without reverse edges:

• early decisions become permanent


• greedy choices can block better solutions


• max-flow can fail

In real life, debt simplification must allow:

“Let me undo that payment and send the money elsewhere instead.”

Reverse edges make that possible.

They are not an implementation trick — they are mathematically required.

What does “optimal” mean here?

This is important.

Optimal does NOT mean:

• fair to everyone


• equal distribution


• preserving original transactions

Optimal means:

The maximum possible amount of money reaches the sink without violating constraints.

This is system-level optimality, not individual optimality.

Some people may:

• pay less than they “could”


• receive later


• be bypassed entirely

And that’s perfectly correct.

Why max-flow fits simplify debt perfectly

This is not a trick or coincidence.

Debt simplification is inherently a routing problem.

Final takeaway

Ford–Fulkerson is not about pipes and water.

It is about:

Routing a conserved quantity optimally under constraints.

Money is conserved.

Debt simplification is routing.

Max-flow fits naturally.

Once I saw that, the algorithm finally made sense.

Note: Technically, minimizing the number of transactions is an NP-hard problem. Max-Flow ensures all money is settled correctly, but getting the absolute minimum number of edges usually requires additional greedy heuristics on top of this. For the purpose of this article, we are focusing on the flow conservation!