Feasible Circulation and Network Flow: A Step-by-Step Guide for CSCI 570 Homework 5

Introduction to Feasible Circulation in Networks

In many real-world applications—from supply chain logistics to tournament scheduling—we encounter networks where flow must satisfy both lower and upper bounds on edges, as well as supply/demand at vertices. This tutorial walks you through the key steps to determine if a feasible circulation exists, using a classic problem similar to CSCI 570 Homework 5. We'll also connect these ideas to current trends like AI-driven scheduling apps and large-scale event planning.

Step 1: Remove Lower Bounds

Given a network with lower bounds l(e) and capacities c(e) on each edge, the first step is to transform the problem into one without lower bounds. For each edge e = (u, v) with lower bound l(e), we subtract l(e) from both the capacity and the demand of vertices. Specifically:

• New capacity: c'(e) = c(e) - l(e)
• Update demands: demand(u) += l(e) (since flow leaves u), demand(v) -= l(e) (since flow enters v)

After this transformation, we have a standard circulation problem with only upper bounds (capacities).

Step 2: Solve the Circulation Problem Without Lower Bounds

To check feasibility, we add a super source s and super sink t. For each vertex i with demand d(i):

• If d(i) > 0, add edge s → i with capacity d(i)
• If d(i) < 0, add edge i → t with capacity -d(i)

Then compute the max flow from s to t. A feasible circulation exists if and only if the max flow saturates all edges from s (i.e., total supply equals max flow value).

Step 3: Determine Feasibility in the Original Graph

If the max flow value equals the total supply, then a feasible circulation exists. We can reconstruct the original flow by adding back the lower bounds. Otherwise, no feasible circulation exists.

Real-World Application: Scheduling the US Open

Imagine scheduling the first round of a tennis tournament like the US Open. With 2N players, constraints include seed differences, court availability, timeslot limits, and player preferences. This can be modeled as a circulation problem:

• Create nodes for top-half players, bottom-half players, matches, timeslots, and courts.
• Edges represent possible assignments with lower bounds (e.g., each match must have one top and one bottom player) and capacities (e.g., max matches per court).
• Player preferences become demand or supply adjustments.

By solving the circulation, we can determine if a feasible schedule exists—similar to how AI scheduling tools optimize event logistics today.

Linear Programming for Resource Allocation

In another scenario, a company must allocate time and special substance (SS) to produce steel, brass, and pewter. This is a linear programming (LP) problem:

Maximize: total tons = x_steel + x_brass + x_pewter
Subject to:
  2*x_steel + 3*x_brass + 1*x_pewter ≤ 8 (time)
  5*x_steel + 4*x_brass + 6*x_pewter ≤ 20 (SS)
  x_steel, x_brass, x_pewter ≥ 0

To maximize SS usage while meeting other constraints, we can adjust the objective or add constraints like x_steel + x_pewter ≥ 2. LP solvers are widely used in finance and supply chain optimization.

Transportation Problem: Minimizing Shipping Costs

A cement supplier with warehouses in cities A and B must ship to customers in C and D. This is a classic transportation LP:

Minimize: 1*x_AC + 2*x_AD + 3*x_BC + 4*x_BD
Subject to:
  x_AC + x_AD ≤ 70 (supply at A)
  x_BC + x_BD ≤ 80 (supply at B)
  x_AC + x_BC ≥ 50 (demand at C)
  x_AD + x_BD ≥ 60 (demand at D)
  all variables ≥ 0

Such problems are fundamental in logistics and are solved using network flow or simplex methods.

Duality and Vertex Cover

The dual of a linear program provides insight into the problem's structure. For example, the dual of the resource allocation LP above can be interpreted as pricing the resources. In integer programming, the vertex cover problem asks for the smallest set of vertices touching every edge:

Minimize: sum_{v in V} x_v
Subject to: x_u + x_v ≥ 1 for every edge (u,v)
  x_v ∈ {0,1}

This is NP-hard, but approximation algorithms exist.

NP-Completeness and Reductions

Many scheduling and optimization problems are NP-complete. For instance, graph 5-coloring is NP-complete (reduction from 3-coloring), and Longest Path reduces from Hamiltonian Path. Understanding these reductions is crucial for algorithm design. In practice, we often use heuristics or approximation algorithms for such problems.

Conclusion

Mastering feasible circulation and network flow opens doors to solving complex scheduling, logistics, and resource allocation problems. Whether you're working on homework or designing AI-driven scheduling systems, these techniques are invaluable. For more practice, try applying these steps to the US Open scheduling problem or your own projects.