A POLYNOMIAL COMBINATORIAL ALGORITHM FOR GENERALIZED MINIMUM COST FLOW, KEVIN D. WAYNE Eyal Dushkin –

REMINDER – GENERALIZED FLOWS u 10 v V received 8 units of flow!

REMINDER - GENERALIZED MINIMUM COST FLOWS

PROBLEM HISTORY

POLYNOMIAL COMBINATORIAL ALGORITHM FOR GENERALIZED MINIMUM COST FLOW (2003) We solve the generalized minimum cost circulation problem, in which all supplies and demands are zero We present combinatorial algorithms which solve the problem in a polynomial time These algorithms are strongly polynomial approximation schemes for the minimum cost circulation problem

GENERALIZED MINIMUM COST CIRCULATION

ASSUMPTIONS

RESIDUAL NETWORKS u v u v 4020

CIRCULATION DECOMPOSITION The gain of a cycle is the product of the gain factors of arcs participating in that cycle A unit-gain cycle is a cycle whose gain is equal to one A flow-generating cycle is a cycle whose gain is greater than one A flow-absorbing cycle is a cycle whose gain is less than one

CIRCULATION DECOMPOSITION A unit-gain cycle is a cycle whose gain is equal to one A flow-generating cycle is a cycle whose gain is greater than one A flow-absorbing cycle is a cycle whose gain is less than one

CIRCULATION DECOMPOSITION A bicycle is a flow-generating cycle, a flow-absorbing cycle, and a (possibly trivial) path from the first to the second

CIRCULATION DECOMPOSITION

CIRCUITS AND COSTS

CIRCUIT CANCELING ALGORITHM We start with a feasible circulation g = 0 and then repeatedly cancel a negative cost residual circuit Klein's cycle-canceling (in non-generalized networks):  Initialize g = 0  Repeat:  Cancel a negative cost circuit in Gg  Update g  until optimal Complexity: Very bad! NP-hard even to detect a unit-gain cycle …

CIRCUIT CANCELING ALGORITHM (2 ND TRY) u v u Cost = = -45 Mean Cost = ( )/3 = -15

CIRCUIT CANCELING ALGORITHM (3 RD TRY) u v u

CIRCUIT CANCELING ALGORITHM

ALGORITHM CORRECTNESS

MINIMUM RATIO CIRCUIT ALGORITHM In this section we discuss 3 matters: 1. Detecting a circuit 2. Detecting a negative cost circuit 3. Finding a minimum ratio circuit

DETECTING A CIRCUIT A circuit is either a bicycle or a unit-gain cycle Recall: a bicycle is a flow-generating cycle and a flow-absorbing cycle connected by a path from the first to the second Detecting a circuit: Step 1 - Detecting a bicycle Step 2 - Remove the bicycles and detect unit-gain cycles

STEP1 - DETECTING A BICYCLE First find a subset of nodes that leads to a flow-absorbing cycle or participate in such one

STEP1 - DETECTING A BICYCLE /2 61 3/4 1

STEP1 - DETECTING A BICYCLE 3/ / /4 1

STEP1 - DETECTING A BICYCLE 3/ / /4 1

STEP1 - DETECTING A BICYCLE 3/4 1 1/ /4 1

STEP1 - DETECTING A BICYCLE 9/16 1 1/ /4 1 nth-step

STEP1 - DETECTING A BICYCLE / /4 1 (2n-1)th-step

STEP1 - DETECTING A BICYCLE / /4 1 (2n)th-step

STEP1 - DETECTING A BICYCLE

STEP2 - DETECTING A UNIT-GAIN CYCLE 1. Detect unit-gain cycles in the subgraph induced by V\N – On Board 2. Detect unit-gain cycles in the subgraph induced by N – On Board

MINIMUM RATIO CIRCUIT ALGORITHM

DETECTING A NEGATIVE COST CIRCUIT

2VPI (2 Variables Per Inequality) Feasibility

2VPI Feasibility

MINIMUM RATIO CIRCUIT ALGORITHM

FINDING A MINIMUM RATIO CIRCUIT

FINDING A MINIMUM RATIO CIRCUIT – ANAYLSIS (NO PROOF)

MINIMUM RATIO CIRCUIT ALGORITHM

ALGORITHM CORRECTNESS

SCALING VERSION The bottleneck computation in the former was detecting min ratio circuits Idea: Cancel approximately min ratio circuits Improvement: Cancel negative cost circuits instead of min ratio circuits (factor n speeds up)

SCALING VERSION

ALGORITHMS SUMMARY Approximation Algorithms 1 st AlgorithmFaster Scaling Version What About Exact Algorithms ?

ROUNDING TO A VERTEX

ALGORITHMS SUMMARY Approximation Algorithms 1 st AlgorithmFaster Scaling Version Exact Algorithms 1 st AlgorithmFaster Scaling Version

QUESTIONS? Eyal Dushkin –