a polynomial combinatorial algorithm for generalized minimum cost flow, kevin d. wayne eyal dushkin...

A POLYNOMIAL COMBINATORIAL ALGORITHM FOR GENERALIZED

MINIMUMCOST FLOW, KEVIN D. WAYNE

Eyal Dushkin – 03.06.13

REMINDER – GENERALIZED FLOWS We are given a graph

We associate a positive with every arc

Assume that if 1 unit of flow was sent from node along the arc then units of flow arrived at node

Notice that the model is a generalization of the minimal cost flow problem, with every arc having the multiplier

u10 v

V received 8 units of flow!

REMINDER - GENERALIZED MINIMUM COST FLOWS G=(V,E) is a directed graph

Capacity function

Gain function

Cost function

Balance function

n = |V| ; m = |E|

Minimize

Subject to:

And

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

Balance function

A circulation is a nonnegative function that satisfies the flow conservation constraints:

It is feasible if it satisfies the capacity constraints:

Define

The problem is to find a feasible generalized circulation of minimum cost

ASSUMPTIONS

Costs and capacities are integral

Gain factors are ratios of two integers p/q

Let B the biggest integer among the above (costs, capacities, p, q) , we assume

RESIDUAL NETWORKS

Residual Capacity

Gain Factor

Cost

u v

u v40

20

𝛾 (𝑒 )=12,𝑐 (𝑒 )=17 ,𝑢𝑔 (𝑒 )=60

𝛾 (𝑒 )=2,𝑐 (𝑒)=−34 ,𝑢𝑔 (𝑒 )=20

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

A circuit is a circulation that sends flow only along the arcs of a single unit-gain cycle or bicycle

Decomposition Lemma (without proof):

A generalized circulation g can be decomposed into components such that and each component is a circuit that is positive only on arcs with

Optimality Condition (without proof):

optimal --- no negative cost residual unit-gain cycles or bicycles

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 = 0Repeat:

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 (2ND TRY)

The mean cost of a cycle is

Repeatedly cancelling the minimum mean cost residual cycle is a strongly polynomial for non-generalized flow

Unfortunately for generalized flow, the complexity analysis changes

u v

u

−20

−10−15

Cost = -15-20-10 = -45Mean Cost = (-15-20-10)/3 = -15

CIRCUIT CANCELING ALGORITHM (3RD TRY)

We are interested in cycles with very negative cost and with large residual capacity

Let a time function

The ratio of a cycle is - the ratio of its cost to time

Consider

For traditional networks this gives a weakly polynomial algorithm

u v

u

−20

−10−15

𝑢𝑔 (𝑒 )=10

𝑢𝑔 (𝑒 )=1𝑢𝑔 (𝑒 )=2

= 28.125

CIRCUIT CANCELING ALGORITHM (3RD TRY)

We define the ratio of a circuit x to be:

where and c(x)=

Examples on board

CIRCUIT CANCELING ALGORITHM

ALGORITHM CORRECTNESS

Lemma 1 - Let g be a feasible circulation. Let x be a negative cost circuit in and let denote its ratio. Cancelling circuit x improves the objective function value by at least

Proof: On Board

Lemma 2 – Let g be a feasible circulation that is not optimal. Let denote the minimum ratio circuit value in . Then

Proof: On Board

ALGORITHM CORRECTNESS

Lemma 1 - Cancelling circuit x improves the objective function value by at least

Lemma 2 –

Lemma 3 – There are cancellations of min ratio circuits

Proof: On Board

ALGORITHM CORRECTNESS

Lemma 3 – There are cancellations of min ratio circuits

Giving a time subroutine for computing a minimum ratio circuit,

the minimum ratio circuit-cancelling algorithm computes an

generalized minimum cost circulation in a strongly Polynomial time!

MINIMUM RATIO CIRCUIT ALGORITHM

In this section we discuss 3 matters:1. Detecting a circuit2. Detecting a negative cost circuit3. 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

𝜋 𝑖 (𝑣 )=min (𝜋 𝑖−1 (𝑣 ) ,𝛾 (𝑣 ,𝑤 ) 𝜋𝑖− 1 (𝑤 ))1

1

1

1

1

1

5

1

1/26 1

3/4

1

STEP1 - DETECTING A BICYCLE

𝜋 𝑖 (𝑣 )=min (𝜋 𝑖−1 (𝑣 ) ,𝛾 (𝑣 ,𝑤 ) 𝜋𝑖− 1 (𝑤 ))3/4

1

1

1

1/2

1

5

1

1/26 1

3/4

1

STEP1 - DETECTING A BICYCLE

𝜋 𝑖 (𝑣 )=min (𝜋 𝑖−1 (𝑣 ) ,𝛾 (𝑣 ,𝑤 ) 𝜋𝑖− 1 (𝑤 ))3/4

1

3/4

1

1/2

1

5

1

1/26 1

3/4

1

STEP1 - DETECTING A BICYCLE

𝜋 𝑖 (𝑣 )=min (𝜋 𝑖−1 (𝑣 ) ,𝛾 (𝑣 ,𝑤 ) 𝜋𝑖− 1 (𝑤 ))3/4

3/4

3/4

1

1/2

1

5

1

1/26 1

3/4

1

STEP1 - DETECTING A BICYCLE

𝜋 𝑖 (𝑣 )=min (𝜋 𝑖−1 (𝑣 ) ,𝛾 (𝑣 ,𝑤 ) 𝜋𝑖− 1 (𝑤 ))9/16

9/16

9/16

1

1/2

1

5

1

1/26 1

3/4

1

nth-step

STEP1 - DETECTING A BICYCLE

𝜋 𝑖 (𝑣 )=min (𝜋 𝑖−1 (𝑣 ) ,𝛾 (𝑣 ,𝑤 ) 𝜋𝑖− 1 (𝑤 ))0.31

6

0.42

0.316

1

1/2

1

5

1

1/26 1

3/4

1

(2n-1)th-step

STEP1 - DETECTING A BICYCLE

𝜋 𝑖 (𝑣 )=min (𝜋 𝑖−1 (𝑣 ) ,𝛾 (𝑣 ,𝑤 ) 𝜋𝑖− 1 (𝑤 ))0.31

6

0.316

0.316

1

1/2

1

5

1

1/26 1

3/4

1

(2n)th-step

STEP1 - DETECTING A BICYCLE

Then detect flow-generating cycles on the sub-graph induced by N, with the update rule:

If we find a flow-generating cycle, then by the definition of we can extract 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

In this section we discuss 3 matters:1. Detecting a circuit a2. Detecting a negative cost circuit3. Finding a minimum ratio circuit

DETECTING A NEGATIVE COST CIRCUIT

Any circulation can be decomposed into circuits, thus it’s sufficient to detect a negative cost circulation.

That is, we want to find a solution to the system:

(I)

Now let variables, and lets look at:

(II)

2VPI (2 Variables Per Inequality) Feasibility

(I)

(II)

System (I) + (II) can’t both be feasible!

There exists a negative cost circulation if and only if the system (II) is infeasible

Thus is it sufficient to have a 2VPI feasibility subroutine

2VPI Feasibility

Fortunately, researchers have previously developed such specialized 2VPI feasibility algorithms

Moreover, all of these algorithms either return a feasible solution to (II) or output a minimal certificate of infeasibility

Hochbaum et al. [21], and Cohen and Megiddo ( מגידו (נמרודdesigned an algorithm in .

MINIMUM RATIO CIRCUIT ALGORITHM

In this section we discuss 3 matters:1. Detecting a circuit a2. Detecting a negative cost circuit a 3. Finding a minimum ratio circuit

FINDING A MINIMUM RATIO CIRCUIT

1. Estimate with a guess

2. Linearize the objective function with the parametric cost function

3. Find a negative cost circuit with respect to cost function : 3.1. If there is a negative cost circuit, then over estimate 3.2. If every non-trivial circuit has positive cost, then is too small 3.3. Else

FINDING A MINIMUM RATIO CIRCUIT – ANAYLSIS (NO PROOF)

3. Find a negative cost circuit with respect to cost function : 3.1. If there is a negative cost circuit, then over estimate 3.2. If every non-trivial circuit has positive cost, then is too small 3.3. Else

3.1 -

By incorporating the parametric search techniques of Megiddo ( מדיגו (נמרודand using parallel implementation, we can do it in -

how much in binary search?

MINIMUM RATIO CIRCUIT ALGORITHM

In this section we discuss 3 matters:1. Detecting a circuit a2. Detecting a negative cost circuit a 3. Finding a minimum ratio circuit a on O(n) Processors

ALGORITHM CORRECTNESS

The minimum ratio circuit-cancelling algorithm computes an

generalized minimum cost circulation in a strongly Polynomial time!

~𝑂 (𝑚𝑛3 )

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

SCALING VERSION

Lemma 4 – There are at most 2m cancelations per scaling phase

Proof: On Board

Lemma 5 – Suppose at the end of a scaling phase we have , where g is the current circulation. Then g is Proof: On Board

SCALING VERSION

Lemma 4 – There are at most 2m cancelations per scaling phase

Lemma 5 – Suppose at the end of a scaling phase we have , where g is the current circulation. Then g is

Lemma 6 – The scaling algorithm finds an circulation in time.Proof: On Board

ALGORITHMS SUMMARY

Approximation Algorithms

1st Algorithm Faster Scaling Version

What About Exact Algorithms ?

ROUNDING TO A VERTEX

Lemma 7 – Let g be a feasible generalized circulation and let Then g is a vertex if and only if the subgraph induced by A has no unit-gain cycles or bicycles

Giving feasible circulation g, we can detect circuits in g in time and cancel them; each cancelation saturates at least one new arc in A, resulted in a time

ROUNDING TO A VERTEX

Lemma 8 – Giving a generalized circulation g we can determine an optimal circulation in n) time

Proof: On Board.

ALGORITHMS SUMMARY

Approximation Algorithms

1st Algorithm Faster Scaling Version

Exact Algorithms

1st Algorithm Faster Scaling Version

QUESTIONS? Eyal Dushkin – 03.06.13