Faster algorithms for min-delay time semi-matching

Bundit LaekhanukitC&O, University of Waterloo

joint work withJ. Fakcharoenphol and D. Nanongkai

   

Agenda

● Definition of min-delay semimatching problem● Previous works & Our results● Unit cost

– Reduction to min-cost flow

– An algorithm for unit cost

● General cost– Reduction to weighted bipartite matching

– Sketch of an algorithm for general cost

   

Semimatching on bipartite graphs

   

U V

Matching MU V

Semimatching M'

No restrictionincident to= 1 M'-edge

incident to≤ 1 M-edge

   

U V

u1

x

u2

y

3

z

u4

Jobs Machines

u1 u2 u3

Wait for being processed1 unit of time

u3 Wait for u1 and itself to be processed1 + 1 units of time

Unit cost

   

U V

u1

x

u2

y

3

z

u4

Jobs Machines

u1 u2 u3

u3

Delay time at machine a = 1 + 2 + 3 units of time

Unit cost

   

U V

u1

x

u2

y

3

z

u4

Jobs Machines

u3

Total delay time (cost of semimatching) = 6 + 1 units of time

Delay time is flow time in scheduling terminology

Unit cost

   

U V

u1

x

u2

y

3

z

u4

Jobs Machines

u1 u2 u3

u3

Delay time at machine x = 1 + (1+2) + (1+2+3)units of time

2

1

3

2 1 3

u2 u1 u3

1 2 3Reorder to minimize delay

2

General cost

   

Previous works & Our results

   

Previous Works & Our results

● Unit cost

Harvey, Ladner, Lovász, and Tamir (2003) :

O(nm)-algorithm using modified Hungarian method

Our algorithm : O(n1/2m log n)

● General cost

Bruno, Coffman, and Sethi (1974) :

O(n3)-algorithm using a reduction to bipartite matching

Our algorithm : O(nm log n)

n : number of vertices, m : number of edges

   

Unit cost

   

Harvey et. al. reduction from unit cost semimatching to min-cost flow

   

U V

u1

x

u2

y

3

z

u4

u3

A reduction to min-cost flow

   

U V

u1

x

u2

y

3

z

u4

u3

s t

C1

C2

C3

Capacity = 1, Cost = 0

Infinite capacity

Capacity = 1, Cost = 0

Cost=1

Cost=2

Cost=3

A reduction to min-cost flow

Cost centers

   

U V

u1

x

u2

y

3

z

u4

u3

s t

C1

C2

C3

Capacity = 1, Cost = 0

Infinite capacity

Capacity = 1, Cost = 0

Cost=1

Cost=3

Cost=2

Cost centers

A reduction to min-cost flow

min-cost flow = min-delay semimatching

   

An algorithm for unit cost

   

Running timeDirectly apply min-cost flow : O(nm log2 n)

Previous best (Harvey et. al. 2003) : O(nm)

Our algorithm : O(n1/2m log n)

   

Use the reduction to min-cost flow and divide & conquer technique

   

High Level Details

U tV

LowCost

s

HighCost

Start from any semimatching and construct flow

   

High Level Details

U tV

LowCost

s

HighCost

Push flow to lowercost centers

Source and Sink can be ignored

   

High Level Details

U V LowCost

HighCost

Divide into two subproblems and recursively solve them

U V

No residual path from G2 to G1

G1

G2

   

U V

Start from an arbitrary semimatching

u1

u2

3

u4

u3

u5

u4

x

y

z

   

U V

Construct a corresponding flow

u1

u2

3

u4

u3

u5

u4

Cost=1

Cost=3

Cost=2

Cost=4

t

x

y

z

C1

C2

C3

C4

s

Cost centers

   

U V

A flow is not a cheapest

u1

u2

3

u4

u3

u5

u4

Cost=1

Cost=3

Cost=2

Cost=4

ts

C1

C2

C3

C4

x

y

z

Cost centers

   

U V

Find a max-flow to lower cost centers

u1

u2

3

u4

u3

u5

u4

x

y

z

C1

C2

C3

C4

T'

S'

Cost centers

Source and Sink are ignored

   

U V

An improved flow

u1

u2

3

u4

u3

u5

u4

x

y

z

C1

C2

C3

C4

Cost centers

   

Divide & recursively solve subproblems

u1

u2

3

u4

u3

u5

u4

x

y

z

C1

C2

C3

C4

Reachable from C3 , C4 by residual paths

Not reachable from C3 , C4 by residual paths

G1

G2

Separated by min-cut

   

Divide & recursively solve subproblems

u1

u2

3

u4

u3

u5

u4

x

y

z

C1

C2

C3

C4

Solve subproblems

Push flow in G1

G1

G2

No flow path in G2

   

A min-cost flow

u1

u2

3

u4

u3

u5

u4

x

y

z

C1

C2

C3

C4

ts

   

A min-delay semimatching

u1

u2

3

u4

u3

u5

u4

x

y

z

   

Running time

   

Running time analysis

● Push flow to lower cost centers O(n1/2m)

(Dinitz's blocking flow algorithm with analysis similar to Kazanov 1973, Even & Tarjan 1974)

● Total running time

T(n, m, c) = T(n1, m

1, c/2) + T(n

2, m

2, c/2) + O(n1/2m)

= . . . = O(n1/2m log n)

n = number of vertices, m = number of edges

c = number of cost centers ≤ n

   

General cost

   

Bruno et. al. reduction toweighted bipartite matching

   

U V

u1

x

u2

y

3

z

u4

Jobs Machines

u3

Delay time at machine a = 1 + (1+2) + (1+2+3)= 3 · 1 + 2 · 2 + 1 · 3units of time

2

1

3

u2 u1 u3

1 2 3

2

A reduction to bipartite matching

   

U V

u1

x

u2

y

3

z

u4

u3

2

1

3

2

3

5

x1

2 · 1

1 · 1

3 · 1

x2

2 · 2

1 · 2

3 · 2

x3

2 · 3

1 · 3

3 · 3

A reduction to bipartite matching

   

u1

x3u2

y13

z1u4

u3

639

2

3

5

x1213

x24

26

z2

410

A reduction to bipartite matching

   

u1

x3u2

y13

z1u4

u3

x1

x2

z2

10

639

2

3

5

213

42

6

4

Equivalent to assign u1, u2 to x

A reduction to bipartite matching

min-weight matching = min-delay semimatching

   

An algorithm for general cost

   

Running timeDirectly apply matching algorithm : O(nm2)

Previous best (Bruno et. al. 1974) : O(n3)

Our algorithm : O(nm log n)

   

Use the reduction to bipartite matching exploit structure of potential function

   

U

z

y

High level details

x

Copies of vertices of V

Reduce to bipartite matching

   

U

z

y

High level details

xAugment matching byshortest augmenting paths(Edmond & Karp 1970,Tomizawa 1971)

   

U

z

y

High level details

xCost of matching edgesare negative

- 10

7

Assign potentialfunction to make non-negativereduced cost

   

U

z

y

High level details

xCost of matching edgesare negative

- 10

7

Use distancelabels fromprevious round as potentialfunction

   

u1

xk

x1

xi

Distances has a break point

Distances areweakly decreasing

Distances areweakly increasing

Dual variables forx1 ,x2, ..., xk havea break point

xj is a copy of x with edge costs multiplied by j

   

U

z

y

Data structure for fast query

x

Create datastructures ateach set forfast distancequery

Use a data structureexploiting the propertyto find a shortest augmenting path

   

Running time analysis

● Using a special data structure, a shortest augmenting path can be found in O(n + m log n)

● Augmenting the matching n times● Total running time O(nm log n)

   

Summary

● Unit cost– Previous best running time O(nm)

– Our running time O(n1/2m log n)

● General cost– Previous best running time O(n3)

– Our running time O(nm log n)

   

Open ProblemsIs it possible to meet the running time of bipartite matching?

● Unit cost– The best running time O(n1/2m) (Hopcroft & Karp 1973)

– Our algorithm O(n1/2m log n)

● General cost– The running time

● O(nm) (Edmond & Karp 1970, Tomizawa 1971)● O(n1/2m log nWmax) (Gabow & Tajan 1989)

– Our algorithm O(nm log n)

   

Thank you for your attention

   

Questions ?