Combinatorial Auctions: A Survey

1
Combinatorial Auctions A Survey

• Sven de Vries Rakesh Vohra (2000)

2
Contents

1. Introduction
2. CAP
3. Decentralized Methods

3
Introduction(1)

• Complimentarities between different assets
• Bidders have preferences not just for particular
  items but for sets of bundels of items
• Traveling to LA
• (restaurants and hotels for the intermediate
  cities, car)
• or (airline ticket, taxi)
• Auctions where bidders submit bids on
  combinations recently been aroused
• Jackson(1976),Caplice(1996),Rothkopf(1998),Fujishi
  ma(1999),Sandholm(1999)
• Increases in computing power

4
Introduction(2)

• Tools
• SBIDS by SAITECH-INC
• OptiBid by Logistics.com
• Combinatorial Auction Problem (CAP)
• Selecting the winning set of bids.
• Can be formulated as an Integer Program

5

1. Introduction
2. CAP
3. Decentralized Methods

6
CAP

1. CAP
2. SPP
3. Solvable Instances of SPP
4. Exact Methods
5. Approximate Methods

7
CAP(1)
CAP (Combinatorial Auction Problem) -Selecting
the winning set of bids-
Difficulty
Resolution

• Each bidder must submit a bid for every subset
  of objects he is interested in
• How to transmit this bidding function in a
  succinct way to the auctioneer

• To restrict the kinds of combinations that
  bidders may bid on

• How to decide which collection of bids to accept

- Solving CAP
8
CAP(2)
CAP (Combinatorial Auction Problem) -Selecting
the winning set of bids-
Difficulty
Resolution

• Each bidder must submit a bid for every subset
  of objects he is interested in
• How to transmit this bidding function in a
  succinct way to the auctioneer

• To restrict the kinds of combinations that
  bidders may bid on

• How to decide which collection of bids to accept

- Solving CAP
9
CAP(3)

• Notations
• N the set of bidders
• M the set of m distinct objects
• S subset of M
• bj(S) the bid that agent j in N has announced
  he is willing to pay for S

10
CAP(4)

• CAP formula

11
CAP(4)

• CAP formula
• x(S) 1 the highest bid on the set S is to be
  accepted
• 0 no bid on the set S are accepted

12
CAP(4)

• CAP formula
• no object in M is
  assigned to
• more than one
  bidder

13
CAP(4)

• CAP formula
• Call this formulation CAP1

14
CAP(5)

• Superadditive
• for
  all j?N and A,B?M
• such that
• CAP1 correctly models CAP when the bid functions
  bj are all superadditive
• The goods complement each other.
• When goods are substitutes, CAP1 is incorrect.
• Why ?
• Superadditive formula doesnt hold for some
  j,A,B.
• An optimal solution to CAP1 may assign A,B to
  bidder j and incorrectly record a revenue of
  bj(A)bj(B) rather than

15
CAP(6)

• How to obviate this difficulty ?
• Through the introduction of dummy good g
• bj(A) gt bj(A?g)
• bj(B) gt bj(B?g)
• bj(A?B) remains the same
• M gt M?g
• If A is assigned to j, then B cannot be assigned
  to j.
• Through the formula CAP2

16
CAP(7)

• CAP2 formulation

• CAP1 formulation

17
CAP(8)

• CAP2 formulation

No bidder receives more than one subset
18
CAP(9)

• CAP2 formulation

Overlapping sets of goods are never assigned
19
CAP(10)

• Assumption of CAP1,CAP2
• There is at most one copy of each object.
• Extending the formulation
• The case when there are multiple copies of the
  same object and each bidder wants at most one
  copy of each object
• The right hand sides of the contraints in CAP1,
  CAP2 take on values larger than 1.
• The case when there are multiple copies and the
  bidder may want more than one copy of the same
  object
• Multi-unit combinatorial auctions (Leyton-Brown
  2000)

20
CAP

1. CAP
2. SPP
3. Solvable Instances of SPP
4. Exact Methods
5. Approximate Methods

21
SPP(1)

• Set Packing Problem
• Given a ground set M of elements and a collection
  V of subsets with non-negative weights, find the
  largest weight collection of subsets that are
  pairwise disjoint.

22
SPP(2)

• Set Packing Problem
• Given a ground set M of elements and a collection
  V of subsets with non-negative weights, find the
  largest weight collection of subsets that are
  pairwise disjoint.
• Notation
• x(j) 1 if the j-th set in V with weight c(j) is
  selected
• 0 otherwise
• a(i,j) 1 if the j-th set in V contains element
  i?M
• 0 otherwise

23
SPP(3)

• Notation
• x(j) 1 if the j-th set in V with weight c(j) is
  selected
• 0 otherwise
• a(i,j) 1 if the j-th set in V contains element
  i?M
• 0 otherwise
• SPP Formulation

24
SPP(3)

• Notation
• x(j) 1 if the j-th set in V with weight c(j) is
  selected
• 0 otherwise
• a(i,j) 1 if the j-th set in V contains element
  i?M
• 0 otherwise
• SPP Formulation

• CAP Formulation

25
SPP(3)

• Notation
• x(j) 1 if the j-th set in V with weight c(j) is
  selected
• 0 otherwise
• a(i,j) 1 if the j-th set in V contains element
  i?M
• 0 otherwise
• SPP Formulation

• CAP Formulation

26
SPP(4)
Other related Prolems
Set Partitioning Problem (SPA)
Set Covering Problem (SCP)
27
SPP(5)
Set Partitioning Problem (SPA)

• Bidders are sellers (rather than buyers).
• Trucking companies bidding for the opportunity
  to ship goods from a particular warehouse to
  retail outlet.

28
SPP(6)
Set Covering Problem (SCP)

• Auction problems in procurement rather than
  selling terms.
• Scheduling of crews for railways.

29
Complexity of SPP

• No polynomial time algorithm for SPP is known.
• Any algorithm for the CAP that uses directly the
  bids for the sets, must scan the bids and the
  number of such bids could be exponential in M.
• M the number of variables
• gt V the number of solutions to check 2M
• SPP NP-hard (NP-complete)
• Effective solution procedures for CAP
• The number of distinct bids is not large
• Be structured in computationally useful ways.

30
CAP

1. CAP
2. SPP
3. Solvable Instances of SPP
4. Exact Methods
5. Approximate Methods

31
Solvable Instances of SPP

1. Total Unimodularity
2. Balanced Matrices
3. Perfect Matrices
4. Graph Theoretic Methods
5. Using Preferences

32
Solvable Instances of SPP

• Usual way in which instances SPP can be solved by
  a polynomial algorithm
• When the extreme points of the polyhedron
• are all integral, i.e. 0-1.
• In these cases, we can simply drop the
  integrality requirement from the SPP and solve it
  as a linear program
• A polyhedron with all integral extreme points is
  called integral.

33
Total Unimodularity(TU) (1)

• A matrix is TU if the determinant of every square
  submatrix is 0,1 or 1.
• A TU ? At TU
• If Aa(i,j)i?M,j?V is TU, then all extreme
  point of the polyhedron P(A) are integral.
• There is a polynomial time algorithm to decide
  whether a matrix is TU.

34
Total Unimodularity(TU) (2)

• Theorem 2.1) Let B be a matrix each of whose
  entries is 0,1 or -1. Suppose each subset S of
  columns of B can be divided into two sets L and R
  such that
• then B is TU. The converse is also true.
• Theorem 2.2) All 0-1 matrices with the
  consecutive ones property are TU.
• A 0-1 matrix has the consecutive ones property if
  the non-zero entries in each column occur
  consecutively.

35
Total Unimodularity(TU) (3)

• For example,
• Objects to be auctioned parcels of land along a
  shore line
• Shore line is important it imposes a linear
  order on the parcels
• Make a restriction to bid only contiguous parcels
• The most interesting combinations would be
  contiguous, in the bidders eyes.
• Two computational consequences.
• Number of distinct bids would be limited by a
  polynomial in the number of objects.
• The constraint matrix A of the CAP would have the
  consecutive ones property in the columns.

36
Balanced Matrices(1)

• A 0-1 matrix B is balanced if it has no square
  submatrix of odd order with exactly two 1s in
  each row and column.
• Theorem 2.3) Let B be a balanced 0-1 matrix. Then
  the following linear program
• has an integral optimal solution whenever the
  c(j)s are integral.

37
Balanced Matrices(2)

• For example,
• Consider a tree T with a distance function d.
• v vertex of T
• N(v,r) set of all vertices in T that are within
  distance r of v.
• The vertices represent parcels of land connected
  by a read network with no cycles.
• Bidders bid for subsets of parcels which is to be
  of the form N(v,r).
• Row of the constraint matrix for each vertex
• Column for each set of the form N(v,r)
• This constraint matrix is balanced.

38
Perfect Matrices

• If the contraints matrix A can be identified with
  the vertex-clique adjacency matrix of what is
  known as a perfect graph, then SPP can be solved
  in polynomial time.
• A simple graph G is perfect if, for every induced
  subgraph H of G, the number of vertices in a
  maximum clique is
• , the chromatic number of H, is the
  minumum k for which H is k-colorable.

39
Graph Theoretic Methods

• There are situations where P(A) is not integral
  yet the SPP can be solved in polynomial time
  because the contraint matrix A admits a graph
  theoretic interpretation in terms of an easy
  problem.
• When each column of the matrix A contains at most
  two 1s. gt maximum weight matching problem
• (can be solved in polynomial time)
• At most two 1s per row of A gt NP-hard
• When A has the circular ones property.
• A 0-1 has the circular ones property if the
  non-zero entries in each column (row) are
  consecutive
• First and last entries in each column (row) are
  treated consecutive
• Note the resemblance to the consecutive ones
  property

40
Graph Theoretic Methods

• There are situations where P(A) is not integral
  yet the SPP can be solved in polynomial time
  because the contraint matrix A admits a graph
  theoretic interpretation in terms of an easy
  problem.
• When each column of the matrix A contains at most
  two 1s. gt maximum weight matching problem
• (can be solved in polynomial time)
• At most two 1s per row of A gt NP-hard
• When A has the circular ones property.
• gt A can be identified with the vertex-clique
  adjacency matrix of a circular arc graph.
• gt maximum weight independent set problem for a
  circular arc graph. (can be solved in poly time)

41
Using Preferences(1)

• Restrictions in the preference orderings of the
  bidders
• Suppose that bidders come in two types
• Type one bj(.) g1(.)
• Type two bj(.) g2(.)
• where g1 and g2 are non-decreasing integer
  valued supermodular functions
• The dual of CAP2 is

42
Using Preferences(1)

• Restrictions in the preference orderings of the
  bidders
• Suppose that bidders come in two types
• Type one bj(.) g1(.)
• Type two bj(.) g2(.)
• where g1 and g2 are non-decreasing integer
  valued supermodular functions
• The dual of CAP2 is

This Problem is an instance of the polymatroid
intersection problem. (polynomially solvable)
43
Using Preferences(1)

• Restrictions in the preference orderings of the
  bidders
• Suppose that bidders come in two types
• Type one bj(.) g1(.)
• Type two bj(.) g2(.)
• where g1 and g2 are non-decreasing integer
  valued supermodular functions
• Using the method to solve problems with three or
  more types of bidders is not possible.
• It is known in those cases that the dual problem
  above admits fractional extreme points.
• The problem of finding an in integer optimal
  solution for the intersection of three or more
  polymatroids is NP-hard.

44
Using Preferences(2)

• Restrictions in the preference orderings of the
  bidders
• When each of the bj(.) have the gross substitutes
  property, CAP2 reduces to a sequence of matroid
  partition problems, each of which can be solved
  in polynomial time.

45
CAP

1. CAP
2. SPP
3. Solvable Instances of SPP
4. Exact Methods
5. Approximate Methods

46
Exact Methods(1)

• The upper bound on the optimal solution value is
  obtained by solving a relaxation of the
  optimization problem.
• Replace the given problem by one with a larger
  feasible region that is more easily solved.
• Lagrangean relaxation
• Will be discussed later
• Linear programming relaxation
• Only the integrality constraints are relaxed

47
Exact Methods(2)

• Exact methods
• Branch and bound
• Cutting planes
• Hybrid called branch and cut

48
Exact Methods(2)

• Exact methods
• Branch and bound
• At each stage, after solving the LP, a fractional
  variable xj is selected and two subproblems are
  set up, one where xj1 and the other where xj0.
  (Branch)
• Solve the LP relaxation of the two subproblems.
• From each subproblem with a nonintegral solution
  we branch again to generate two subproblems and
  so on.
• By comparing the LP bound across nodes in
  different branches of the tree, one can prune
  some branches in advance. (Bound)
• Cutting planes
• Hybrid called branch and cut

49
Exact Methods(3)

• Exact methods
• Branch and bound
• Cutting planes
• Find linear inequalities (cuts) that are violated
  by a solution of a given relaxation but are
  satisfied by all feasible zero-one solution.
• If one adds enough cuts, one is left with
  integral extreme points.
• Hybrid called branch and cut

50
Exact Methods(4)

• Exact methods
• Branch and bound
• Cutting planes
• Hybrid called branch and cut
• Works like branch and bound, but tightens the
  bounds in every node of the tree by adding cuts.
• Since even small instances of the CAP1 may
  involve a huge number of columns (bids), this
  method needs to be augmented with another method
  known as column generation.
• (It works by generating a column when needed
  rather than all at once.)

51
Exact Methods(5)

• How successful exact approaches are
• Being able to find an optimal solution to an
  instance of SPA with 1,053,137 variables and 145
  constraints in under 25 minutes.
• Major impetus behind the desire to solve large
  instances of SPA(SPC) quickly has been the
  airline industry.
• Assinging crews to routes can be formulated as an
  SPA.
• The rows of the SPA correspond to flight legs.
• The columns correstpond to a sequence of flight
  legs that would be assigned to a crew.

52
CAP

1. CAP
2. SPP
3. Solvable Instances of SPP
4. Exact Methods
5. Approximate Methods

53
Approximate Methods

• Probably every heuristic approach for solving
  general integer programming problems has been
  applied to the SPP.
• Greedy, Interchange/steepest ascent approach,
  genetic algorithms, probabilistic search,
  simulated annealing, neural networks
• Give up on finding the optimal solution.
• Rather one seeks a feasible solution fast and
  hopes that it is near optimal.
• How close to optimal is the solution ?
• Worst-case analysis
• Probabilistic analysis
• Empirical testing

54

1. Introduction
2. CAP
3. Decentralized Methods

55
Decentralized Methods

1. Duality in Integer Programming
2. Lagrangean Relaxation

56
Decentralized Methods

• One way of reducing some of the computational
  burden in solving the CAP.
• Auctioneer sets prices for the objects
• Agents announce which sets of objects they
  will purchase ar the posted prices
• If two or more agents compete for the same
  object, the auctioneer adjusts the price vector.
• Bidders save from specifying their bids for
  every possible combination
• auctioneer saves from having to process each
  bid function

57
Duality in Integer Programming(1)

• Decentralized approach
• Auctioneer chooses a feasible solution.
• Bidders are asked to submit improvements.
• Auctioneer agrees to share a portion of the
  revenue gain with the bidder.
• Above method can be viewed as instances of dual
  based procedures for solving an integer program.

58
Duality in Integer Programming(2)

• The (superadditive) dual to SPP
• the problem of finding a superadditive,
  non-decreasing function such
  that
• If the primal integer program has the integrality
  property, there is an optimal integer solution to
  its LP relaxation, and the dual function F will
  be linear,i.e.,

59
Duality in Integer Programming(3)

• The (superadditive) dual to SPP
• If the primal integer program has the integrality
  property, there is an optimal integer solution to
  its LP relaxation, and the dual function F will
  be linear,i.e.,
• The dual becomes

60
Duality in Integer Programming(3)

• The (superadditive) dual to SPP
• If the primal integer program has the integrality
  property, there is an optimal integer solution to
  its LP relaxation, and the dual function F will
  be linear,i.e.,
• The dual becomes

Superadditive dual reduces to the dual of the
linear programming relaxation of SPP. yi can be
interpreted as the price of the object i.
61
Duality in Integer Programming(4)

• Solving the superadditive dual problem is as hard
  as solving the original primal problem.
• It is possible to reformulate the superadditive
  dual problem as a linear program.
• The number of variables is exponential in the
  size of the original problem.
• For small or specially structured problems, this
  can provide some insight.
• In general, one relies on the solution to the LP
  dual and uses its optimal value to guide the
  search for an optimal solution to the original
  primal integer program.
• gt Lagrangean Relaxation

62
Lagrangean Relaxation(1)

• Relax some of the constraints of the original
  problem by moving them into the objective
  function with a penalty term.
• Infeasible solutions allowed but penalized in
  proportion to the amount of infeasibility.

63
Lagrangean Relaxation(2)

• Recall the SPP
• Notation
• ZLP optimal objective function value to LP
  relaxation of SPP. (Note that Z ZLP)
• s.t.

64
Lagrangean Relaxation(3)

• Theorem3.2)
• Computing Z(?) is easy.
• Simply set x(j)1 if
• 0 otherwise
• since

65
Lagrangean Relaxation(3)

• Theorem3.2)
• Computing Z(?) is easy.
• Using subgradient algorithm, finding ? which
  minimizes Z(?) can be done.
• Therefore, ZLP can be found in a fast procedure.
• Lagrangean relaxation is not guaranteed to find
  the optimal solution to the underlying problem.
• It finds an optimal solution to a relaxation of
  it.
• The resulting solution may not be too infeasible,
  so could be fudged into a feasible solution
  without a great reduction in objective function
  value.

66
Lagrangean Relaxation(4)

• Market Interpretation
• Auctioneer chooses a price vector ? for the
  objects.
• Bidders submit bids.
• If the highest bid c(j) for the jth bundle
  exceeds
• this bundle is tentatively assigned to that
  bidder.
• SAA (simultaneous ascending auction)
• Bidders bid simultaneously in rounds.
• Bids must be increased by a specified minimum
  from one round to the next.
• Bidders adjust prices which is different from the
  way of Lagrangean Relaxation.
• Exposure problem occurs.

67
Lagrangean Relaxation(5)

• Exposure Problem
• Bidders pay too much for individual items or
  bidders with preferences for certain bundles drop
  out early to limit losses.
• For example,
• A bidder A values the bundle of goods i and j at
  100 but each at 0.
• In SAA, A has to submit high bids on i and j to
  secure them.
• Suppose that it loses the bidding on i.
• A is left standing with a high bid j which A
  valued at 0.
• Any auction scheme that relies on prices for
  individual items will face this problem.

68
Lagrangean Relaxation(6)

• AUSM (Adaptive User Selection Mechanism)
• Asynchronous in that bids on subsets can be
  submitted at any time.
• Difficult to connect to the Lagrangean ideas.
• Iterative auction scheme
• Hybrid of the SAA and AUSM
• Easier to connect to the Lagrangean framework.
• Bidders submit bids on packages rather than on
  individual items.

69
The End

• Even if the researcher does not find
• what was initially expected,
• the pursuit of a personally important topic is
  still rewarding and
• generally produces continuing researches.