Combinatorial Auctions: A Survey Sven de Vries & Rakesh Vohra (2000)
Contents 1. Introduction 2. CAP 3. Decentralized Methods
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), Fujishima( 1999), Sandholm(1999) – Increases in computing power
Introduction(2) • Tools – ‘SBIDS’ by SAITECH-INC – ‘Opti. Bid’ by Logistics. com • Combinatorial Auction Problem (CAP) – Selecting the winning set of bids. – Can be formulated as an Integer Program
1. Introduction 2. CAP 3. Decentralized Methods
CAP 1. CAP 2. SPP 3. Solvable Instances of SPP 4. Exact Methods 5. Approximate Methods
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
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
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
CAP(4) • CAP formula :
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
CAP(4) • CAP formula : – : no object in M is assigned to more than one bidder
CAP(4) • CAP formula : • Call this formulation CAP 1
CAP(5) • Superadditive : – for all j∈N and A, B⊂M such that • CAP 1 correctly models CAP when the bid functions bj are all superadditive – The goods complement each other. • When goods are substitutes, CAP 1 is incorrect. – Why ? • Superadditive formula doesn’t hold for some j, A, B. • An optimal solution to CAP 1 may assign A, B to bidder j and incorrectly record a revenue of bj(A)+bj(B) rather than
CAP(6) • How to obviate this difficulty ? – Through the introduction of dummy good g • bj(A) => bj(A∪{g}) bj(B) => bj(B∪{g}) bj(A∪B) remains the same M => M∪{g} • If A is assigned to j, then B cannot be assigned to j. – Through the formula CAP 2
CAP(7) • CAP 2 formulation • CAP 1 formulation
CAP(8) • CAP 2 formulation No bidder receives more than one subset
CAP(9) • CAP 2 formulation Overlapping sets of goods are never assigned
CAP(10) • Assumption of CAP 1, CAP 2 – 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 CAP 1, CAP 2 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)
CAP 1. CAP 2. SPP 3. Solvable Instances of SPP 4. Exact Methods 5. Approximate Methods
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.
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
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
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
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
SPP(4) Other related Prolems Set Partitioning Problem (SPA) Set Covering Problem (SCP)
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.
SPP(6) Set Covering Problem (SCP) – Auction problems in procurement rather than selling terms. – Scheduling of crews for railways.
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 => |V| : the number of solutions to check = 2|M| • SPP : NP-hard (NP-complete) • Effective solution procedures for CAP – The number of distinct bids is not large – Be structured in computationally useful ways.
CAP 1. CAP 2. SPP 3. Solvable Instances of SPP 4. Exact Methods 5. Approximate Methods
Solvable Instances of SPP 1. Total Unimodularity 2. Balanced Matrices 3. Perfect Matrices 4. Graph Theoretic Methods 5. Using Preferences
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.
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 A={a(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.
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.
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.
Balanced Matrices(1) • A 0 -1 matrix B is balanced if it has no square submatrix of odd order with exactly two 1’s 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.
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.
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.
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 1’s. => maximum weight matching problem (can be solved in polynomial time) • At most two 1’s per row of A => 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
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 1’s. => maximum weight matching problem (can be solved in polynomial time) • At most two 1’s per row of A => NP-hard – When A has the circular ones property. => A can be identified with the vertex-clique adjacency matrix of a circular arc graph. => maximum weight independent set problem for a circular arc graph. (can be solved in poly time)
Using Preferences(1) • Restrictions in the preference orderings of the bidders – Suppose that bidders come in two types • Type one : bj(. ) = g 1(. ) • Type two : bj(. ) = g 2(. ) where g 1 and g 2 are non-decreasing integer valued supermodular functions The dual of CAP 2 is :
Using Preferences(1) • Restrictions in the preference orderings of the bidders – Suppose that bidders come in two types • Type one : bj(. ) = g 1(. ) • Type two : bj(. ) = g 2(. ) where g 1 and g 2 are non-decreasing integer valued supermodular functions The dual of CAP 2 is : This Problem is an instance of the polymatroid intersection problem. (polynomially solvable)
Using Preferences(1) • Restrictions in the preference orderings of the bidders – Suppose that bidders come in two types • Type one : bj(. ) = g 1(. ) • Type two : bj(. ) = g 2(. ) where g 1 and g 2 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.
Using Preferences(2) • Restrictions in the preference orderings of the bidders – When each of the bj(. ) have the gross substitutes property, CAP 2 reduces to a sequence of matroid partition problems, each of which can be solved in polynomial time.
CAP 1. CAP 2. SPP 3. Solvable Instances of SPP 4. Exact Methods 5. Approximate Methods
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
Exact Methods(2) • Exact methods – Branch and bound – Cutting planes – Hybrid called branch and cut
Exact Methods(2) • Exact methods – Branch and bound 1. At each stage, after solving the LP, a fractional variable xj is selected and two subproblems are set up, one where xj=1 and the other where xj=0. (Branch) 2. Solve the LP relaxation of the two subproblems. 3. From each subproblem with a nonintegral solution we branch again to generate two subproblems and so on. 4. 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
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 zeroone solution. If one adds enough cuts, one is left with integral extreme points. – Hybrid called branch and cut
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 CAP 1 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. )
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.
CAP 1. CAP 2. SPP 3. Solvable Instances of SPP 4. Exact Methods 5. Approximate Methods
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
1. Introduction 2. CAP 3. Decentralized Methods
Decentralized Methods 1. Duality in Integer Programming 2. Lagrangean Relaxation
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
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.
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. ,
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 :
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.
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. => Lagrangean Relaxation
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.
Lagrangean Relaxation(2) • Recall the SPP: • Notation – ZLP : optimal objective function value to LP relaxation of SPP. (Note that Z ≤ ZLP) – s. t.
Lagrangean Relaxation(3) • Theorem 3. 2) – Computing Z(∧) is easy. • Simply set x(j)=1 if 0 otherwise since
Lagrangean Relaxation(3) • Theorem 3. 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.
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.
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.
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.
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.
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