Matroids, Secretary Problems, and Online Mechanisms Nicole Immorlica, Microsoft Research Joint work with Robert Kleinberg and Moshe Babaioff

The Secretary Problem Company wants to hire a secretary  There are n secretaries available, each of whom will accept any offer they receive  Each secretary i has an inherent value v i  Secretaries interview in a random order, revealing their value at the interview  Hiring decision must be made at the interview Question: Can the company design an interviewing procedure to guarantee that it hires the (approximately) best secretary?

The Secretary Algorithm Algorithm:  Observe first n/e elements. Let v=maximum.  Pick the next element whose value is > v. Theorem: Pr(picking max elt. of S) > 1/e.* Proof: Select best elt. if i’th best elt is best in first 1/e elts and best elt is first among best (i-1) elts. Happens with probability (1/e) ¢ (1-1/e) i ¢ (1/i). * Elements come in a random order. Threshold time t = n/etime t = n i’th bestbest 2 nd best through (i-1) st best

Generalized Secretary Problems Input  Set of secretaries {1, …, n}, each has a value v i  Feasible or independent family of subsets of {1, …, n} Secretaries arrive in random order, and alg. must decide online whether to select each secretary Goal is to select maximum weight feasible set Performance measure is competitive ratio: E[weight of selected set]/[weight of max ind. set]

Example Multicast in a network Each node wants an edge-disjoint path to source Value: $8 Value: $10 Value: $7 Value: $12

Special Cases Standard secretary problem: independent sets are all singletons Thm [Dynkin ‘63]: There is an algorithm with competitive ratio (1/e). k-Secretary problem: independent sets are all sets of size at most k Thm [Kleinberg ‘05]: There is an algorithm with competitive ratio 1- Θ (k -1/2 )

Matroid Secretary Problems Defn.: A matroid consists of a universe of elements and a family of distinguished subsets called independent sets which satisfy:  Subsets of independent sets are independent.  Exchange property: If S,T are independent and |S| < |T| then S U {t} is independent for some t in T. A matroid secretary problem is a generalized secretary problem in which the independent sets form a matroid. The standard and k-secretary problems are matroid secretary problems.

More Examples Gammoid Matroids:  Elements (customers) are sources in a graph  Set S of sources is independent if there exist edge-disjoint paths routing each source in S to the sink Graphical Matroids:  Elements are the edges of an undirected graph G = (V;E)  Set of edges is independent if it does not contain a cycle Truncated Partition Matroids of rank k:  Elements (items) are partitioned into m sets  Set of elements is independent if it contains at most one item from each partition and at most k items in total (production constraint)

Open Question Is there a constant-competitive secretary algorithm for all matroids? Intuition:  In matroids, a single mistake can only ruin your chance of picking one element of the best set  If algorithm could discard a previously selected element, matroid properties guarantee the greedy algorithm always selects optimal set. Thm.: If independent sets are allowed to be an arbitrary set system closed under containment, no algorithm can be constant-competitive.

Our Results 1. O(log k)-competitive algorithm for general matroids, where k is the rank competitive algorithm for graphical matroids. 3. 4d-competitive algorithm for transversal matroids, where d is the max size of an agent’s set of desired items. 4. If M has a c-competitive algorithm, then every truncation of M has a 48c-competitive algorithm.

Lower bound for general set systems

Proof of lower bound Suppose the algorithm makes its first selection at time t, and that it chooses an element x in Si. All future selections must be elements of Si. Let Ti be the subset of Si which have not yet been observed at time t. The values of the elements of Ti are independent of all the information observed up to time t; there are less than k elements in Ti and each of them has expected value 1/k, so the expected combined value of all remaining elements is less than 1. The only element selected up to time t is x, whose value is at most 1. This proves that the expected value of the set selected by the algorithm is less than 2. Let j = k/(2 ln(k)). Probability that at least one Si has j elements of value 1 is 1-o(1). Therefore the maximum-weight element of I has weight lower bounded by j=(ln n /2 ln ln n)

O(log k)-competitive algorithm 1. Assume the algorithm knows an integer s between log(k)-1 and log(k).* 2. Sample the first n/2 elements without selecting any of them. Let v* be the maximum value observed so far. Pick random r in {1,…,s}. 3. Set threshold value w = v*/2 r. 4. From then on, select every element independent of previous selections whose value is at least w. * This assumption is not needed. We can estimate s using the rank of the sample.

Single Threshold Algorithms An algorithm which computes a threshold value v and stopping time  and then selects every feasible element after  whose value is at least v  O(log k)-competitive algorithm is single threshold Counterexample: single threshold algorithms are not constant competitive (lower-bound)  Partition matroid with k sets of size n/k  Set i has (k-1) values =1/(c i ) and 1 value =1/i

Greedy Algorithms Algorithm  Observe a constant fraction of the input without selecting any element  Compute a maximum weight basis among elements observed so far  Select any feasible element which can be exchanged with an element in the basis to improve its weight Counterexample: greedy algorithms can not be constant competitive … … 1 Node i Weight n-i Weight i

Open Questions Is there a constant-competitive algorithm for general matroids? If so, is it e-competitive? Relaxations:  Matroid structure known in advance.  Values assigned randomly to the matroid elements. Special cases:  Transversal matroids, gammoid matroids  Is the class of constant-competitive matroids closed under contraction?