Matroids, Secretary Problems, and Online Mechanisms

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

2. 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 vi • 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?

3. 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. 2nd best through (i-1)st best i’th best best Threshold time t = n/e time t = n

4. Generalized Secretary Problems • Input • Set of secretaries {1, …, n}, each has a value vi • 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]

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

6. 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)

7. 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.

8. 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)

9. 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 alg. could discard a previously selected element, matroid properties guarantee the greedy alg. 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.

10. Our Results • O(log k)-competitive algorithm for general matroids, where k is the rank. • 16-competitive algorithm for graphical matroids. • 4d-competitive algorithm for transversal matroids, where d is the max size of an agent’s set of desired items. • If M has a c-competitive algorithm, then every truncationof M has a 48c-competitive algorithm.

11. O(log k)-competitive algorithm • Assume the algorithm knows an integer s between log(k)-1 and log(k).* • 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}. • Set threshold value w = v*/2r. • 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.

12. 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 • Partition matroid with k sets of size n/k • Set i has (k-1) elts of value 1/(ci) and 1 elt of value 1/i

13. 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 Weight i … Node i Weight n-i …

14. 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?