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Matroids, Secretary Problems, and Online Mechanisms Nicole Immorlica, Microsoft Research Joint work with Robert Kleinberg and Moshe Babaioff
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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?
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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
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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]
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Example Multicast in a network Each node wants an edge-disjoint path to source Value: $8 Value: $10 Value: $7 Value: $12
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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 )
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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.
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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)
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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.
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Our Results 1. O(log k)-competitive algorithm for general matroids, where k is the rank. 2. 16-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.
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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.
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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
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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
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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?
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