Research Statement José A. Soto (2011/03/15)

Transcription

1 Research Statement José A. Soto (2011/03/15) Research in Mathematics is a never ending task that requires one to cross the boundaries between different disciplines and areas of knowledge. My main research area is Combinatorial Optimization. However, in order to develop tools to study and ultimately solve different problems in this area, it has been necessary for me to explore the interface between some pure and applied areas of Discrete Mathematics. On the pure side, my research interests include Combinatorics, Polyhedral Theory, Partial Order Theory, Matroids and Submodular Functions, Computational Complexity and Discrete Probability. On the more applied side, they focus on the Design and Analysis of deterministic and randomized Algorithms for online and offline problems arising from Combinatorial Optimization, Scheduling and Computational Geometry. The following pages include a short summary of my past research, and a concrete overview of some of my current and future research goals. The two first problems I describe are related to matroids. Many different topics in Mathematics naturally leads to matroids. They are structures that generalize certain aspects of linear independence in vector spaces and some aspects of graph theory. As independently shown by Rado[32], Gale[13] and Edmonds[10], Matroids also characterize those hereditary set systems for which optimizing a linear objective function is attained via a natural greedy algorithm. They are used in a wide variety of applications, ranging from the simple minimum spanning tree problem[24], to more complex ones in network coding[9], rigidity of joint frameworks[40], and kidney exchange mechanisms[33]. Matroid secretary problems Secretary problems arise from real world sequential decision-making tasks. The most basic of these problems deals with choosing the maximum value out of a stream of n elements, under the condition that every element must be immediate selected or rejected at the moment its value is revealed. Examples of tasks modeled like this include selling an item to the highest bidder, and hiring the most qualified person applying for a job for example a secretary position (reason why this problem got its, in my opinion, unfortunate name). Babaioff et al.[3] proposed in 2007 a generalization of this problem. In the matroid secretary problem, we wish to select many elements of a stream under three conditions: First, the elements are presented in uniformly random order. Second, the decision of accepting or rejecting an element is made at the moment it is revealed. And third, at every moment the set of selected elements must be an independent set of a given matroid. Our objective is to maximize the the total selected value. The matroid secretary problem has a natural interpretation in the context of online auctions: We can regard the algorithm as an auctioneer having many identical items, and the elements as agents arriving at random times, each one having a different valuation for the item. The goal of the algorithm is to assign the items to the agents as they arrive while maximizing the total social welfare, subject to some restrictions. In many situations, these restrictions can be modeled by matroid constraints. What is the best strategy for deciding which element to accept? This question has different answers under different assumptions about the values of the elements presented. A natural model, and the most studied one, is the adversarial-value model in which the elements select their value adversarially. A second, much simpler model, is the full-information model in which the values of all the elements are selected i.i.d. with respect to a given distribution. In the middle of these two models sits the random

2 assignment model in which an adversary selects a collection of values which are then randomly assigned to the elements of the stream, independent of the random order in which they are presented. Babaioff et al. conjectured that for every matroid there exists a constant competitive algorithm for each of the previously defined models. This conjecture has been partially proved in the adversarialvalue model (and, as a consequence, in the other two models) only for specific classes of matroids, such as uniform, graphical, transversal and laminar[22, 2, 3, 8, 23, 1]. However, for general matroids only O(log r)-competitive ratio algorithms have been proposed[3], where r is the rank of the matroid. I have solved affirmatively Babaioff et al. s conjecture for both the random assignment model and the full-information model, exhibiting a constant competitive algorithm for any matroid (see[36]). This algorithm, unlike others in the area, makes use of a non-standard result in Matroid Theory: Every matroid admits a decomposition, known as its principal partition, into other matroids that are wellbehaved for online algorithms. I expect this novel technique to be used in different contexts and applications. I have also developed constant competitive algorithms in the adversarial-value model for special cases, such as cographic matroids, low-density matroids and sparse linear matroids. These algorithms are based on simple, but powerful facts, coming from Graph Theory and polyhedral representations of matroids. Furthermore, I have also given an alternative O(log r)-competitive algorithm for general matroids. This algorithm, unlike the previous ones achieving this competitive ratio, does not need to use the actual numerical values of the elements: It is enough for the algorithm to have the ability to compare elements already seen. This is a desirable property since the features revealed by the elements may be of qualitative type (for example the qualifications of a person applying for a job), but the actual value or profit arising from the elements may be an unknown (increasing) function of the features revealed. As a matter of fact, all the algorithms I have proposed for variants of the matroid secretary problem, have the mentioned desirable property. Regarding future work on this area, I intend to work on the following questions. 1. Prove or disprove the existence of a constant competitive algorithm for the matroid secretary problem in the adversarial-value model. This still remains as the main open problem for this area. 2. Is there an approximate notion of principal partition for hereditary systems other than matroids? An affirmative answer to this question for the particular case of matroid intersection would allow me to extend my constant competitive algorithm for the matroid secretary problem on the random assignment model to settings in which the feasible sets are independent sets in two or more matroids, such as matchings or arborescences. This strategy, however, can not be directly extended to arbitrary hereditary systems, as it is known that no algorithm for the secretary problem in arbitrary hereditary systems can achieve an O(log n/ log log n) competitive ratio even for the full information model. 3. Matroid secretary problems where the order of the elements is not uniform. Recently, Vondrák and Oveis Gharan have informed me that a simple variation of the algorithms I have proposed for the matroid secretary problem on the random assignment model achieves a constant competitive ratio for a stronger model. In this model, the elements receive their values just like in the random assignment model but are revealed in an adversarial predetermined order. The following related question has been transmitted to me by Immorlica in a recent conference: Does the ability of choosing the order in which the elements are presented help in the adversarial-value model? 2

3 Weighted Matroid Matching in Strongly Based Orderable Matroids Consider a graph G and a matroid on the set of vertices of G. A matching M on the graph G is said to be feasible for the matroid if the collection of vertices covered by M form an independent set in the matroid. Given a weight function w: E +, the weighted matroid matching problem consists on finding a feasible matching of maximum weight. The weighted matroid matching problem generalizes two important problems in combinatorial optimization: the weighted matching problem in a graph and the weighted matroid intersection problem. Although both of these problems are polynomially time solvable, the matroid matching problem is already untractable even in its unweighted version[21, 27]: there are instances where checking that no feasible matching of a given size exists requires an exponential number of queries to an oracle for independence in a matroid. This implies that the problem is outside oracle co-np. The aforementioned instances can be modified to show a more familiar proof of NP-completeness. For the unweighted version, Lovász[27] has given polynomial time algorithms for linear matroids and Lee et al.[26] have developed a polynomial time approximation scheme (PTAS) for general matroids using local search techniques. For arbitrary weights, the situation is less developed: We only have polynomial time algorithms for the case where the matroid is a gammoid[39]. In a recent work[35], I have given a PTAS based on local search techniques for the weighted version on a special class of matroids, containing gammoids and not comparable with linear matroids: strongly base orderable matroids. These are matroids satisfying a strong exchange property. Namely, for every pair of independent sets I and J having the same cardinality, there is a bijectionπ: I J, such that for all K I, the setπ(k) (I\ K) is also an independent set. I have also shown that for strongly based orderable matroids, the problem is still NP-hard, and lies outside oracle co-np. The main open problem I intend to address in this area is to determine whether or not the proposed algorithm will also work on general matroids. An affirmatively answer for this question would settle the complexity of this important problem. Jump number and the maximum independent set of rectangles problems Claudio Telha and I[37] have recently given a surprising connection between the jump number problem on a certain class of partially ordered sets (posets) and the maximum independent set problem of a collection of rectangles in the plane. In what follows I describe our results. Although the jump number problem is a purely combinatorial problem arising from partial order theory, it admits a natural scheduling interpretation: Given a collection of jobs having some precedence constraints (a poset), schedule all the jobs on a single machine obeying the precedences, in such a way that the total setup cost is minimized. Here, the setup cost of a job is 0 if the job immediately before it in the schedule is constrained to precede it, or 1 otherwise (we say in this case that there was a jump in the schedule). It is known that this problem is NP-hard even for bipartite posets[31]. We have studied a particular class of posets: bicolored 2D-posets (also known as two directional orthogonal ray posets). These are obtained geometrically as follows: Given a sets of red points R and blue points B in the plane, a red element r precedes a blue element b if r is located below and to the left of b. These are the only precedences included in the poset. Telha and I have shown that solving the jump number problem on these posets is equivalent to finding a maximum cardinality family of disjoint orthogonal rectangles having a red point as bottom-left corner and a blue point as top-right corner. Furthermore, we have shown that the problem described above can be solved in polynomial time, and we have given a combinatorial algorithm running in time O(n ω ), where 2<ω<2.376 is the exponent 3

4 for the matroid multiplication problem. Finally, we have also shown that a weighted version of the jump number problem is NP-hard for bicolored 2D-posets and we have given combinatorial algorithms to solve this problem exactly for certain subclasses. These results have many consequences. First, we have extended the class of bipartite posets for which the jump number problem is polynomially solvable. Previously, the largest class of this type was the class of posets having convex comparability graphs (a subclass of bicolored 2D-posets). In fact, we have improved tremendously the running time for computing the jump number on these posets: Before our work, only an O(n 9 )-algorithm for the jump number problem was known for this class[7]. By relating our work with some previous results of Györi, we have given an alternative O(n 2 )-algorithm for posets with convex comparability graphs. Second, we have shown that for comparability graphs of bicolored 2D-posets there is a min-max relation between two problems related to the jump number problem: the maximum cross-free matching problem and the minimum biclique cover problem. The minimum biclique cover problem of a graph is equivalent to the problem of computing the boolean rank of its adjacency matrix A: This is finding the minimum value k for which the n m matrix A can be written as the boolean product of a n k matrix P and a k m matrix Q. The boolean rank is used, for example, to find lower bounds for communication complexity[25]. As a corollary, we have also expanded the class of matrices for which the boolean rank can be computed exactly in polynomial time. Third, we have shown a min-max relation for families of rectangles arising from bicolored 2Dposets: the maximum size of a disjoint subfamily of rectangles is equal to the minimum number of points needed to hit every rectangle in the family. This fact solves an open problem in computational geometry proposed in a recent Dagstuhl Seminar. Fourth, we have related the previous min-max relations to other relations arising from apparently unrelated problems in combinatorial optimization: the minimum rectangle cover and the maximum antirectangle of an orthogonal biconvex board, studied by Chaiken et al.[6], the minimum base of a family of intervals and the maximum independent set of point-interval pairs, studied by Györi[17] and Lubiw[28]; and the minimum edge-cover and the maximum half-disjoint family of set-pairs, studied by Frank and Jordán[12]. Our min-max relations can be seen in a certain way as both a generalization of Györi s result and as a non-trivial application of Frank and Jordán s result. Regarding future work in this area, I intend to work on the following problem. Determine the complexity of the jump number problem on two dimensional Posets and on k- colored 2D-posets. Two dimensional posets can be defined geometrically as follows. The elements are a collection of points in the plane and the precedences are given by the natural partial order on the plane: a point precedes another, if the first is below and to the left of the second. Many problems that are hard for the general case become polynomially solvable for two-dimensional posets. This has led Bouchitté and Habib[4] to conjecture that the jump number problem is polynomial time solvable in this class. Interestingly enough, as first observed by Ceroi[5], for two-dimensional posets we can interpret the jump number as the problem of finding a maximum weight independent set of rectangles having their corners on the set of points, for certain assignment of weights. Telha and I have run some experiments which have led us to believe that a natural linear program formulation for this problem has very tight integrality gap. Hopefully, we can exploit this approach to give an approximation algorithm for this problem. I also hope to use our previous approach to give polynomial time algorithms for a natural extension to bicolored 2D-posets, in which the points are assigned three or more colors instead. 4

5 Symmetric Submodular Function Minimization under Hereditary Constraints Submodularity is observed in a wide family of problems. The rank function of a matroid, the cut function of a (weighted, directed or undirected) graph, the entropy of a set of random variables, or the logarithm of the volume of the parallelepiped formed by a set of vectors are all examples of submodular functions. As many combinatorial optimization problems are particular cases of it, the problem of minimizing submodular functions is considered a fundamental problem in combinatorial optimization. A submodular function f is usually given by an oracle which, given a set S, returns f(s). Grötschel, Lovász and Schrijver[15, 16] have shown that this problem can be solved in strongly polynomial time using the ellipsoid method. Later, a collection of combinatorial strongly polynomial algorithms were developed by several authors[11, 19, 18, 30, 34, 20]. However if we impose simple additional constraints, the problem immediately becomes intractable. For example, the problem of minimizing a submodular function over all the sets having cardinality at most k is NP-hard to approximate within an o( V /log V ) factor, as shown by Svitkina and Fleischer[38]. Nevertheless, Goemans and I[14] have shown that we can efficiently find a non-empty 1 minimizer for any symmetric submodular functions over an hereditary family of subsets of V. This is, over a family closed under inclusion. Common examples of hereditary families include cardinality families (sets of at most k elements), knapsack families (given a cost function on V, consider the sets having total cost smaller than a fixed budget), matroid families (independent sets of a matroid), matching families of graphs and hypergraphs (collections of disjoint edges), hereditary properties of a graph (collection of vertices such that the associated induced subgraph satisfy some hereditary property such as being a clique, independent set, being triangle-free, planar, etc.) and any intersection of the previous families. As the canonical example of a symmetric submodular function is the cut capacity function of a nonnegatively weighted undirected graph, our algorithm allows us to, for instance, find a minimum unbalanced cut in a graph. That is, a set of vertices of cardinality bounded a priori, that minimizes the number of edges crossing from the set to its complement. As an additional feature, our algorithm is able to return all inclusionwise minimal minimizers in O( V 3 )-time. It is necessary to mention that after the completion of this work, we were informed of an independently discovered algorithm of Nagamochi[29] which is able to perform a slightly stronger feat: It can return all extreme sets of a symmetric submodular function in O( V 3 )-time. Here, an extreme set is a set whose f -value is strictly smaller than the one of any of its nontrivial subsets. Nevertheless, we have been able to extend our algorithm to work on a wider class of functions than the ones for which Nagamochi s algorithm was designed. The approach given in my joint work with Goemans allows us to minimize set functions as long we can devise a procedure that efficiently finds a so called pendant pair on any fusion of the function f : 2 V we are minimizing. Nagamochi s approach to solve this problem, on the other hand, requires a procedure to efficiently find flat pairs on any fusion of f. Goemans and I have shown that for certain functions that extend symmetric submodular functions, known as Rizzi functions, we can efficiently find pendant pairs for all their fussions. However, it is unknown if this class of functions admits flat pairs. Among the open problems on this area that I intend to address in the future, are to prove or disprove the question above and to determine a wider class of set functions for which the minimization under hereditary constraints can be solved in polynomial time. 1 The technical restriction of a non-empty minimizer is required because, for any symmetric submodular function, the empty set is already a minimizer. 5

6 References [1] M. Babaioff, M. Dinitz, A. Gupta, N. Immorlica, and K. Talwar. Secretary problems: weights and discounts. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages , [2] M. Babaioff, N. Immorlica, D. Kempe, and R. Kleinberg. A knapsack secretary problem with applications. In Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 16 28, [3] M. Babaioff, N. Immorlica, and R. Kleinberg. Matroids, secretary problems, and online mechanisms. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages , [4] V. Bouchitte and M. Habib. The calculation of invariants of ordered sets. In I. Rival, editor, Algorithms and Order, volume 255 of NATO Science Series C, pages Kluwer Academic Publishers, [5] S. Ceroi. Ordres et géométrie plane: Application au nombre de sauts. PhD thesis, Université Montpellier II, [6] S. Chaiken, D. J. Kleitman, M. Saks, and J. Shearer. Covering regions by rectangles. SIAM J. Algebra Discr., 2(4): , [7] E. Dahlhaus. The computation of the jump number of convex graphs. In V. Bouchitté and M. Morvan, editors, ORDAL 1994, volume 831 of LNCS, pages , Heidelberg, Springer. [8] N. B. Dimitrov and C. G. Plaxton. Competitive weighted matching in transversal matroids. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Part I, pages , [9] R. Dougherty, C. Freiling, and K. Zeger. Networks, matroids, and non-shannon information inequalities. Information Theory, IEEE Transactions on, 53(6): , [10] J. Edmonds. Matroids and the greedy algorithm. Mathematical Programming, 1: , /BF [11] L. Fleischer and S. Iwata. A push-relabel framework for submodular function minimization and applications to parametric optimization. Discrete Applied Mathematics, 131(2): , Sept [12] A. Frank and T. Jordán. Minimal edge-coverings of pairs of sets. J. Comb. Theory, Ser. B, 65(1):73 110, [13] D. Gale. Optimal assignments in an ordered set: An application of matroid theory. Journal of Combinatorial Theory, 4(2): , [14] M. X. Goemans and J. A. Soto. Symmetric submodular function minimization under hereditary family constraints. Manuscript. ArXiv version in

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8 [31] W. R. Pulleyblank. Alternating cycle free matchings. Technical Report CORR 82-18, University of Waterloo - Dept. of Combinatorics and Optimization, [32] R. Rado. Note on independence functions. Proceedings of the London Mathematical Society, 3(1):300, [33] A. E. Roth, T. Sönmez, and M. U. Ünver. Pairwise kidney exchange. Journal of Economic Theory, 125(2): , [34] A. Schrijver. A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory. Series B, 80(2): , [35] J. A. Soto. A simple PTAS for weighted matroid matching on strongly base orderable matroids. To appear in LAGOS 11: VI Latin-American Algorithms, Graphs and Optimization Symposium., [36] J. A. Soto. Matroid secretary problem in the random assignment model. In SODA 11: 22th Annual ACM-SIAM Symposium on Discrete Algorithms, pages , [37] J. A. Soto and C. Telha. Jump number of two-directional orthogonal ray graphs. To appear in IPCO 11: Integer Programming & Combinatorial Optimization., [38] Z. Svitkina and L. Fleischer. Submodular approximation: Sampling-based algorithms and lower bounds. In Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, pages , [39] P. Tong, E. Lawler, and V. Vazirani. Solving the weighted parity problem for gammoids by reduction to graphic matching. Progress in combinatorial optimization (Waterloo, Ont.), pages , [40] W. Whiteley. The union of matroids and the rigidity of frameworks. SIAM Journal on Discrete Mathematics, 1(2): ,