An official website of the United States government
We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a $$1 - 1/e - \epsilon$$ approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, $$1 - 1/e - \epsilon$$ approximation, using a nearly-linear number of function evaluations and arithmetic operations.
more »
« less
Probing to Minimize
We develop approximation algorithms for set-selection problems with deterministic constraints, but random objective values, i.e., stochastic probing problems. When the goal is to maximize the objective, approximation algorithms for probing problems are well-studied. On the other hand, few techniques are known for minimizing the objective, especially in the adaptive setting, where information about the random objective is revealed during the set-selection process and allowed to influence it. For minimization problems in particular, incorporating adaptivity can have a considerable effect on performance. In this work, we seek approximation algorithms that compare well to the optimal adaptive policy. We develop new techniques for adaptive minimization, applying them to a few problems of interest. The core technique we develop here is an approximate reduction from an adaptive expectation minimization problem to a set of adaptive probability minimization problems which we call threshold problems. By providing near-optimal solutions to these threshold problems, we obtain bicriteria adaptive policies. We apply this method to obtain an adaptive approximation algorithm for the Min-Element problem, where the goal is to adaptively pick random variables to minimize the expected minimum value seen among them, subject to a knapsack constraint. This partially resolves an open problem raised in [Goel et al., 2010]. We further consider three extensions on the Min-Element problem, where our objective is the sum of the smallest k element-weights, or the weight of the min-weight basis of a given matroid, or where the constraint is not given by a knapsack but by a matroid constraint. For all three of the variations we explore, we develop adaptive approximation algorithms for their corresponding threshold problems, and prove their near-optimality via coupling arguments.
more »
« less
- PAR ID:
- 10326998
- Editor(s):
- Braverman, Mark
- Date Published:
- Journal Name:
- 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a 1 - 1/e - epsilon approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, 1 - 1/e - epsilon approximation, using a nearly-linear number of function evaluations and arithmetic operations.more » « less
-
We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $$1-1/e-\epsilon$$ approximation for monotone functions and a $$1/e-\epsilon$$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $$O(\log^2{n}/\epsilon^3)$$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $$1/e-\epsilon$$ approximation using $$O(\log(n/\epsilon) \log(1/\epsilon) \log(n+m)/ \epsilon^2)$$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $$1-1/e-\epsilon$$ approximation in $$O(\log(n/\epsilon)\log(m)/\epsilon^2)$$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective (Mahoney et al., 2016). Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.more » « less
-
There has been a long-standing interest in computing diverse solutions to optimization problems. In 1995 J. Krarup [28] posed the problem of finding k-edge disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer k, an approximation factor c, and an instance I of an optimization problem, we aim to obtain a set of k solutions to I that a) are all c approximately-optimal for I and b) maximize the diversity of the k solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. Given a metric on the space of solutions, and the diversity measure as the sum of pairwise distances between solutions, we first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to optimized subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem. As applications of our result, we present polynomial time approximation algorithms for several problems such as diverse c-approximate maximum matchings, shortest paths, global min-cut, and minimum weight bases of a matroid. The last result gives us diverse c-approximate minimum spanning trees, advancing a step towards achieving diverse c-approximate TSP tours. We also explore the connection to the field of multiobjective optimization and show that the class of problems to which our result applies includes those for which the associated DUALRESTRICT problem defined by Papadimitriou and Yannakakis [35], and recently explored by Herzel et al. [26] can be solved in polynomial timore » « less
-
There has been a long-standing interest in computing diverse solutions to optimization problems. In 1995 J. Krarup [28] posed the problem of finding k-edge disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer k, an approximation factor c, and an instance I of an optimization problem, we aim to obtain a set of k solutions to I that a) are all c approximately-optimal for I and b) maximize the diversity of the k solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. Given a metric on the space of solutions, and the diversity measure as the sum of pairwise distances between solutions, we first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to optimized subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem. As applications of our result, we present polynomial time approximation algorithms for several problems such as diverse c-approximate maximum matchings, shortest paths, global min-cut, and minimum weight bases of a matroid. The last result gives us diversec-approximate minimum spanning trees, advancing a step towards achieving diverse c-approximate TSP tours. We also explore the connection to the field of multiobjective optimization and show that the class of problems to which our result applies includes those for which the associated DUALRESTRICT problem defined by Papadimitriou and Yannakakis [35], and recently explored by Herzel et al. [26] can be solved in polynomial time.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.ITCS.2022.120
