27,295 research outputs found
Towards a better approximation for sparsest cut?
We give a new -approximation for sparsest cut problem on graphs
where small sets expand significantly more than the sparsest cut (sets of size
expand by a factor bigger, for some small ; this
condition holds for many natural graph families). We give two different
algorithms. One involves Guruswami-Sinop rounding on the level- Lasserre
relaxation. The other is combinatorial and involves a new notion called {\em
Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which
we show exists in the input graph. Both algorithms run in time . We also show similar approximation algorithms in graphs with
genus with an analogous local expansion condition. This is the first
algorithm we know of that achieves -approximation on such general
family of graphs
Conditional Hardness of Earth Mover Distance
The Earth Mover Distance (EMD) between two sets of points A, B subseteq R^d with |A| = |B| is the minimum total Euclidean distance of any perfect matching between A and B. One of its generalizations is asymmetric EMD, which is the minimum total Euclidean distance of any matching of size |A| between sets of points A,B subseteq R^d with |A| <= |B|. The problems of computing EMD and asymmetric EMD are well-studied and have many applications in computer science, some of which also ask for the EMD-optimal matching itself. Unfortunately, all known algorithms require at least quadratic time to compute EMD exactly. Approximation algorithms with nearly linear time complexity in n are known (even for finding approximately optimal matchings), but suffer from exponential dependence on the dimension.
In this paper we show that significant improvements in exact and approximate algorithms for EMD would contradict conjectures in fine-grained complexity. In particular, we prove the following results:
- Under the Orthogonal Vectors Conjecture, there is some c>0 such that EMD in Omega(c^{log^* n}) dimensions cannot be computed in truly subquadratic time.
- Under the Hitting Set Conjecture, for every delta>0, no truly subquadratic time algorithm can find a (1 + 1/n^delta)-approximate EMD matching in omega(log n) dimensions.
- Under the Hitting Set Conjecture, for every eta = 1/omega(log n), no truly subquadratic time algorithm can find a (1 + eta)-approximate asymmetric EMD matching in omega(log n) dimensions
Conditional Hardness of Earth Mover Distance
The Earth Mover Distance (EMD) between two sets of points A, B subseteq R^d with |A| = |B| is the minimum total Euclidean distance of any perfect matching between A and B. One of its generalizations is asymmetric EMD, which is the minimum total Euclidean distance of any matching of size |A| between sets of points A,B subseteq R^d with |A| <= |B|. The problems of computing EMD and asymmetric EMD are well-studied and have many applications in computer science, some of which also ask for the EMD-optimal matching itself. Unfortunately, all known algorithms require at least quadratic time to compute EMD exactly. Approximation algorithms with nearly linear time complexity in n are known (even for finding approximately optimal matchings), but suffer from exponential dependence on the dimension.
In this paper we show that significant improvements in exact and approximate algorithms for EMD would contradict conjectures in fine-grained complexity. In particular, we prove the following results:
- Under the Orthogonal Vectors Conjecture, there is some c>0 such that EMD in Omega(c^{log^* n}) dimensions cannot be computed in truly subquadratic time.
- Under the Hitting Set Conjecture, for every delta>0, no truly subquadratic time algorithm can find a (1 + 1/n^delta)-approximate EMD matching in omega(log n) dimensions.
- Under the Hitting Set Conjecture, for every eta = 1/omega(log n), no truly subquadratic time algorithm can find a (1 + eta)-approximate asymmetric EMD matching in omega(log n) dimensions
Asymptotically Optimal Algorithms for Pickup and Delivery Problems with Application to Large-Scale Transportation Systems
The Stacker Crane Problem is NP-Hard and the best known approximation
algorithm only provides a 9/5 approximation ratio. The objective of this paper
is threefold. First, by embedding the problem within a stochastic framework, we
present a novel algorithm for the SCP that: (i) is asymptotically optimal,
i.e., it produces, almost surely, a solution approaching the optimal one as the
number of pickups/deliveries goes to infinity; and (ii) has computational
complexity O(n^{2+\eps}), where is the number of pickup/delivery pairs
and \eps is an arbitrarily small positive constant. Second, we asymptotically
characterize the length of the optimal SCP tour. Finally, we study a dynamic
version of the SCP, whereby pickup and delivery requests arrive according to a
Poisson process, and which serves as a model for large-scale demand-responsive
transport (DRT) systems. For such a dynamic counterpart of the SCP, we derive a
necessary and sufficient condition for the existence of stable vehicle routing
policies, which depends only on the workspace geometry, the stochastic
distributions of pickup and delivery points, the arrival rate of requests, and
the number of vehicles. Our results leverage a novel connection between the
Euclidean Bipartite Matching Problem and the theory of random permutations,
and, for the dynamic setting, exhibit novel features that are absent in
traditional spatially-distributed queueing systems.Comment: 27 pages, plus Appendix, 7 figures, extended version of paper being
submitted to IEEE Transactions of Automatic Contro
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