Planar 3-dimensional assignment problems with Monge-like cost arrays

Given an $n\times n\times p$ cost array $C$ we consider the problem $p$-P3AP which consists in finding $p$ pairwise disjoint permutations $\varphi_1,\varphi_2,\ldots,\varphi_p$ of $\{1,\ldots,n\}$ such that $\sum_{k=1}^{p}\sum_{i=1}^nc_{i\varphi_k(i)k}$ is minimized. For the case $p=n$ the planar 3-dimensional assignment problem P3AP results. Our main result concerns the $p$-P3AP on cost arrays $C$ that are layered Monge arrays. In a layered Monge array all $n\times n$ matrices that result from fixing the third index $k$ are Monge matrices. We prove that the $p$-P3AP and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that in the layered Monge case there always exists an optimal solution of the $p$-3PAP which can be represented as matrix with bandwidth $\le 4p-3$. This structural result allows us to provide a dynamic programming algorithm that solves the $p$-P3AP in polynomial time on layered Monge arrays when $p$ is fixed.Comment: 16 pages, appendix will follow in v

Recognition of d-dimensional Monge arrays

AbstractIt is known that the d-dimensional axial transportation (assignment) problem can easily be solved by a greedy algorithm if and only if the underlying cost array fulfills the d-dimensional Monge property. In this paper the following question is solved: Is it possible to find d permutations in such a way that the permuted array becomes a Monge array? Furthermore we give an algorithm which constructs such permutations in the affirmative case. If the cost array has the dimensions n1×n2×⋯×nd with n1⩽n2⩽⋯⩽nd, then the algorithm has time complexity O(d2n2⋯nd(n1+lognd)). By using this algorithm a wider class of d-dimensional axial transportation problems and in particular of the d-dimensional axial assignment problems can be solved efficiently

Work-Optimal Parallel Minimum Cuts for Non-Sparse Graphs

We present the first work-optimal polylogarithmic-depth parallel algorithm for the minimum cut problem on non-sparse graphs. For $m\geq n^{1+\epsilon}$ for any constant $\epsilon>0$, our algorithm requires $O(m \log n)$ work and $O(\log^3 n)$ depth and succeeds with high probability. Its work matches the best $O(m \log n)$ runtime for sequential algorithms [MN STOC 2020, GMW SOSA 2021]. This improves the previous best work by Geissmann and Gianinazzi [SPAA 2018] by $O(\log^3 n)$ factor, while matching the depth of their algorithm. To do this, we design a work-efficient approximation algorithm and parallelize the recent sequential algorithms [MN STOC 2020; GMW SOSA 2021] that exploit a connection between 2-respecting minimum cuts and 2-dimensional orthogonal range searching.Comment: Updates on this version: Minor corrections for the previous and our resul

Re-Use Dynamic Programming for Sequence Alignment: An Algorithmic Toolkit

International audienceThe problem of comparing two sequences S and T to determine their similarity is one of the fundamental problems in pattern matching. In this manuscript we will be primarily concerned with sequences as our objects and with various string comparison metrics. Our goal is to survey a methodology for utilizing repetitions in sequences in order to speed up the comparison process. Within this framework we consider various methods of parsing the sequences in order to frame their repetitions, and present a toolkit of various solutions whose time complexity depends both on the chosen parsing method as well as on the string-comparison metric used for the alignment

The Monge array--an abstraction and its applications

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1991.Includes bibliographical references (p. 211-219).by James KIimbrough Park.Ph.D

The Earth Mover\u27s Distance Through the Lens of Algebraic Combinatorics

The earth mover\u27s distance (EMD) is a metric for comparing two histograms, with burgeoning applications in image retrieval, computer vision, optimal transport, physics, cosmology, political science, epidemiology, and many other fields. In this thesis, however, we approach the EMD from three distinct viewpoints in algebraic combinatorics. First, by regarding the EMD as the symmetric difference of two Young diagrams, we use combinatorial arguments to answer statistical questions about histogram pairs. Second, we adopt as a natural model for the EMD a certain infinite-dimensional module, known as the first Wallach representation of the Lie algebra su(p,q), which arises in the Howe duality setting in Type A; in this setting, we show how the second fundamental theorem of invariant theory generalizes the northwest corner rule\u27\u27 from optimal transport theory, yielding a simple interpretation of the partial matching\u27\u27 case of the EMD via separation into invariants and harmonics. Third, we reapproach partial matching in the context of crystal bases of Types A, B, and C, which leads us to introduce a variation of the EMD in terms of distance on a crystal graph. Having exploited these three approaches, we generalize all of our EMD results to an arbitrary number of histograms rather than only two at a time. In the final chapter, we observe a combinatorial connection between generalized BGG resolutions arising in Type-A Howe duality and certain non-holomorphic discrete series representations of the group SU(p,q)