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Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision

Sumanta Ghosh, Rohit Gurjar, Mary Wootters, Laura Sanità
2021
We study the matroid intersection problem from the parallel complexity perspective.  ...  We make a significant progress on this question by giving a pseudo-deterministic parallel (NC) algorithm for the search version that uses an oracle access to the weighted decision.  ...  Their main result was, what they call, a pseudo-deterministic NC algorithm for bipartite matching, which followed from this reduction. 41:4 Pseudo-Deterministic NC Reduction from search-MI to weighted-decision-MI  ... 
doi:10.4230/lipics.approx/random.2021.41 fatcat:eaprv3jq2ffqvedfkymshjt2gq

Matching Is as Easy as the Decision Problem, in the NC Model

Nima Anari, Vijay V. Vazirani, Michael Wagner
2020 Innovations in Theoretical Computer Science  
Algorithms were also obtained for the generalization of bipartite matching to the linear matroid intersection problem by [11] , and to a further generalization of isolating a vertex of a polytope with  ...  ., is there a deterministic fast parallel algorithm for it?  ...  This leads to a legitimate question: why not attempt to reduce, in NC, the search problem to the second decision problem?  ... 
doi:10.4230/lipics.itcs.2020.54 dblp:conf/innovations/AnariV20 fatcat:g3ekmspa4rdddj3tkdyxmixyiq

Matching is as Easy as the Decision Problem, in the NC Model [article]

Nima Anari, Vijay V. Vazirani
2020 arXiv   pre-print
., is there a deterministic fast parallel algorithm for it?  ...  We give what appears to be the culmination of this line of work: An NC algorithm for finding a minimum-weight perfect matching in a general graph with polynomially bounded edge weights, provided it is  ...  RNC algorithms for linear matroid intersection and for finding a vertex of a polytope with faces given by totally unimodular constraints.  ... 
arXiv:1901.10387v5 fatcat:hxqczx6hybglboy55hxiurpqby

Sensor Planning for Large Numbers of Robots [article]

Micah Corah
2021 arXiv   pre-print
However, greedy algorithms typically force robots to make decisions sequentially so that planning time grows with the number of robots.  ...  To assist search and rescue teams and improve response times, teams of camera-equipped aerial robots can engage in tasks such as mapping buildings and locating victims.  ...  an action from a block of a partition matroid).  ... 
arXiv:2102.04054v1 fatcat:zgg75lo6wfdwvekn5twqpmi7oe

Scalable Distributed Planning for Multi-Robot, Multi-Target Tracking [article]

Micah Corah, Nathan Michael
2021 arXiv   pre-print
Our prior work proposed a distributed planner (RSP) which reduces this number of sequential steps to a constant, even for large numbers of robots, by allowing robots to plan in parallel while ignoring  ...  some of each others' decisions.  ...  Then, robots assigned to the same step plan in parallel with access (via N in i ) to some or all decisions from prior steps.  ... 
arXiv:2107.08550v1 fatcat:o2x3veb36vejlm5xpu35sbsq3m

Limits to parallel computation: P-completeness theory

1996 ChoiceReviews  
Consequently, the promise of parallel computation, namely that applying more processors to a problem can greatly speed its solution, appears to be broken by the entire class of P -complete problems.  ...  That is, algorithm designers have failed to find NC algorithms, feasible highly parallel solutions that take time polynomial in the logarithm of the problem size while using only a polynomial number of  ...  Hint: Note that the problem as stated is a search problem. It is in FP if the weights are encoded in unary. The reduction is from LFMIS, Problem A.2.1 [119] .  ... 
doi:10.5860/choice.33-3959 fatcat:qjoueeu225gr7jvwdf6gkoxfbq

Metabolic Networks, Thermodynamic Constraints, and Matroid Theory [article]

Arne C. Reimers, Universitätsbibliothek Der FU Berlin, Universitätsbibliothek Der FU Berlin
2014
With the definition of k-module, I also show a decomposition theorem for general polyhedra using the concept of matroid branch-width.  ...  These insights I then use to develop a bi-level optimization method to analyze a growth behavior of the green alga Chlamydomonas reinhardtii. Another area covered by this work are flux modules.  ...  The density D is a variable. 3. We assume that the light input is fixed to intensity in . 4.  ... 
doi:10.17169/refubium-15314 fatcat:dbtnieyknrfbxp3hqbwcnj2fnm

14th International Symposium on Mathematical Programming

1990 Mathematical programming  
If we use a decomposition approach in order to solve a minimization problem we often get an objective function in such a w a y that its domain dom 6 = n is not given explicitely to us.  ...  to the well known serious steps und null steps of bundle methods a third type of steps is used to generate the model of dom .  ...  Geelen James Ferdinand Keywords: matchings -matroid -polyhedra We present a common generalization of weighted matching and weighted matroid intersection, and give results that imply most of the important  ... 
doi:10.1007/bf01580875 fatcat:3jtclwmntzgjxkqs5uecombdaa

Rigidity through a Projective Lens

Anthony Nixon, Bernd Schulze, Walter Whiteley
2021 Applied Sciences  
Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance.  ...  These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.  ...  Some key collaborators from the 1970s to the early 2000s included Janos Baracs, a Hungarian trained architectural engineer who lived within projective geometry up to at least 4 dimensions, Henry Crapo  ... 
doi:10.3390/app112411946 fatcat:25pxvf5mezavdi7fb46gramgby

Minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights

Adam Kasperski, Paweł Zieliński
2010 European Journal of Operational Research  
Possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones.  ...  This paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights.  ...  In order to show its NP-completeness, he proposed a reduction from Poss Longest Path. Unfortunately this reduction is incorrect. We now give a correct reduction. Theorem 2.  ... 
doi:10.1016/j.ejor.2009.01.044 fatcat:v3g4wi2csfa7pjafisjpgbfhzi

Cumulative index volumes 1–92

2000 Discrete Applied Mathematics  
., Legendre sums and codes related to QR , A., Finding a feasible course schedule using Tabu search 35 (1992) 255-270 Hertz, A., B.  ...  Novick, The 2-quasi-greedy algorithm for cardinality constrained matroid bases 13 (1986) 277-286 Glover, F., Tabu search for nonlinear and parametric optimization (with links to genetic algorithms) 49  ... 
doi:10.1016/s0166-218x(00)00210-9 fatcat:vvomn3k7hbbmliptwaczxk6wi4

Submodularity In Machine Learning and Artificial Intelligence [article]

Jeff Bilmes
2022 arXiv   pre-print
We discuss a variety of ways to produce a submodular function useful for machine learning, including heuristic hand-crafting, learning or approximately learning a submodular function or aspects thereof  ...  We then turn to how submodularity is useful in machine learning and artificial intelligence.  ...  It is also possible to define constraints consisting of an intersection of matroids, meaning that the solution must be simultaneously independent in multiple distinct matroids.  ... 
arXiv:2202.00132v1 fatcat:sp4b3ww3ajdxvfgigp7xw4f4yq

Computing Walrasian equilibria: fast algorithms and structural properties

Renato Paes Leme, Sam Chiu-wai Wong
2018 Mathematical programming  
A key technical ingredient is to regularize a convex programming formulation of the problem in a way that subgradients are cheap to compute.  ...  The conference version is attached to this submission for the records.  ...  First, we show a simple reduction from the welfare problem for gross substitutes to this problem.  ... 
doi:10.1007/s10107-018-1334-9 fatcat:fytimwept5dejbxc4i5qvvpk4e

Introduction to the Theory of Complexity

V. J. Rayward-Smith
1995 Journal of the Operational Research Society  
and functions 3 3. the union of the elements of P is equal to A. 3, 5, 8, 12, 16, 25, 50}.  ...  Although every precaution has been taken in the preparation of this work, the authors shall not have any liability to any person or entity with respect to any loss or damage caused or alleged to be caused  ...  Acknowledgements This book originated from a course on Algorithms and Complexity given at the University of Rome 'La Sapienza' by D.P. Bovet and P. Crescenzi since 1986.  ... 
doi:10.1057/jors.1995.211 fatcat:rx2m7i3vyreaflkxrbmuh5kbhu

Computing Walrasian Equilibria: Fast Algorithms and Structural Properties [article]

Renato Paes Leme, Sam Chiu-wai Wong
2016 arXiv   pre-print
A key technical ingredient is to regularize a convex programming formulation of the problem in a way that subgradients are cheap to compute.  ...  We present the first polynomial time algorithm for computing Walrasian equilibrium in an economy with indivisible goods and general buyer valuations having only access to an aggregate demand oracle, i.e  ...  First, we show a simple reduction from the welfare problem for gross substitutes to this problem.  ... 
arXiv:1511.04032v2 fatcat:tharcfrx65b4xnr7y4nonpxuei
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