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Set-to-real map with diminishing returns
mathematics, a submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between a set of inputs
Submodular_set_function
Class of mathematical functions
(strictly) supermodular then f is called (strictly) submodular. A function that is both submodular and supermodular is called modular. This corresponds
Supermodular_function
Function from sets to numbers
mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values
Set_function
Problem in combinatorial optimization
flow out, and have minimum total cost. In submodular flow, as well, one is given a submodular set function on sets of vertices of the graph. Instead of obeying
Submodular_flow
Mapping function
notion of measure in mathematics Submodular set function – Set-to-real map with diminishing returns Subadditive set function τ-additivity – Property of certain
Sigma-additive_set_function
Maximum size of an independent set of the matroid
may be axiomatized. Matroid rank functions form an important subclass of the submodular set functions. The rank functions of matroids defined from certain
Matroid_rank
submodular set function is subadditive (the family of non-negative submodular functions is strictly contained in the family of subadditive functions)
Subadditive_set_function
submodular agent has a utility function that is a submodular set function. This means that the agent's utility has decreasing marginals. Submodular utilities
Welfare_maximization
Computer-based method for summarizing a text
naturally model summarization problems are TextRank and PageRank, Submodular set function, Determinantal point process, maximal marginal relevance (MMR)
Automatic_summarization
Israeli computer scientist
knapsack problems, interval scheduling, and the optimization of submodular set functions. She is a professor of computer science at the Technion – Israel
Hadas_Shachnai
Generalization of binary functions
a pseudo-Boolean function. The submodular set functions can be viewed as a special class of pseudo-Boolean functions, which is equivalent to the condition
Pseudo-Boolean_function
Statistical law in machine learning
their inherent diminishing returns nature, value data based on a submodular set function which was shown in a paper on this topic. Training cost is typically
Neural_scaling_law
Economic theory
of resources Self-organized criticality – Concept in physics Submodular set function – Set-to-real map with diminishing returns Sunk-cost fallacy – Unrecoverable
Diminishing_returns
Refinement of perfect matching theorems
−sur(G; X)] In a bipartite graph G = (X+Y, E), the surplus function is a submodular set function: for every two subsets X1, X2 of X: sur G ( X 1 ∪ X 2
Deficiency_(graph_theory)
Sequence of locally optimal choices
Fisher, M.L. (1978). "An analysis of approximations for maximizing submodular set functions—I". Mathematical Programming. 14 (1): 265–294. doi:10.1007/BF01588971
Greedy_algorithm
Measure of difference between two points
been defined over sets, through a submodular set function which is known as the discrete analog of a convex function. The submodular Bregman divergences
Bregman_divergence
{\displaystyle u} is a subadditive set function. Assuming u ( ∅ ) {\displaystyle u(\emptyset )} is non-negative, every submodular function is subadditive. However
Utility functions on indivisible goods
Utility_functions_on_indivisible_goods
Ratio in Mathematical Optimization
is bounded in several cases. For example, when the cost function is a submodular set function (as in the above example), the correlation gap is at most
Correlation_gap
_{i}v(T_{i})} . Every submodular set function is XOS, and every XOS function is a subadditive set function. See also: Utility functions on indivisible goods
Fractionally subadditive valuation
Fractionally_subadditive_valuation
Problem in computer science
submodular set functions I, Mathematical Programming 14 (1978), 265–294 Hochbaum, Dorit S. (1997). "Approximating Covering and Packing Problems: Set Cover
Maximum_coverage_problem
Concept in graph theory and network analysis
the result of the greedy algorithm. For non-negative monotone submodular set functions, such as GED-Walk centrality, this greedy algorithm is already
Group_centrality
Concept in economics
functions on indivisible goods Independent goods Submodular set function Supermodular set function Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang
Additive_utility
Multiset analogue of matroids
In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also a generalization
Polymatroid
demonstrated that for any submodular set function, i.e. valuation function where the marginal value of an item decreases as the set of items grows (the law
Budget-feasible_mechanism
Combinatorial optimization method for pseudo-Boolean functions
the target function with a similar but submodular approximation, e.g. by removing all non-submodular terms or replacing them with submodular approximations
Quadratic pseudo-Boolean optimization
Quadratic_pseudo-Boolean_optimization
valuation is a submodular set function. The converse is not necessarily true. This is shown by the example on the right. The utility is submodular since it
Gross substitutes (indivisible items)
Gross_substitutes_(indivisible_items)
allows for simpler optimization while maintaining sparsity. See: Submodular set function Besides the norms discussed above, other norms used in structured
Structured sparsity regularization
Structured_sparsity_regularization
Israeli computer scientist
Kulik, Ariel; Shachnai, Hadas; Tamir, Tami (2009), "Maximizing submodular set functions subject to multiple linear constraints", in Mathieu, Claire (ed
Tami_Tamir
Combinatorial optimization method for a family of functions of discrete variables
of functions that can be optimised through graph cuts, such as submodular quadratic functions. Graph cut optimization can be extended to functions of
Graph_cut_optimization
Economical computational problem
game is called submodular if (a) the set of feasible joint decisions is a sublattice; (b) the cost function of each player is submodular and has antitone
Nash_equilibrium_computation
American operations researcher (born 1937)
Fisher, M. L. (1978), "An analysis of approximations for maximizing submodular set functions I", Mathematical Programming, 14 (1): 265–294, doi:10.1007/BF01588971
George_Nemhauser
Mathematics textbook on algebraic combinatorics
theory", as well as matchings in graphs, include incidence matrices, submodular set functions, independent matchings in matroids, the Birkhoff–von Neumann theorem
Combinatorics:_The_Rota_Way
British mathematician (born 1945)
Fisher (1978). "An analysis of approximations for maximizing submodular set functions I". Mathematical Programming A. 14: 265–294. doi:10.1007/BF01588971
Laurence_Wolsey
Textbook on the theory of matroids
through their independent sets, but equivalences with definitions using circuits, matroid rank, and submodular set function are also presented, as are
Independence Theory in Combinatorics
Independence_Theory_in_Combinatorics
is an extreme case of a submodular set function. It is characteristic of items that are pure substitute goods. Utility functions on indivisible goods Matching
Unit_demand
second-price auction) in each round. Case 4: submodular bidders. The bidders' valuations are arbitrary submodular set functions (note that additive and unit-demand
Sequential_auction
Method to solve optimization problems
programming.) Edmonds, Jack; Giles, Rick (1977). "A Min-Max Relation for Submodular Functions on Graphs". Studies in Integer Programming. Annals of Discrete Mathematics
Linear_programming
Design concept in economics
than when serving each agent individually (i.e., the cost is a submodular set function). As a typical example, consider two agents, Alice and George,
Cost-sharing_mechanism
Game where groups of players may enforce cooperative behaviour
choosing an appropriate permutation π {\displaystyle \pi } . Submodular and supermodular set functions are also studied in combinatorial optimization. Many of
Cooperative_game_theory
individuals in the group. The assignment valuations are a subset of the submodular valuations. Suppose there are three items and two agents who value the
Assignment_valuation
Correlation inequality
distributive lattice. Now, if Φ {\displaystyle \Phi } is a submodular potential (i.e., a family of functions Φ Λ : S Λ ⟶ R ∪ { ∞ } , {\displaystyle \Phi _{\Lambda
FKG_inequality
Abstraction of linear independence of vectors
r(A\cup B)+r(A\cap B)\leq r(A)+r(B)} . That is, the rank is a submodular function. (R4) For any set A {\displaystyle A} and element x {\displaystyle x} , we
Matroid
Join-meet algebra on matroid flats
matroid must be monotonic (adding an element to a set can never decrease its rank) and it must be submodular, meaning that it obeys an inequality similar to
Geometric_lattice
Theory of generalized measures in mathematics
(OWA) operator. Submodular fuzzy measures result in convex functions, while supermodular fuzzy measures result in concave functions when used to define
Fuzzy_measure_theory
Award for advancements in discrete mathematics
minimizing submodular functions," Journal of the ACM, 48 (4): 761–777, 2001. Alexander Schrijver, "A combinatorial algorithm minimizing submodular functions in
Fulkerson_Prize
Function in algorithmic game theory
approximations are known for special cases, such as submodular valuations (this is called the "submodular welfare problem"). Some algorithms use only a value
Demand_oracle
Set system used in greedy optimization
Theory of Greedy Algorithms Archived 2016-03-04 at the Wayback Machine Submodular Functions and Optimization Matchings, Matroids and Submodular Functions
Greedoid
Theorem in order and lattice theory
lattice and let f : L → L be an order-preserving (monotonic) function with respect to ≤. Then the set of fixed points of f in L forms a complete lattice under
Knaster–Tarski_theorem
Subdivision into few independent sets
948–957. doi:10.1137/0215066. ISSN 0097-5397. Edmonds, Jack (1970), "Submodular functions, matroids, and certain polyhedra", Combinatorial Structures and their
Matroid_partitioning
1016/0024-3795(79)90018-1. Edmonds, J.; R. Giles (1977). "A min-max relation for submodular functions on graphs". Annals of Discrete Mathematics. 1: 185–204. Schrijver
Total_dual_integrality
Computer vision algorithm
NP-complete problem in the general case. For some families of cost functions (e.g. submodular functions) a solution with strong optimality properties can be found
Semi-global_matching
Integer matrices with +1 or −1 determinant; invertible over the integers. GL_n(Z)
Fujishige, Satoru (1984), "A System of Linear inequalities with a Submodular Function on (0, ±1) Vectors", Linear Algebra and Its Applications, 63: 253–266
Unimodular_matrix
Process in machine learning and statistics
multinomial logit (RMNL) Auto-encoding networks with a bottleneck-layer Submodular feature selection Local learning based feature selection. Compared with
Feature_selection
Weighted tree representing s-t cuts of a graph
(\{u\},\{v\})\in E_{T}} by (u, v). Output T. Using the submodular property of the capacity function c, one has c ( X ) + c ( Y ) ≥ c ( X ∩ Y ) + c ( X ∪
Gomory–Hu_tree
Edges crossing all dicuts in a directed graph
MR 0499529 Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming (Proc. Workshop, Bonn
Dijoin
American/Canadian mathematician and computer scientist
Programming. 1: 127–136. doi:10.1007/BF01584082. Edmonds, Jack (1970). "Submodular functions, matroids, and certain polyhedra". In R. Guy; H. Hanam; N. Sauer;
Jack_Edmonds
Shared independent set of two matroids
Matroid partitioning - a related problem. Edmonds, Jack (1970), "Submodular functions, matroids, and certain polyhedra", in R. Guy; H. Hanam; N. Sauer;
Matroid_intersection
Operation in graph theory
cut-rank functions are pivot-equivalent, and so locally equivalent bipartite graphs are also pivot-equivalent. The cut-rank function is submodular since
Local_complementation
Mathematical and computational problem
equivalent to a submodular bin packing problem, in which the "load" in each bin is not equal to the sum of items, but to a certain submodular function of it. In
Bin_packing_problem
MR 4915164 Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming (Proc. Workshop, Bonn
Woodall's_conjecture
Class of statistical modeling methods
HMMs. If the CRF only contains pair-wise potentials and the energy is submodular, combinatorial min cut/max flow algorithms yield exact solutions. If exact
Conditional_random_field
Methodology for creation of markets
preferences: Goods are substitutes if and only if the indirect utility function is submodular. Ausubel and Milgrom (2006a, 2006b) exposit and elaborate on these
Market_design
Utility function used in economic equations
in which the budget is infinite. Every budget-additive valuation is a submodular valuation. Garg, Jugal; Hoefer, Martin; Mehlhorn, Kurt (January 2018)
Budget-additive_valuation
Vertex partition in a directed graph
MR 0499529 Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming (Proc. Workshop, Bonn
Dicut
Criterion for evaluating fairness of electoral systems
candidate rule. PJR+ can be verified in polynomial time by reduction to submodular optimization - in contrast to PJR which is coNP-hard to verify. EJR+ can
Justified_representation
Matroid that can be represented over all fields
matroid through an independence oracle. Fujishige, Satoru (2005), Submodular Functions and Optimization, Annals of Discrete Mathematics, Elsevier, p. 24
Regular_matroid
or Topkis, D. M. (1979): “Equilibrium Points in Nonzero-Sum n-Person Submodular Games,” SIAM Journal of Control and Optimization, 17, 773–787. Quah, J
Monotone_comparative_statics
Fair item allocation problem
are matroid rank functions (i.e., submodular with binary marginals), the set of absolute-leximin allocations is equivalent to the set of max-product allocations;
Egalitarian_item_allocation
Criterion of fair item allocation
independent sets of a partition matroid. Barman and Biswas present an algorithm reducing the problem to a problem with no constraints but with submodular valuations
Maximin_share
Graph representing faces of another graph
MR 1857074. Gabow, Harold N. (1995), "Centroids, representations, and submodular flows", Journal of Algorithms, 18 (3): 586–628, doi:10.1006/jagm.1995
Dual_graph
buyers, and on the type of auction used for each individual item. Case 1: submodular buyers, second-price auctions, complete information: There exists a pure
Price_of_anarchy_in_auctions
Thought experiments
See: Topkis, D. M. (1979): “Equilibrium Points in Nonzero-Sum n-Person Submodular Games,” SIAM Journal of Control and Optimization, 17, 773–787; as well
Comparative_statics
solution is not necessarily EF1; but if the agents' utilities are at least submodular, the max-product solution satisfies a weaker property called Marginal-Envy-Freeness
Efficient approximately fair item allocation
Efficient_approximately_fair_item_allocation
Theory in microeconomics
{\displaystyle u} continuous, non-negative, non-decreasing, symmetric and submodular. Studying optimal search from a given distribution of prices led economists
Search_theory
entropic function is a concept arising in information theory. It represents the possible values of Shannon's information entropy that subsets of one set of
Entropic_vector
Fair division problem for discrete items
{\displaystyle 1-{\tfrac {1}{e}}} even when all agents have the same submodular utility function. Algorithm: Kawase and Sumita present an algorithm that, given
Fair_item_allocation
American mathematician (1924–2021)
paper on this topic "On greedy algorithms, partially ordered sets and submodular functions," co-authored with Dietrich, appeared in 2003. Hoffman visited
Alan_J._Hoffman
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