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In mathematics, a submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between...

In the theory of combinatorial optimization, submodular flow is a general class of optimization problems that includes as special cases the minimum-cost...

submodular optimization. For example, the set cover problem is a special case of submodular optimization, since the set cover function is submodular....

submodular agent has a utility function that is a submodular set function. This means that the agent's utility has decreasing marginals. Submodular utilities...

solution at least half the value of the optimal solution. Solutions for submodular maximization are approximated using a greedy algorithm which yields a...

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...

f} is submodular then QPBO produces a global optimum equivalently to graph cut optimization, while if f {\displaystyle f} contains non-submodular terms...

(strictly) supermodular then f is called (strictly) submodular. A function that is both submodular and supermodular is called modular. This corresponds...

non-negative, every submodular function is subadditive. However, there are non-negative subadditive functions that are not submodular. For example, assume...

German computer scientist whose research in machine learning includes submodular optimization in computer vision and deep learning for graph neural networks...

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...

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...

{\displaystyle S} ⁠. Every non-negative submodular set function is subadditive (the family of non-negative submodular functions is strictly contained in the...

function with a similar but submodular one, for instance truncating all non-submodular terms or replacing them with similar submodular expressions. Such approach...

that we say the cost game is convex if the characteristic function is submodular. Convex cooperative games have many nice properties: Supermodularity trivially...

of 3, 6, 12, 15, 30 and 60 basic modules. For smaller dimensions, the submodular increments 1⁄4 M and 1⁄2 M are preferred. The numbers 300 and 600 were...

the maximum cut problem as maximizing a pseudo-Boolean function. The submodular set functions can be viewed as a special class of pseudo-Boolean functions...

defined over sets, through a submodular set function which is known as the discrete analog of a convex function. The submodular Bregman divergences subsume...

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...

well-known integral LPs include the matching polytope, lattice polyhedra, submodular flow polyhedra, and the intersection of two generalized polymatroids/g-polymatroids...

axiomatized. Matroid rank functions form an important subclass of the submodular set functions. The rank functions of matroids defined from certain other...

ratio achieved by the generic greedy algorithm used for maximization of submodular functions with a cardinality constraint. The maximum coverage problem...

weighted graph can be found in polynomial time, and is a special case of the submodular flow problem. In planar graphs, dijoins and feedback arc sets are dual...

allocation of resources Self-organized criticality – Concept in physics Submodular set function – Set-to-real map with diminishing returns Sunk-cost fallacy –...

including knapsack problems, interval scheduling, and the optimization of submodular set functions. She is a professor of computer science at the Technion...

(B\setminus A)+2\mu (A\cap B).} However, the related properties of submodularity and subadditivity are not equivalent to each other. Note that modularity...

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...

multinomial logit (RMNL) Auto-encoding networks with a bottleneck-layer Submodular feature selection Local learning based feature selection. Compared with...

approximations are known for special cases, such as submodular valuations (this is called the "submodular welfare problem"). Some algorithms use only a value...

−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 )...

Lisa Fleischer, Satoru Fujishige, and Alexander Schrijver for showing submodular minimization to be strongly polynomial. 2006: Manindra Agrawal, Neeraj...

Donald M. (1979-11-01). "Equilibrium Points in Nonzero-Sum n -Person Submodular Games". SIAM Journal on Control and Optimization. 17 (6): 773–787. doi:10...

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...

made seminal contributions in duality theory for integer programming, submodular optimization, the group-theoretic approach and polyhedral analysis of...

due to 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...

with Satoru Iwata, Lisa Fleischer, and Satoru Fujishige) for showing submodular minimization to be strongly polynomial. He won the INFORMS Frederick W...

{\displaystyle 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...

several advantages: It works with arbitrary utility functions - not only submodular ones. It does not even require monotonicity of preferences. It works with...

if and only if it is both submodular and supermodular. Utility functions on indivisible goods Independent goods Submodular set function Supermodular set...

 223–246 Fujishige, Satoru (1984), "A System of Linear inequalities with a Submodular Function on (0, ±1) Vectors", Linear Algebra and Its Applications, 63:...

simple algorithm for 1/10-fraction MMS for the more challenging case of submodular valuations - based on round-robin item allocation. Ghodsi, Hajiaghayi...

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...

graphs from the point of view of matchings. He introduced polymatroids, submodular flows with Richard Giles, and the terms clutter and blocker in the study...

constant there is an FPTAS using Woeginger technique. For agents with submodular utility functions: Golovin gave an ( m − n + 1 ) {\displaystyle (m-n+1)}...

Theory of Greedy Algorithms Archived 2016-03-04 at the Wayback Machine Submodular Functions and Optimization Matchings, Matroids and Submodular Functions...

MR 1857074. Gabow, Harold N. (1995), "Centroids, representations, and submodular flows", Journal of Algorithms, 18 (3): 586–628, doi:10.1006/jagm.1995...

are approximations in special cases) EF1, and approximately n {\displaystyle n} -maximin-share PE With submodular utilities, allocation is PE and MEF1....

another name for submodular function of two discrete variables. Precisely, A is a Monge matrix if and only if A[i,j] is a submodular function of variables i...

al. (2011) Gabow, Harold N. (1995), "Centroids, representations, and submodular flows", Journal of Algorithms, 18 (3): 586–628, doi:10.1006/jagm.1995...

{\displaystyle u} continuous, non-negative, non-decreasing, symmetric and submodular. Studying optimal search from a given distribution of prices led economists...

Goods are substitutes if and only if the indirect utility function is submodular. Ausubel and Milgrom (2006a, 2006b) exposit and elaborate on these ideas...

HMMs. If the CRF only contains pair-wise potentials and the energy is submodular, combinatorial max-flow min-cut algorithms yield exact solutions. If exact...

(adding an element to a set can never decrease its rank) and it must be submodular, meaning that it obeys an inequality similar to the one for semimodular...

that is a distributive lattice. Now, if Φ {\displaystyle \Phi } is a submodular potential (i.e., a family of functions Φ Λ : S Λ ⟶ R ∪ { ∞ } , {\displaystyle...

( E ) + g ( F ) {\displaystyle g(E\cup F)+g(E\cap F)\geq g(E)+g(F)} ; submodular if for any E , F ∈ C {\displaystyle E,F\in {\mathcal {C}}} , we have g...

studies of approximation algorithms for facility location problems and for submodular optimization. Nemhauser, together with Leslie Trotter, showed in 1975...

MR 4915164 Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming (Proc. Workshop,...

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...

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...

{\displaystyle (\{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...

problem in the general case. For some families of cost functions (e.g. submodular functions) a solution with strong optimality properties can be found in...

u(A)=\max _{x\in A}u(\{x\})} A unit-demand function is an extreme case of a submodular set function. It is characteristic of items that are pure substitute goods...

in which the budget is infinite. Every budget-additive valuation is a submodular valuation. Garg, Jugal; Hoefer, Martin; Mehlhorn, Kurt (January 2018)...

S2CID 9065000 Kulik, Ariel; Shachnai, Hadas; Tamir, Tami (2009), "Maximizing submodular set functions subject to multiple linear constraints", in Mathieu, Claire...

x_{j}\partial x_{i}}}} is negative, that is, if Π {\displaystyle \,\Pi } is submodular. In their original paper, Bulow et al. use a simple model of competition...

MR 0499529 Edmonds, Jack; Giles, Rick (1977), "A min-max relation for submodular functions on graphs", Studies in integer programming (Proc. Workshop,...

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...

second-price auction) in each round. Case 4: submodular bidders. The bidders' valuations are arbitrary submodular set functions (note that additive and unit-demand...

a fixed budget. With Eric Balkanski, Singer developed algorithms for submodular optimization that evaluated multiple candidates in parallel, reducing...

approach allows for simpler optimization while maintaining sparsity. See: Submodular set function Besides the norms discussed above, other norms used in structured...

1515/9783110868647, ISBN 978-3-11-011213-9, MR 1292462 Zimmermann, Irene (1989), "Submodular subgroups in finite groups", Mathematische Zeitschrift, 202 (4): 545–557...

paper on this topic "On greedy algorithms, partially ordered sets and submodular functions," co-authored with Dietrich, appeared in 2003. Hoffman visited...

was initiated by Yaron Singer in 2010. Singer demonstrated that for any submodular set function, i.e. valuation function where the marginal value of an item...

Quinn, Christopher John (2022). "An Explore-then-Commit Algorithm for Submodular Maximization under Full-Bandit Feedback". Uncertainty in Artificial Intelligence...

948–957. doi:10.1137/0215066. ISSN 0097-5397. Edmonds, Jack (1970), "Submodular functions, matroids, and certain polyhedra", Combinatorial Structures...

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...

problem. Matroid partitioning - a related problem. Edmonds, Jack (1970), "Submodular functions, matroids, and certain polyhedra", in R. Guy; H. Hanam; N. Sauer;...

smaller 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...

neither is the result of the greedy algorithm. For non-negative monotone submodular set functions, such as GED-Walk centrality, this greedy algorithm is already...

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...

\dots ,n\}} . The Shannon inequality says that an entropic vector is submodular: 3 ) H ( X I ) + H ( X J ) ≥ H ( X I ∪ J ) + H ( X I ∩ J ) {\displaystyle...

but equivalences with definitions using circuits, matroid rank, and submodular set function are also presented, as are sums, minors, truncations, and...

MR 0258678 Gabow, Harold N. (1995), "Centroids, representations, and submodular flows", Journal of Algorithms, 18 (3): 586–628, doi:10.1006/jagm.1995...

ISBN 978-3-540-44180-9. Moulin, Hervé; Shenker, Scott (2001). "Strategyproof sharing of submodular costs:budget balance versus efficiency". Economic Theory. 18 (3): 511...

the matroid through an independence oracle. Fujishige, Satoru (2005), Submodular Functions and Optimization, Annals of Discrete Mathematics, Elsevier,...

bipartite graphs are also pivot-equivalent. The cut-rank function is submodular since it can be shown that ρ G ( X ) + ρ G ( Y ) ≥ ρ G ( X ∪ Y ) + ρ G...

non-equivalent measure theoretic definition of "valuation" that is given below. submodular if μ ( E ∪ F ) + μ ( E ∩ F ) ≤ μ ( E ) + μ ( F ) {\displaystyle \mu (E\cup...

{\displaystyle 1-{\tfrac {1}{e}}} even when all agents have the same submodular utility function. Algorithm: Kawase and Sumita present an algorithm that...

) {\displaystyle v(S)\leq \sum _{i=1}^{k}\alpha _{i}v(T_{i})} . Every submodular set function is XOS, and every XOS function is a subadditive set function...

Mathematics Genealogy Project. Frank, András (1998). "Applications of relaxed submodularity". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. III. pp. 343–354...

theory", as well as matchings in graphs, include incidence matrices, submodular set functions, independent matchings in matroids, the Birkhoff–von Neumann...