The Distributions of Functions Related to Parametric Integer Optimization

Oertel, Timm; Paat, Joseph; Weismantel, Robert

doi:10.1137/19m1275954

Cited by 12 publications

(5 citation statements)

References 33 publications

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“…Of course, an even more compelling extension would involve mixed -integer decision sets. This will require a deep appreciation of parametric mixed-integer linear programming, a topic that remains of keen interest in the integer programming community (see, for instance, Eisenbrand and Shmonin (2008), Oertel et al (2020), Gribanov et al (2020)). In this case, the integer programming theory necessary to study the diversity maximization problem is still being developed.…”

Section: Discussionmentioning

confidence: 99%

Optimizing for Strategy Diversity in the Design of Video Games

Hanguir¹,

Ma²,

Ryan³

2021

Preprint

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We consider the problem of designing video games (modeled here by choosing the structure of a linear program solved by players) so that players with different resources play diverse strategies. In particular, game designers hope to avoid scenarios where players use the same "weapons" or "tactics" even as they progress through the game. We model this design question as a choice over the constraint matrix A and cost vector c that seeks to maximize the number of possible supports of unique optimal solutions (what we call loadouts) of Linear Programs max{c x | Ax ≤ b, x ≥ 0} with nonnegative data considered over all resource vectors b.We provide an upper bound on the optimal number of loadouts and provide a family of constructions that have an asymptotically optimal number of loadouts. The upper bound is based on a connection between our problem and the study of triangulations of point sets arising from polyhedral combinatorics, and specifically the combinatorics of the cyclic polytope. Our asymptotically optimal construction also draws inspiration from the properties of the cyclic polytope. Our construction provides practical guidance to game designers seeking to offer a diversity of play for their plays.

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Section: Discussionmentioning

confidence: 99%

Optimizing for Strategy Diversity in the Design of Video Games

Hanguir¹,

Ma²,

Ryan³

2021

Preprint

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“…where dist ( A) measures distance using the • ∞ norm. The recent work of Oertel et al [14] considers a random model that allows b to vary but keeps A fixed. More precisely, for a given positive integer t, the vector b is chosen uniformly at random from {−T , .…”

Section: Related Workmentioning

confidence: 99%

Proximity bounds for random integer programs

Celaya

Henk

2022

Math. Program.

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We study proximity bounds within a natural model of random integer programs of the type $$\max \;\varvec{c}^{\top }\varvec{x}:\varvec{A}\varvec{x}=\varvec{b},\,\varvec{x}\in {\mathbb {Z}}_{\ge 0}$$ max c ⊤ x : A x = b , x ∈ Z ≥ 0 , where $$\varvec{A}\in {\mathbb {Z}}^{m\times n}$$ A ∈ Z m × n is of rank m, $$\varvec{b}\in {\mathbb {Z}}^{m}$$ b ∈ Z m and $$\varvec{c}\in {\mathbb {Z}}^{n}$$ c ∈ Z n . In particular, we seek bounds for proximity in terms of the parameter $$\Delta (\varvec{A})$$ Δ ( A ) , which is the square root of the determinant of the Gram matrix $$\varvec{A}\varvec{A}^{\top }$$ A A ⊤ of $$\varvec{A}$$ A . We prove that, up to constants depending on n and m, the proximity is “generally” bounded by $$\Delta (\varvec{A})^{1/(n-m)}$$ Δ ( A ) 1 / ( n - m ) , which is significantly better than the best deterministic bounds which are, again up to dimension constants, linear in $$\Delta (\varvec{A})$$ Δ ( A ) .

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“…Sparsity of solutions to linear Diophantine equations is relevant for the theory of compressed sensing for integer-valued signals [17,18,24], motivated by many applications in which the signal is known to have integer entries, for instance, in wireless communication [31] and in the theory of error-correcting codes [10]. Support minimization was also investigated in connection to integer optimization [2,16,29,30]. Also, numerous applications to combinatorial optimization problems have been explored.…”

Section: Introductionmentioning

confidence: 99%

Sparse representation of vectors in lattices and semigroups

et al. 2021

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We study the sparsity of the solutions to systems of linear Diophantine equations with and without non-negativity constraints. The sparsity of a solution vector is the number of its nonzero entries, which is referred to as the $$\ell _0$$ ℓ 0 -norm of the vector. Our main results are new improved bounds on the minimal $$\ell _0$$ ℓ 0 -norm of solutions to systems $$A\varvec{x}=\varvec{b}$$ A x = b , where $$A\in \mathbb {Z}^{m\times n}$$ A ∈ Z m × n , $${\varvec{b}}\in \mathbb {Z}^m$$ b ∈ Z m and $$\varvec{x}$$ x is either a general integer vector (lattice case) or a non-negative integer vector (semigroup case). In certain cases, we give polynomial time algorithms for computing solutions with $$\ell _0$$ ℓ 0 -norm satisfying the obtained bounds. We show that our bounds are tight. Our bounds can be seen as functions naturally generalizing the rank of a matrix over $$\mathbb {R}$$ R , to other subdomains such as $$\mathbb {Z}$$ Z . We show that these new rank-like functions are all NP-hard to compute in general, but polynomial-time computable for fixed number of variables.

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The Distributions of Functions Related to Parametric Integer Optimization

Cited by 12 publications

References 33 publications

Optimizing for Strategy Diversity in the Design of Video Games

Optimizing for Strategy Diversity in the Design of Video Games

Proximity bounds for random integer programs

Sparse representation of vectors in lattices and semigroups

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