On Block-Structured Integer Programming and Its Applications

Chen, Lin

doi:10.1007/978-3-030-16194-1_7

Cited by 4 publications

(4 citation statements)

References 19 publications

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“…As shown explicitly by Koutecký et al [26] and somewhat implicitly by Eisenbrand et al [9], ILP Optimization can be solved in fixed-parameter time when parameterized by A ∞ and td D (A). For more detailed discussion of algorithmic implications and theory of block-structured integer programs we refer the reader to a recent survey [6].…”

Section: Parameterization By the Dual Treedepthmentioning

confidence: 99%

“…Proof of Lemma 11. We proceed by induction on the number of rows of A using the recursive definition (6). For the base case-when A has one row-we may use the following well-known bound.…”

Section: Lemma 11 For Any Matrix a With Integer Entries It Holds Thatmentioning

confidence: 99%

“…Suppose then that A is not block-decomposable. By (6), there exists a row a of A such that td D (A\a ) = td D (A) − 1. Then, by Claim 13 and the inductive assumption, we have…”

Section: Lemma 11 For Any Matrix a With Integer Entries It Holds Thatmentioning

confidence: 99%

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Tight complexity lower bounds for integer linear programming with few constraints

Knop¹,

Pilipczuk²,

Wrochna³

2018

Preprint

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We consider the standard ILP Feasibility problem: given an integer linear program of the form {Ax = b, x 0}, where A is an integer matrix with k rows and columns, x is a vector of variables, and b is a vector of k integers, we ask whether there exists x ∈ N that satisfies Ax = b. Each row of A specifies one linear constraint on x; our goal is to study the complexity of ILP Feasibility when both k, the number of constraints, and A ∞, the largest absolute value of an entry in A, are small.Papadimitriou [32] was the first to give a fixed-parameter algorithm for ILP Feasibility under parameterization by the number of constraints that runs in time ((. This was very recently improved by Eisenbrand and Weismantel [11], who used the Steinitz lemma to design an algorithm with running time (k∞ , which was subsequently improved by Jansen and Rohwedder [19] We prove that for {0, 1}-matrices A, the running time of the algorithm of Eisenbrand and Weismantel is probably optimal: an algorithm with running time k) would contradict the Exponential Time Hypothesis (ETH).This improves previous non-tight lower bounds of Fomin et al. [12].We then consider integer linear programs that may have many constraints, but they need to be structured in a "shallow" way. Precisely, we consider the parameter dual treedepth of the matrix A, denoted tdD(A), which is the treedepth of the graph over the rows of A, where two rows are adjacent if in some column they simultaneously contain a non-zero entry. It was recently shown by Koutecký et al. [26] that ILP Feasibility can be solved in time 1) . We present a streamlined proof of this fact and prove that, again, this running time is probably optimal: even assuming that all entries of A and b are in {−1, 0, 1}, the existence of an algorithm with running time 2 2 o(td D (A))• (k + ) O(1) would contradict the ETH.

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Section: Parameterization By the Dual Treedepthmentioning

confidence: 99%

“…Proof of Lemma 11. We proceed by induction on the number of rows of A using the recursive definition (6). For the base case-when A has one row-we may use the following well-known bound.…”

Section: Lemma 11 For Any Matrix a With Integer Entries It Holds Thatmentioning

confidence: 99%

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Tight complexity lower bounds for integer linear programming with few constraints

Knop¹,

Pilipczuk²,

Wrochna³

2018

Preprint

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“…Ganian et al [18] and Eiben et al [11] considered treewidth and cliquewidth instead, assuming additional bounds on coefficients and variable domains. For more detailed discussion of algorithmic implications and theory of block-structured integer programs, we refer the reader to a recent survey [7].…”

Section: Parameterization By the Dual Treedepthmentioning

confidence: 99%

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

Knop

Pilipczuk

Wrochna

2020

ACM Trans. Comput. Theory

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• (k + + log b ∞) O(1). We present a streamlined proof of this fact and prove that, again, this running time is probably optimal: even assuming that all entries of A and b are in {−1, 0, 1}, the existence of an algorithm with running time 2 2 o (td D (A)) • (k +) O(1) would contradict the exponential time hypothesis. CCS Concepts: • Theory of computation → Fixed parameter tractability; Integer programming; This work is a part of projects CUTACOMBS, PowAlgDO (M. Wrochna), and TOTAL (M. Pilipczuk), which have received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreements 714704, 714532, and 677651, respectively). The authors acknowledge the support of the OP VVV MEYS funded project CZ.

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The complexity landscape of decompositional parameters for ILP: Programs with few global variables and constraints

Dvořák

Eiben

Ganian

et al. 2021

Artificial Intelligence

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On Block-Structured Integer Programming and Its Applications

Cited by 4 publications

References 19 publications

Tight complexity lower bounds for integer linear programming with few constraints

Tight complexity lower bounds for integer linear programming with few constraints

Tight Complexity Lower Bounds for Integer Linear Programming with Few Constraints

The complexity landscape of decompositional parameters for ILP: Programs with few global variables and constraints

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