A Sieving Algorithm for Approximate Integer

Programming

Daniel Dadush, CWI

Integer Programming (IP)

variables, constraints

𝐾

𝑐

Classic Problem in Discrete Optimization:

Danzig, Fulkerson, Johnson `54Gomory `58, Lenstra `82, Kannan `87, …

Complexity & Algorithms

IP is NP-Complete (even with just binary variables).We do not expect polynomial time algorithms.

Question: What is worst case complexity of IP?

H. Lenstra `83: IP can be solved in time

using space, where is the input length.

Kannan `87: Running time can be reduced to

Current Fastest: time, space [D. `12].

min s.t.

Complexity & Algorithms

H. Lenstra `83: IP can be solved in time

using space, where is the input length.

Kannan `87: Running time can be reduced to

Developed tools for lattice problems which have found applications in Cryptanalysis, Cryptography, Coding Theory, Number Theory, ….

min s.t.

Complexity & Algorithms

Integer Programming Feasibility

0

𝐾

ℤ𝑛

Find feasible solution to or decide that no solution exists. (continuous relaxation)

𝑦

0

𝐾

ℤ𝑛

Find feasible solution to or decide that no solution exists.Total enumeration is “easy”

( time).

Binary vs General Integer

0

𝐾

ℤ𝑛

Find feasible solution to or decide that no solution exists.No easy analog of total enumeration.

𝑦

Binary vs General Integer

Binary vs General Integer

0

𝐾

ℤ𝑛

Find feasible solution to or decide that no solution exists.Need to adapt to the geometry of .

𝑦

0

𝐾

ℤ𝑛

Find feasible solution to or decide that no solution exists.Question: Is there a time algorithm for IP?

𝑦

Binary vs General Integer

Pick , . Find solution to or decide system is infeasible for .

0

𝐾

ℤ𝑛

𝑐

𝑦 ′(1+𝜖 ) scaling  of 𝐾 about𝑐

Relaxing the Model

Relaxing the Model

Pick , . Find solution to or decide system is infeasible for .

0

𝐾

ℤ𝑛

𝑦 ′(1+𝜖 ) scaling  of 𝐾 about𝑐

𝑐

Result: Approximate Feasibility

Algorithm: the center of mass of ,

either (1) finds in scaling of about c,

or (2) decides that is integer free,

using time, space and randomness.

𝐾 𝑐

𝑥

Result: Approximate Optimization

Algorithm: objective ,

either (1) decides that is integer free or

(2) finds in (“blowup” of ),

satisfying ,

using time, space, randomness.

𝐾𝑥

𝑣

Center of MassConvex body

Center of Mass:

Algorithm:

are iid uniform from [Dyer-Frieze-Kannan

89].

𝑐𝐾

𝑥1

𝑥3𝑥6

𝑥2𝑥5𝑥4

𝑥7

Center of MassConvex body

Center of Mass:

Crucial Property [Milman-Pajor `00]:

(near symmetry)

𝐾𝑐

2𝑐−𝐾

Algorithm: Given . Using time and space(1)either finds , or

𝐾

Kinchine’s Flatness Theorem

𝑥

Algorithm: Given . Using time and space(1)either finds , or(2)find satisfying

𝐾

ytx=0 ytx=1 ytx=2

𝑦

Kinchine’s Flatness Theorem

[Kinchine 48, Babai 86, Lenstra-Lagarias-Schnorr 87, Hastad 88, Kannan-L0vasz 88, Banasczyk 96, Rudelson 00]

Best known bounds for Flatness Constants:1. Ellipsoids: 2. General Bodies:

[Kinchine 48, Lenstra-Lagarias-Schnorr 87, Kannan-L0vasz 88, Banasczyk et al 99, Rudelson 00]

𝐾𝐸

Kinchine’s Flatness Theorem

Symmetric convex body (.

-norm:

0

𝑥𝐾

𝑠𝐾

-

1. (triangle inequality) 2. (homogeneity)3. (symmetry)

is unit ball of

Norms and Convex Bodies

Convex body containing origin in its interior.

-norm: 𝑥

𝐾𝑠𝐾

Norms and Convex Bodies

0

1. (triangle inequality) 2. (homogeneity)3. (symmetry)

is unit ball of

Convex body containing origin in its interior.

-norm: 𝑥

0𝐾

𝑠𝐾

Norms and Convex Bodies

1. (triangle inequality) 2. (homogeneity)3. (near-symmetry)

is unit ball of

Shortest Vector Problem (SVP)Given: norm in .Goal: Find minimizing .

-

𝑦

0𝐾

ℤ𝑛

Closest Vector Problem (CVP)Given: target , norm in .Goal: Find minimizing .

𝑦𝑥𝐾

ℤ𝑛

Main Tools

Algorithms: Near symmetric norm .

(1) Finds shortest non-zero integer vector under

using time, space, and randomness.

𝐾0𝑦

Main Tools

Algorithms: Near symmetric norm .

(1) Finds shortest non-zero integer vector under

using time, space, and randomness.

(2) , . Finds -approximate closest integer vector

to t under using “…”.

𝐾 𝑡

𝑦

Algorithm:1. Estimate center of mass of (via uniform

sampling).2. Solve -approximate Closest Vector Problem

with target under (near symmetric).3. If , return y.

Else, return .

Approx. IP Approx. CVP

ℤ𝑛𝐾𝑦

𝑐

The Randomized Sieve

General Idea: Sample exponentially many “perturbed” integer points, combine them to get closer & closer (shorter & shorter) integer vectors.

Ajtai-Kumar-Sivakumar `0o+`01: Developed randomized sieving strategy for SVP and -CVP for the norm.

Blomer-Naewe `07: Refined and extended AKS approach to get SVP and -CVP for norms.

Arvind-Joglekar `09: SVP for symmetric norms.

This paper: SVP & -CVP for near symmetric norms.

Recent work[D. Vempala `12, D. `12]: Deterministic time, space algorithm for (1+ CVP.

Conclusions1. Presented approximate IP model and gave

single exponential time algorithm to solve it.

2. Generalized the AKS randomized sieve to nearly all norms. Open Problems

1. Achieve same time complexity using space?

2. time algorithm for IP?

Thank You!