Parameterized Complexity of Even Set (and others)

Parameterized Complexity of Even Set (and others)

Dániel Marx

Hungarian Academy of Sciences Dagstuhl Seminar 19041

January 25, 2019

[1] Bingkai Lin, SODA 2015, JACM 2018 [2] Bonnet, Egri, Lin, M. , unpublished

[3] Bhattacharyya, Ghoshal, Kartik C.S., Manurangsi, ICALP 2018

L

BELM BGKM

BBEGKLMM

Hitting Set and Variants

MINIMUM EXACT

HITTING

UNIQUE HITTING ODD

EVEN

ODD/EVEN

{ SET

} } { }

{ × ×

All of them are known to be W[1]-hard, except MINIMUM EVEN SET (note: solution is required to be nonempty).

Dual form: SET COVER and friends.

EVEN SET definitions

• Hypergraphs (EVEN HITTING SET)

• Bipartite Graphs (RED-BLUE DOMINATING SET)

• Linear Algebra (MINIMUM DEPENDENT SET)

• Find a nonempty vector with Hamming weight at most such that .

• Matroid Theory (SHORTEST CIRCUIT):

• Find a circuit of size at most in a binary matroid

• Coding Theory (MINIMUM DISTANCE):

• Find a nonempty codeword of Hamming weight at most .

•

Coding theory

}

Generating matrix: ^{�∈ �}

h �

Codewords:

Fact:

distance, systematic code, Sphere packing bound,…

W[1]-hardness of ODD SET

�₁ �₂ �₃ �₄

�_1,2 �_1,3 �_1,4 �_2,3 �_2,4 �_3,4

Odd Set ? Even Set

Gap Odd Set Gap Even Set

Odd Set ? Even Set

Odd Set ? Even Set

Gap Odd Set Gap Even Set

BICLIQUE

• Given a bipartite graph and integer , find two sets and of size that are fully adjacent to each other.

• Should have been on the Downey-Fellows list.

• Easy to give an incorrect proof (which works only for PARTITIONED BICLIQUE)

• Theorem: BICLIQUE is W[1]-hard

[Bingkai Lin, SODA’15, JACM’18]

•

Template graph

-threshold property for a bipartite graph on and :

• For any vertices in have at most common neighbors.

• For any choice of sets , we can find one vertex in each set such that these vertices have at least common neighbors.

•

There are randomized constructions for such graphs with and (deterministic construction is worse).

The reduction

clique

vertices with common neighbors

no clique

Any vertices has common

neighbors

Hardness of BICLIQUE

• We reduced -CLIQUE to -BICLIQUE.

• Theorem: BICLIQUE is W[1]-hard and no algorithm assuming randomized ETH.

• Approximation version:

ONE SIDED BICLIQUE

Find vertices maximizing the common intersection

• Hard to distinguish between optimum of and

•

Randomized construction

• Random graph on vertices with edge probability

• Claim 1: probability of vertices with common vertices is at most .

• Claim 2: probability of a given set of vertices do not have common neighbors is less than .

•

LINEAR DEPENDENT SET

• Given vectors , find vectors that are linearly dependent.

s.t. has Hamming weight at most

• Hard to approximate for any constant: proof by reduction from ONE SIDED BICLIQUE.

•

Encoding vertices

Vertex set

vectors in such that

• any are linearly independent

• any are linearly dependent

Vertex set B

vectors in such that

• any are linearly independent

• any are linearly dependent (Vandermonde matrices)

•

Reduction

edge

�(�_�) block

�(�_�) block

REALLY!

REALLY!

F(e)

Reduction

biclique

no biclique

COLORED LINEAR DEPENDENT SET

Vectors partitioned into color classes, distinguish between

• exists dependent set of vectors, one from each color,

• no dependent set of vectors (or arbitrary colors!) Simple reduction from the uncolored version using

Color Coding

(Turing or not)

•

MAXIMUM LIKELIHOOD DECODING

Given and , find of Hamming weight at most such that .

= ODD/EVEN SET

Reduction from COLORED LINEAR DEPENDENT SET.

Tool: as -bit vectors.

•

Reduction

Colored dependent set

MLD solution of weight

no dependent set of size

No MLD solution of weight

NEAREST CODEWORD

Given and find such that has Hamming weight at most

• Sparse version: in a YES instance, exists solution of Hamming weight at most .

• Inapproximability by an easy reduction from MAXIMUM LIKELIHOOD DECODING.

•

Minimum Distance (finally!)

SPARSE NEAREST CODEWORD

Given , find such that has Hamming weight at most

•

MINIMUM DISTANCE

Given and , find of such that has Hamming weight at most .

Reduction

codeword at distance

Distance is at most

no codeword at distance

Distance is at least

Increasing the gap

Codes with matrices , the tensor product is the code }

Fact:

Any inapproximability for MINIMUM DISTANCE can be boosted to and hence to any constant.

Theorem: MINIMUM DISTANCE (EVEN SET etc.) is randomized W[1]-hard to approximate for any .

•

Integer Lattice problems

SHORTEST VECTOR

Given a matrix and integer , find a vector with .

NEAREST VECTOR

Given a matrix , vector and integer , find an with .

Theorem: SHORTEST VECTOR for any and NEAREST VECTOR for any is randomized W[1]-hard to approximate for any

constant .

•

Summary

CLIQUE

ONE SIDED BICLIQUE LINEAR DEPENDENT SET

MAXIMUM LIKELIHOOD DECODING (SPARSE) NEAREST CODEWORD

MINIMUM DISTANCE

•

Transferring inapproximability results can be easier than transferring exact hardness!