Kernelization and the Larger Picture of Practical Algorithm - PowerPoint Presentation

Slide1

Kernelization and the Larger Picture of Practical Algorithmics, in Contemporary Context

Michael R. Fellows

Charles Darwin University

Australia

WorKer

, Vienna 2011Slide2

Two thoughts on parameterized complexity and theoretical computer science.

PC is as much about “workflow reform” as about “more fine-grained complexity analysis”

We want to create mathematical tools withExplanatoryPredictive Three kinds of powerEngineering To help us create useful algorithms.Slide3

A classic example of explanatory power:TYPE CHECKING in MLSlide4

Combinatorial optimization problems arise frequently in computational molecular biology …Except in rare cases, the problems are NP-hard, and the performance guarantees provided by polynomial-time approximation algorithms are far too pessimistic to be useful. Average-case analysis of algorithms is also of limited use because the spectrum of real-life problem instances is unlikely to be

representable

by a mathematically tractable probablility distribution. Thus it appears necessary to attack these problems using heuristic algorithms. Although we focus here on computational biology, heuristics are also likely to be the method of choice in many other application areas, for reasons similar to those that we have advanced in the case of biology. -Introduction to “Heuristic algorithms in computational molecular biology,” Richard M. Karp, JCSS 77 (2011) 122-128.Slide5

Karp’s proposed:General heuristic for Implicit Hitting Set problems.

Running example

: DIRECTED FEEDBACK VERTEX SETIn: Digraph DOut: A minimum cardinality set of vertices that “hit” all directed cycles.Explicit versus implicit Hitting Set ProblemsExplicit: List the things that need to be hit.Implicit:

The list is implicit in the digraph description (made explicit, the list might be exponential in size).Slide6

Assumed

Separation oracle

Find an unhit cycle if there is oneP-time algorithm for approx solution of the explicit hitting set problemAlgorithm for optimal solution of the explicit HS problem

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Γ : things to be hit (cycles)

Н

: a hitting set (vertices)Karp’s generic Hitting Set heuristic: Γ ← empty setRepeat:Using the approximation algorithm, construct a hitting set Н for Γ :

Using the separation oracle, attempt to find a circuit that H does not hit; If a circuit is found then add that circuit to Γ

else

Н

←

an optimal hitting set for

Γ

:

Using the separation oracle, attempt to find a circuit

Н

that does not hit;

If a circuit is found

then add the circuit to

Γ

:

else return

Н

and halt

Slide8

The intuition behind Karp’s general heuristic

Quickly identify a (hopefully) small set of important cycles to cover

If these are covered then “probably” all cycles are covered – reasonable to pay for optimal solution at this pointIf this fails, then (win/win) a new important cycle has been discoveredSlide9

Quickly identify a (hopefully) small set of important cycles to cover

What to call this?

“Strategic kernelization”in the space between “implicit” and “explicit” ?Slide10

EXPLICIT DFVS I

In

: digraph D, and a list L of directed cycles in DParameter: kQuestion: Is there a set of at most k vertices that hits every cycle on the list L? OOPS!

While IMPLICIT DFVS I is FPT,Thm: EXPLICIT DFVS I is W[1] – hard.Slide11

v

“V” selected

k– 1 of these

u

R to B

B to R

Backward adjacency test

k

vertex selection gadgets

Forward adjacency test

N(v)Slide12

EXPLICIT DFVS IIIn

: digraph D, list L of directed cycles in D, r

Parameter: | L | = kQuestion: Is there a set of at most r vertices that hits all cycles in L?Thm: This problem is FPTPf: (1) If r > k, then YES (2) r · 2

k dynamic programmingSlide13

Summary so Far

The design of “effective heuristics” is our inevitable primary mission for most problems, as theoretical computer scientists.

General strategic approaches to this task throw up many novel parameterized problems, largely unexplored, as subroutines. Slide14

Plan “B” – Two Principles

We do what we have been doing:

enriching the model when there is tractabilitydeconstructing the proofs when there is intractabilityand there is very very much to be done, for fun and profit.Slide15

Parameterized Algorithmics

Branch out

! To opportunity!Focus on the unvisited core problemsFind a mentor/collaborator/interpreter who is established in the areaReport on NAG and examples

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Stefan and Fran in AustraliaSlide17

Taking Our Own Advice II

A Report on the workshop: Not About Graphs

Darwin, Australia

August 5—8 and 9-13Slide18

Workshop Theme

The focus of the workshop is to investigate opportunities for expanding parameterized complexity into important unreached areas of algorithmic mathematical science (algebra, number theory, analysis, topology, geometry, game theory, robotics, vision, crypto, etc.) beyond areas where it already has a strong presence (graph theory, computational biology, AI, social choice, etc.). This may require new mathematical techniques. The workshop is also focused on identifying and promoting the key unsolved problems in these new directions. Slide19

According to Papadimitriou, every year, several thousand scientific papers use the words “NP-complete” or “NP-hard”.

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Example: Computational Logistics

Trains!

Regular meeting: ATMOS

NP-hard classic problemTRAIN MARSHALLINGIn: Partition Π of [n] Ex. {1, 3}, {2, 4, 5}Parameter

:

k

k

= 2

Question:

Is

k

enough?

1

2

3

|

4

5

·

1

2

3 4 5

YESSlide21

Example: Computational Geometry

Problem

! Most of the classic problems are in P.Not a problem! “enrich the model”In: A set of colored points in the plane.Parameter: kQuestion: Are k lines sufficient to dissect into monochrome regions?Good news: NP-hard!Slide22

Example: Computer Vision

SEGMENTATION

In: matrix of grey-scale valuesParameter: kQuestion: Can the matrix be segmented into < k regions?

3

4

1

2

4

3

2

1

2

4

3

1

3

2

4

3

1

2

4

3

2

1

2

3Slide23

QuestionShould we do this again next year in Germany?

Maybe…Gabor Erdelyi has offered to host.

Proposed acronym: DECONSlide24

Open ProblemHow does kernelization as we know it interact with real practical computing and

heurisitcs

?Slide25

Thank you