Algorithms for hard
problems
Parameterized complexity –
definitions, sample algorithms
Juris Viksna, 2015
Parameterized complexity
Combinatorial "explosion" for NP-hard problems
[Adapted from R.Downey and M.Fellows]
Parameterized complexity
Parameterized complexity attempts to confine
combinatorial "explosion"
[Adapted from R.Downey and M.Fellows]
Parameterized complexity - Definitions
Definition (FPT)
A parametrized problem L  ** is Fixed
Parameter Tractable if there is an algorithm that
for input (x,y)  ** with |x| = k and |y| = n
decides whether (x,y)  L in time f(k) n  ,
where f is an arbitrary function and  is a constant.
Definition does not change if
f(k) n
is replaced by
f(k) + n (!)
Parameterized complexity - Definitions
Mk for every n solves the problem in time f(k) n 
For each k there is a constant ck, such that
f(k) n  > n  + 1 for n ck
M'k:
- simulates Mk for n ck
- looks up value from the table for n ck
M'k solves the problem in time f(k) ck + n a+1
Some parameterized problems
VERTEX COVER
Instance:
Parameter:
A graph G=(V,E)
A positive integer k
Question:
Is there a subset SV, such that |S|=k and for all
{x,y}E either xS or yS?
[Adapted from R.Downey and M.Fellows]
Some parameterized problems
GRAPH GENUS
Instance:
Parameter:
A graph G=(V,E)
A positive integer k
Question:
Does graph G have genus k?
[Adapted from R.Downey and M.Fellows]
Some parameterized problems
GRAPH LINKING NUMBER
Instance:
Parameter:
A graph G=(V,E)
A positive integer k
Question:
Can G be embedded in 3-space such that the
maximum size of a collection of topologically
linked disjoint cycles is bounded by k?
[Adapted from R.Downey and M.Fellows]
Some parameterized problems
VERTEX COVER:
A single algorithm G running in time 2k |G| for each k
[Downey and Fellows 1992]
GRAPH GENUS:
A single algorithm F, which for each fixed k determines
whether graph G has genus k in time O(|G|3)
[Fellows and Langston 1988]
GRAPH LINKING NUMBER:
For each k there is an algorithm k, which determines
whether graph G has linking number k in time O(|G|3)
[Fellows and Langston 1988]
Parameterized complexity - Definitions
A problem L is uniformly FPT, if there is an algorithm F,
a constant c and a function f, such that:
- the running time of F(‹x,k›) is at most f(k)|x|c
- ‹x,k›A iff F(‹x,k›) = 1.
A problem L is strongly uniformly FPT, if L is uniformly
FPT via some F and f, such that f is recursive.
A problem L is nonuniformly FPT if there is a constant c, a
function f and a collection of algorithms {Fk}kN, such that
for each k:
- the running time of Fk(‹x,k›) is at most f(k)|x|c
- ‹x,k›A iff Fk(‹x,k›) = 1.
"klam" values
Algorithms for VERTEX
Tower of 2's of height described by
tower of 2's of height described by
COVER: tower of 2's... (impractical even for k
= 1 :)
• O(f(k) n3)
[Fellows, Langston 1986]
• O(f(k) n2)
[Johnson, 1987]
500k
2
2
FPT if solvable in time f(k)+nc
klam value - the largest k for which
f(k) is less that some universal
"speed limit" U (e.g. U= 1020)
[Adapted from R.Downey and M.Fellows]
Parameterized complexity
• Parametrized tractability
(methods for constructing FPT
algorithms)
• Parametrized intractability
(how to "prove" that there are no
FPT algorithm for a given problem)
Bounded search tree methods
• First, construct a search space (tree), such that the size
search space depends only from parameter k
• Then run some relatively efficient algorithm (say DFS)
on each branch of the tree
For fixed k search space is of constant size, thus we
obtain a FPT algorithm
Bounded search tree methods Vertex cover
, G
{u1}, G-{u1}
{u1,v1}
{v1}, G-{v1}
k+1
{u2,v2}
{v1,u2}, G-{v1,u2}
{v1,v2}, G-{v1,v2}
By considering only graphs with vertices of degree 3 or higher, it
is possible to shrink search space from 2k to (51/4)k
[Balasubramanian et al 1997]
Theorem [Downey and Fellows 1992]
VERTEX COVER is solvable in time O(2k |V(G)|).
Kernel methods
The main idea of the reduction to problem kernel is to
reduce a problem instance I to an equivalent instance I',
where the size of I' is bounded by some function of the
parameter k.
The instance I' is then exhaustively analyzed, and a
solution for I' lifted to a solution to I.
Usually this will give and additive rather that
multiplicative f(k) exponential factor
Kernel methods - Vertex cover
Theorem [Buss 1989]
VERTEX COVER is solvable in time O((2k)k + k |V(G)|).
Observation
Any vertex of degree greater than k must belong to every k-element
vertex cover.
Step 1
Locate all vertices in G of degree greater than k; let p equal the
number of such vertices. If p > k, there is no k-vertex cover.
Otherwise let k' = k – p.
Step 2
Discard all vertices found in step 1 and the edges incident to them.
If the resulting graph G' has more than k'(k+1) vertices, reject.
Kernel methods - Vertex cover
Observation
Any vertex of degree greater than k must belong to every k-element
vertex cover.
Graph with k' vertex cover and vertex degree
bounded by k has up to k'(k+1) vertices
Step 1
Locate all vertices in G of degree greater than k; let p equal the
number of such vertices. If p > k, there is no k-vertex cover.
Otherwise let k' = k – p.
Step 2
Discard all vertices found in step 1 and the edges incident to them.
If the resulting graph G' has more than k'(k+1) vertices, reject.
Doable in O((2k)k) time
Step 3
If G' has no k'-vertex cover, reject. Otherwise, any k'-vertex cover
of G' plus the p vertices from step 1 constitutes a k-vertex cover for G.
Kernel methods - Vertex cover
Kernel method
Search tree method
O((2k)k + k |V(G)|)
O(2k |V(G)|)
O((51/4)k |V(G)|)
If we use reduction to problem kernel first, and then
apply search tree method to problem kernel, we improve
additive bounds to:
O(kn + 2kk2)
[Balasubramanian et al 1992]
O(kn + ((51/4)kk2)
[Balasubramanian et al 1992]
Closest string problem
Closest string problem
Closest string problem