Alternate Parameterizations

Motivation for Parameterized
Complexity
1

Complexity
• Inputs in practice have nice “structure” (read: parameters
beyond just the input size) that can be exploited by
algorithms.
1

Complexity
algorithms.
• This could possibly explain why NP-hard problems (say
SAT) are sometimes solvable well “in practice”.
(Ongoing Program)
1

Complexity
algorithms.
(Ongoing Program)
• Type Checking of ML programs can be done well in practice
though the problem is EXP-hard;
1

Complexity
algorithms.
(Ongoing Program)
though the problem is EXP-hard; One explanation: O(2kn)
time where k is the alternation depth of the program.
1

Complexity
algorithms.
(Ongoing Program)
• However, much of early work on PC (vertex cover, feedback
vertex set – undirected and directed, longest path/cycle,
steiner tree, ..) used the solution size as the parameter.
1

Complexity
algorithms.
(Ongoing Program)
• However, much of early work on PC (vertex cover, feedback
vertex set – undirected and directed, longest path/cycle,
steiner tree, ..) used the solution size as the parameter.
• One possible exception – Treewidth
1

This talk
• Some results and open problems on Alternate
Parameterizations
2

This talk
Parameterizations (Not a comprehensive survey)
2

This talk
Parameterizations (Not a comprehensive survey)
• Parameters are some functions of input or output, but not
JUST output size.
2

Alternate Parameterizations
or
Ecology of Parameters
Venkatesh Raman
The Institute of Mathematical Sciences, Chennai, India.
NMI Workshop on Cryptography and Complexity,
IIT Gandhinagar, November 5, 2016
3

Outline
1 Input Parameterizations
Forest + k vertices
Bipartite + k vertices?
Another way to measure closeness to bipartite graphs
Other Highlights/Results/Open Problems
Parameter Ecology
Input parameterizations within P
2 Output Parameterizations
Motivation
Above guarantee parameterization
Below Guarantee Parameterization
Above guarantee parameterization within P
4

Forest + k vertices
• In trees, and more generally, in forests, VERTEX COVER,
FEEDBACK VERTEX SET, DOMINATING SET, COLORING are all
solvable in polynomial time.
5

Forest + k vertices
• What about in a graph that is a forest + k vertices? Are
these problems FPT when parameterized by k?
5

Forest + k vertices
• Note that k is the size of feedback vertex set,
5

Forest + k vertices
• Note that k is the size of feedback vertex set, and we are
parameterizing these problems by the size of feedback
vertex set of the input graph.
5

VC, MIS, FVS, DomSet, Coloring,
Clique in Forest + k vertices
Deﬁnition
• Input: A graph G and a FVS S of G of size k, an integer
• Parameter: k
• Question: Does G have a VC/DomSet/FVS/Chromatic
number of size at most or IS of size at least ? (Or
essentially solve the optimization problems.)
6

Deﬁnition
• Parameter: k
Theorem: FVS parameterized by FVS is FPT.
6

Deﬁnition
• Parameter: k
Proof:
6

Deﬁnition
• Parameter: k
Proof:
• FVS If ≥ k, then YES,
6

Deﬁnition
• Parameter: k
Proof:
• FVS If ≥ k, then YES,
else ≤ k, apply the O∗(5 ) algorithm, that is O∗(5k).
6

Vertex Cover in Forest + k vertices
S
Y
As Red vertices from S are not picked into solution, the
Red vertices from V S are forced to be picked into solution.
7

S
Y
• Guess the intersection Y of solution with S, delete Y,
7

S
Y
• Pick N(S Y) ∩ (V S) into the solution, delete them
7

S
Y
• Solve the (optimum) VC problem in the remaining forest.
7

VC in F + k vertices where F is a class
where VC is in P
8

VC in F + k vertices where F is a class
where VC is in P
• Solve the (optimum) VC problem in the remaining graph in
F.
S
Y
8

Dominating Set, Chromatic Number in
Forest + k vertices
9

Forest + k vertices
Theorem They are FPT as G has treewidth at most k + 1, and
DomSet, Coloring are FPT parameterized by treewidth
9

Forest + k vertices
• Forest + k edges?
9

Forest + k vertices
• Forest + k edges? Similar results go through as FAS of size
at most k implies FVS of size at most 2k.
9

Kernelization for VC in F+k graphs
where F is a class where VC is in P
10

Kernelization question tends to be much harder.
10

VERTEX COVER in F + k graph
• has an O(k3) kernel when parameterized by FVS (FJR
2013).
• has an O(k5) vertex kernel if F is the class of degree at
most 2 graphs (MRS IPEC 2015);
• has an O(k12) vertex kernel if F is the class of psuedoforests
(each component has at most one cycle) (FS IPEC 2016);
• has an O(kd) vertex kernel if F is the class of bounded
cluster graphs (each component is a cluster of size at most
d); (MRS IPEC 2015)
• but has no polynomial kernel (under complexity
conditions) if F is a collection of all graphs where each
component is a (unbounded size) clique. (FJR 2013)
10

VERTEX COVER in F + k graph
• has an O(k3) kernel when parameterized by FVS (FJR
2013).
• has an O(k5) vertex kernel if F is the class of degree at
most 2 graphs (MRS IPEC 2015);
• has an O(k12) vertex kernel if F is the class of psuedoforests
(each component has at most one cycle) (FS IPEC 2016);
• has an O(kd) vertex kernel if F is the class of bounded
cluster graphs (each component is a cluster of size at most
d); (MRS IPEC 2015)
• but has no polynomial kernel (under complexity
conditions) if F is a collection of all graphs where each
component is a (unbounded size) clique. (FJR 2013)
• A dichotomy result for kernels here? OPEN
10

Parameterized by the size of Odd Cycle
Transversal (I.e. Bipartite + k vertices)
11

• Domset, FVS are hard even for constant k (as they are
NP-hard in bipartite graphs).
11

• For VC, we have seen it FPT. There is a randomized
polynomial sized kernel.
11

• 3-Coloring
11

• 3-Coloring is NP-hard in graphs that are Bipartite + 2
vertices and Bipartite + 3 edges (Cai 2003).
11

Parameterized by the length of the
longest odd cycle
12

longest odd cycle
Another way to measure “closeness” to bipartite graphs.
12

longest odd cycle
• Input: A graph G, k is the length of the longest cycle in G,
an integer
• Parameter: k
• Question: Does G have a VC/MIS/Chromatic number of
size at most or IS of size at least ?
12

longest odd cycle
an integer
• Parameter: k
• MIS can be solved in time nO(k) – Hsu, Ikura and
Nemhauser, Math. Programming, 1981
• MAX-CUT can be solved in time nO(k) – Grötschel and
12

longest odd cycle
an integer
• Parameter: k
• MIS can be solved in time nO(k) – Hsu, Ikura and
• MAX-CUT can be solved in time nO(k) – Grötschel and
• MIS, Max-Cut, Coloring can be solved in time 2O(k)nO(1).
(Rai and Panolan – 2012)
12

Other Highlights/Results/Open
Problems
13

Problems
• VC param by degree 3 modulator: Para-NP-Hard.
13

Problems
• FVS param by degree 3 modulator: Open
13

Problems
• CLIQUE param by VC: (i.e. Clique in Independent Set + k
vertices)
At most one vertex from V S can participate in the solution.
S
13

Problems
• CLIQUE param by VC: (i.e. Clique in Independent Set + k
vertices)
At most one vertex from V S can participate in the solution.
S
FPT but has no polynomial kernel unless NP ⊆ coNP/poly.
13

Parameter Ecology
(ﬁgure from Bart Jansen’s thesis)
Vertex Cover Max Leaf
Distance to
Linear Forest
Genus
Distance
to a Clique
Feedback
Vertex Set
Cutwidth Bandwidth
Topological
Bandwidth
Distance
to Chordal
Distance to
Outerplanar
Pathwidth
Odd Cycle
Transversal
Treewidth
Distance
to Perfect
Chromatic
Number
`2O(h2
)[72,84]
[39]
`  h
1
` 2 O(h2
) [84]
h +
1
`[8]
` 2 O(
p h)
[118]
`  h  `2
[29, 30]
[8]h+
2
`
h + 2
`
`  h + 1 [8]
Figure 1: A hierarchy of parameters, with larger parameters drawn higher. An unlabeled line
between two parameters means that the parameter drawn lowest is never larger than the one
drawn highest. These unlabeled relationships (cf. [109]) follow from inclusions between graph
classes [18] or bounds which can be found in Bodlaender’s survey on treewidth [8]. When a
line between parameters is labeled by a bound, the lower-drawn parameter is represented by `
and the higher-drawn parameter by h.
14

The general ecology program
15

• If a parameter is W-hard (or has no polykernel under
complexity theoretic assumptions) under some
parameterization, try a larger parameter.
15

• If a parameter is FPT (or has a polykernel) under some
parameterization, try a smaller parameter.
15

• If a parameter is FPT (or has a polykernel) under some
parameterization, try a smaller parameter.
Helps understand the role of parameters in the complexity of
the problem.
15

16

LINEAR PROGRAMMING (LP)
16

• LP in d-variables and n-constraints — nO(d) – Simplex
16

• 22O(d)
n – Megiddo 1984 (JACM)
16

• 22O(d)
• d2n + 2O(
√
d log d)
– randomized – Combination of Clarkson
(JACM 1995)/ Kalai (STOC 1992) / Matoušek, Sharir, and
Welzl (Algorithmica 1996)
16

• 22O(d)
• d2n + 2O(
√
d log d)
– randomized – Combination of Clarkson
(JACM 1995)/ Kalai (STOC 1992) / Matoušek, Sharir, and
Welzl (Algorithmica 1996)
•
√
d
O(d)
(log d)3dn – deterministic – Chan (SODA 16)
16

LP Developments
simplex method det. O(n/d)d/2+O(1)
Megiddo [24] det. 2O(2d
)
n
Clarkson [9]/Dyer [14] det. 3d2
n
Dyer and Frieze [15] rand. O(d)3d
(log d)d
n
Clarkson [10] rand. d2
n + O(d)d/2+O(1)
log n + d4
√
n log n
Seidel [26] rand. d!n
Kalai [19]/Matouˇsek, Sharir, and Welzl [23] rand. min{d2
2d
n, e2
√
d ln(n/
√
d)+O(
√
d+log n)
}
combination of [10] and [19, 23] rand. d2
n + 2O(
√
d log d)
Hansen and Zwick [18] rand. 2O(
√
d log((n−d)/d))
n
Agarwal, Sharir, and Toledo [4] det. O(d)10d
(log d)2d
n
Chazelle and Matouˇsek [8] det. O(d)7d
(log d)d
n
Br¨onnimann, Chazelle, and Matouˇsek [5] det. O(d)5d
(log d)d
n
this paper det. O(d)d/2
(log d)3d
n
Table 1: Deterministic and randomized time bounds for linear programming on the real RAM.
2. Clarkson’s paper [10] described a second sampling
algorithm, based on iterative reweighting (or in
more trendy terms, the multiplicative weights update
method). This second algorithm has expected n log n
running time in terms of n, but has better depen-
Given a set H of n hyperplanes in Rd
, ﬁnd a
p that lies on or below all hyperplanes in H,
minimizing a given linear objective function.
All our algorithms rely on the concept of ε
Table taken from the SODA 16 paper of Chan
17

Recent work
• Longest path in interval graphs can be solved in O(n4) time,
but in O(k9n) time if the interval graph is a proper interval
graph + k vertices (GMN IPEC 2015).
18

Recent work
• Diameter in treewidth k graphs can be found in
2O(k lg k)n1+o(1) time, but can’t be solved in 2o(k)n2−ε time
under some plausible assumptions (AWW SODA 2016).
18

Recent work
• Diameter in treewidth k graphs can be found in
2O(k lg k)n1+o(1) time, but can’t be solved in 2o(k)n2−ε time
under some plausible assumptions (AWW SODA 2016).
• O(k3n lg2
n) randomized algorithm to ﬁnd a maximum
matching in a graph of treewidth k (earlier there was a
O(2kn) algorithm) (FLPSW SODA 2017).
18

Some Surveys
• Towards fully multivariate algorithmics: Parameter ecology
and deconstruction of computational complexity, (Fellows,
Jansen and Rosamond, European Journal of Combinatorics
2013)
• Parameter Ecology for Feedback Vertex Set, (Jansen, R.
Vatshelle, Tsingua Science and Technology Journal on
information sciences, 2014).
• Parameterized SAT (Szeider, Encyclopedia of Algorithms
2016).
• Backdoors to Satisfaction (Gaspers and Szeider,
Multivariate Algorithmics Revolution and Beyond, 2012)
19

Output Parameterizations
(Parameters that are functions of output size)
20

Decision versions of some NP-hard
problems
21

problems
VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices and m edges, that has a
perfect matching.
• Question: Does G have a vertex cover of size at most k?
21

problems
perfect matching.
Max 3-SAT (k is the parameter)
• Input: A 3-CNF formula F on n variables and m clauses.
• Question: Does F have an assignment satisfying at least k
clauses?
21

problems
perfect matching.
clauses?
MaxCut (k is the parameter)
• Input: A graph G on n vertices and m edges
• Question: Does G have a cut on at least k edges?
21

problems
perfect matching.
clauses?
MaxCut (k is the parameter)
• Question: Does G have a cut on at least k edges?
Planar MIS (k is the parameter)
• Input: A planar graph G on n vertices and m edges.
• Question: Does G have an independent set of size at least
k?
21

Trivial linear kernels and hence trivial
FPT algorithms
22

FPT algorithms
Max-3SAT If k ≤ m/2, then YES, else m ≤ 2k (as every CNF sat on m
clauses has a satisfying assignment with at least m/2
clauses)
22

FPT algorithms
clauses)
MaxCUT If k ≤ m/2, then YES, else m ≤ 2k (as every graph has a cut
of size at least m/2)
22

FPT algorithms
clauses)
VC-PM If k ≤ n/2, then NO, else n ≤ 2k (as matching size is a
lower bound for vertex cover size)
22

FPT algorithms
clauses)
VC-PM If k ≤ n/2, then NO, else n ≤ 2k (as matching size is a
lower bound for vertex cover size)
lanar-MIS If k ≤ n/4, then YES, else n ≤ 4k (as a planar graph has an
independent set of size at least n/4)
22

What’s happening?
• The problems are trivial for small values of k (as there is a
large lower bound for their solution size),
23

What’s happening?
large lower bound for their solution size), and when brute
force algorithm is applied, k is large.
23

What’s happening?
• No new FPT algorithm
23

What’s happening?
• No new FPT algorithm
• So a better way to parameterize would be
23

24

AG Max 3-SAT (k is the parameter)
• Question: Does F have an assignment satisfying at least
m/2 + k clauses?
24

m/2 + k clauses?
AG Max Cut (k is the parameter)
• Question: Does G have a cut on at least m/2 + k edges?
24

m/2 + k clauses?
AG VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices with a perfect matching.
• Question: Does G have a vertex cover of size at most
n/2 + k?
24

m/2 + k clauses?
AG VC-perfect matching (k is the parameter)
• Input: A graph G on n vertices with a perfect matching.
• Question: Does G have a vertex cover of size at most
n/2 + k?
• Question: Does G have an IS of size at least n/4 + k?
24

Above guarantee parameterization -
FPT Results
AG-MAXSAT: It is FPT to determine whether there is an
assignment satisfying
• at least m/2 + k clauses in a CNF formula or
• at least k more than the expected solution of a random
assignment.
• at least k more than the number of variables
One can generalize some of these results for general CSPs.
25

Above guarantee MaxCUT
AG-MAXCUT: It is FPT to determine whether there is a cut of
size
• at least m/2 + k (MR 1999);
• at least m/2 + (n − 1)/4 + k (CJM 2011); also has a
polynomial kernel, FPT bound is optimal under ETH.
• at least n − 1 + k (??)
26

Above guarantee Vertex Cover
AGVC: It is FPT to determine whethere there is a VC of size
27

• at most MM + k; (LNRRS ACM TALG 2014)
27

• at most LPopt + k (LNRRS ACM TALG 2014)
27

• at most LPopt + k (LNRRS ACM TALG 2014)
• at most 2LPopt − MM + k: O(3k · nO(1)) (GP SODA 2015).
27

FPT or W-hard?
28

FPT or W-hard? Famously OPEN for several years.
28

FPT or W-hard? Famously OPEN for several years.
• In triangle free planar graphs, one can ﬁnd an independent
set of size n/3 + k in FPT time (ESA 2014).
28

Above guarantee parameterization - W
hardness Results
29

hardness Results
• Does a given connected graph have a 2-connected subgraph
with at most n + k edges? (note than n is a guaranteed
lower bound).
29

hardness Results
• Does a given connected graph have a 2-connected subgraph
with at most n + k edges? (note than n is a guaranteed
lower bound).
• It is NP-hard even for k = 0
29

Parameterizing from the extreme
• Input: A graph G
• Parameter: k
• Question: Does G have a vertex cover of size at most n − k?
30

• Parameter: k
This is W[1]-hard as it is the same as k-IS.
30

• Parameter: k
• Parameter: k
• Question: Does G have chromatic number at most n − k?
30

• Parameter: k
• Parameter: k
• Question: Does G have chromatic number at most n − k?
This is FPT.
30

Some Surveys
• Kernelization, constraint satisfaction problems
parameterized above average, Gregory Gutin, Encyclopedia
of Algorithms (2016).
• Kernelization, permutation CSPs parameterized above
average, Gregory Gutin, Encyclopedia of Algorithms
(2016).
• Constraint satisfaction problems parameterized above or
below tight bounds, a survey, Gregory Gutin and Anders
Yeo, in the Multivariate algorithmic revolution and beyond,
2012.
31

within P
Dominating set in tournaments
• Input: A tournament T on n vertices.
• Parameter: k
• Question: Does G have a dominating on k vertices?
32

within P
• Parameter: k
This is W[2]-hard (as in the case of general directed graphs).
32

within P
• Parameter: k
However, every tournament on n vertices has a dominating set
of size at least lg n and can be found in O(n2) time.
32

within P
• Parameter: k
• Parameter: k
• Question: Does G have a dominating on lg n − k vertices?
32

within P
• Parameter: k
• Parameter: k
• Question: Does G have a dominating on lg n − k vertices?
OPEN
32

Finding lg n + k sized dominating sets
• Parameter: k
• Question: Find a dominating set on lg n + k vertices?
33

• Parameter: k
Can be done in O(n2/k) time, and the bound is optimal.
33

• Parameter: k
Can be generalized similarly for distance d-dominating sets.
33

• Parameter: k
Theorem: Any dominating set has a distance 2-dominating set
of size 1 (called a king) that can be found in O(n3/2) time.
33

• Parameter: k
Theorem: Any dominating set has a distance 2-dominating set
of size 1 (called a king) that can be found in O(n3/2) time. We
can ﬁnd a 2-dominating set of size lg n + k in O((n3/2)/k) time
and the bound is optimal.
33

To Conclude
• We have come a long way in making researchers consider
parameterized complexity as a way to deal with
NP-completeness.
34

To Conclude
NP-completeness.
• Hopefully this talk will inspire researchers to investigate
problems (not necessarily NP-complete) through multiple
different parameterizations!
34

To Conclude
NP-completeness.
• Hopefully this talk will inspire researchers to investigate
problems (not necessarily NP-complete) through multiple
different parameterizations!
• Some general dichotomy theorems?
34

Thank You for your Attention
35

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