The (Parameterized) Complexity of Equitable Coloring

The (Parameterized) complexity of Equitable Coloring

Guilherme C. M. Gomes Matheus R. Guedes Vinicius F. dos Santos Carlos Vinicius G. C. Lima

Universidade Federal de Minas Gerais

Grafos e Combinatória - 1a temporada Equitable Coloring 1 / 34 Equitable Coloring

Equitable Coloring

Grafos e Combinatória - 1a temporada Equitable Coloring 2 / 34 Equitable Coloring Equitable (Vertex) Coloring

Can we k-color G such that the size of two color classes diﬀer by ≤ 1?

Grafos e Combinatória - 1a temporada Equitable Coloring 3 / 34 Equitable Coloring Equitable (Vertex) Coloring

Can we k-color G such that the size of two color classes diﬀer by ≤ 1?

Grafos e Combinatória - 1a temporada Equitable Coloring 3 / 34 Equitable Coloring Equitable (Vertex) Coloring

Can we k-color G such that the size of two color classes diﬀer by ≤ 1?

Grafos e Combinatória - 1a temporada Equitable Coloring 3 / 34 Equitable Coloring Equitable (Vertex) Coloring

Can we k-color G such that the size of two color classes diﬀer by ≤ 1?

Grafos e Combinatória - 1a temporada Equitable Coloring 3 / 34 Equitable Coloring Applications

• Municipal garbage collection; • Processor task scheduling; • Communication control; • Server load balancing.

Grafos e Combinatória - 1a temporada Equitable Coloring 4 / 34 Equitable Coloring Conjecture For every connected graph G which is neither a complete graph nor an odd-hole, χ=(G) ≤ ∆(G).

Equitable Coloring Some important stuﬀ I

Equitable chromatic number The smallest integer k such that G is equitably k-colorable is the equitable chromatic number χ=(G).

Grafos e Combinatória - 1a temporada Equitable Coloring 5 / 34 Equitable Coloring Some important stuﬀ I

Equitable chromatic number The smallest integer k such that G is equitably k-colorable is the equitable chromatic number χ=(G).

Equitable Coloring Conjecture For every connected graph G which is neither a complete graph nor an odd-hole, χ=(G) ≤ ∆(G).

Grafos e Combinatória - 1a temporada Equitable Coloring 5 / 34 Hajnal-Szmerédi Theorem Any graph G is equitably k-colorable if k ≥ ∆(G) + 1. Equivalently, ∗ χ=(G) ≤ ∆(G) + 1.

Equitable ∆ Coloring Conjecture For every connected graph G which is not a complete graph, an odd-hole ∗ nor K2n+1,2n+1, for any n ≥ 1, χ=(G) ≤ ∆(G) holds.

Ko-Wei Lih. “Equitable coloring of graphs”. In: Handbook of combinatorial optimization. Springer, 2013, pp. 1199–1248

Equitable Coloring Some important stuﬀ II

Equitable chromatic threshold The smallest integer k such that for every k0 ≥ k, G is equitably 0 ∗ k -colorable is its equitable chromatic threshold χ=(G).

Grafos e Combinatória - 1a temporada Equitable Coloring 6 / 34 Equitable ∆ Coloring Conjecture For every connected graph G which is not a complete graph, an odd-hole ∗ nor K2n+1,2n+1, for any n ≥ 1, χ=(G) ≤ ∆(G) holds.

Ko-Wei Lih. “Equitable coloring of graphs”. In: Handbook of combinatorial optimization. Springer, 2013, pp. 1199–1248

Equitable Coloring Some important stuﬀ II

Equitable chromatic threshold The smallest integer k such that for every k0 ≥ k, G is equitably 0 ∗ k -colorable is its equitable chromatic threshold χ=(G).

Hajnal-Szmerédi Theorem Any graph G is equitably k-colorable if k ≥ ∆(G) + 1. Equivalently, ∗ χ=(G) ≤ ∆(G) + 1.

Grafos e Combinatória - 1a temporada Equitable Coloring 6 / 34 Ko-Wei Lih. “Equitable coloring of graphs”. In: Handbook of combinatorial optimization. Springer, 2013, pp. 1199–1248

Equitable Coloring Some important stuﬀ II

Equitable chromatic threshold The smallest integer k such that for every k0 ≥ k, G is equitably 0 ∗ k -colorable is its equitable chromatic threshold χ=(G).

Hajnal-Szmerédi Theorem Any graph G is equitably k-colorable if k ≥ ∆(G) + 1. Equivalently, ∗ χ=(G) ≤ ∆(G) + 1.

Equitable ∆ Coloring Conjecture For every connected graph G which is not a complete graph, an odd-hole ∗ nor K2n+1,2n+1, for any n ≥ 1, χ=(G) ≤ ∆(G) holds.

Grafos e Combinatória - 1a temporada Equitable Coloring 6 / 34 Equitable Coloring Some important stuﬀ II

Equitable chromatic threshold The smallest integer k such that for every k0 ≥ k, G is equitably 0 ∗ k -colorable is its equitable chromatic threshold χ=(G).

Hajnal-Szmerédi Theorem Any graph G is equitably k-colorable if k ≥ ∆(G) + 1. Equivalently, ∗ χ=(G) ≤ ∆(G) + 1.

Equitable ∆ Coloring Conjecture For every connected graph G which is not a complete graph, an odd-hole ∗ nor K2n+1,2n+1, for any n ≥ 1, χ=(G) ≤ ∆(G) holds.

Ko-Wei Lih. “Equitable coloring of graphs”. In: Handbook of combinatorial optimization. Springer, 2013, pp. 1199–1248

Grafos e Combinatória - 1a temporada Equitable Coloring 6 / 34 Equitable Coloring The story so far...

Class Complexity Trees P Forests P Bipartite NP-complete, even if k = 3 Co-bipartite P Cographs NP-complete, P for each ﬁxed k Bounded Treewidth P Chordal NP-complete Block ? Split P Unipolar ? Interval NP-complete Co-interval P

Grafos e Combinatória - 1a temporada Equitable Coloring 7 / 34 Equitable Coloring After this talk

Class Complexity Trees P Forests P Bipartite NP-complete, even if k = 3 Co-bipartite P Cographs NP-complete, P for each ﬁxed k Bounded Treewidth P Chordal NP-complete Block NP-complete Split P Unipolar P Interval NP-complete Co-interval P

Grafos e Combinatória - 1a temporada Equitable Coloring 8 / 34 a1 For each item of A, build a gadget with some key vertices. a2 All key vertices must have the same color. a3 Key vertices with color i → item

a4 in i-th bin.

0 1 2 3 4

Equitable Coloring Bin Packing

P Can we partition A = {a1,..., an} in k bins such that aj ∈bini aj = B?

Grafos e Combinatória - 1a temporada Equitable Coloring 9 / 34 For each item of A, build a gadget with some key vertices. All key vertices must have the same color. Key vertices with color i → item in i-th bin.

Equitable Coloring Bin Packing

P Can we partition A = {a1,..., an} in k bins such that aj ∈bini aj = B?

0 1 2 3 4

Grafos e Combinatória - 1a temporada Equitable Coloring 9 / 34 Equitable Coloring Bin Packing

P Can we partition A = {a1,..., an} in k bins such that aj ∈bini aj = B?

a1 For each item of A, build a gadget with some key vertices. a2 All key vertices must have the same color. a3 Key vertices with color i → item

a4 in i-th bin.

0 1 2 3 4

Grafos e Combinatória - 1a temporada Equitable Coloring 9 / 34 a1

a2 y 1 y2

0 1 2 3 4 y y4 3 k = 4

Equitable Coloring (a, k)-ﬂowers

Create a + 1 cliques with k − 1 vertices and add one universal vertex.

Grafos e Combinatória - 1a temporada Equitable Coloring 10 / 34 y 1 y2

y y4 3

Equitable Coloring (a, k)-ﬂowers

Create a + 1 cliques with k − 1 vertices and add one universal vertex.

0 1 2 3 4

k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 10 / 34 Equitable Coloring (a, k)-ﬂowers

Create a + 1 cliques with k − 1 vertices and add one universal vertex.

a2 y 1 y2

0 1 2 3 4 y y4 3 k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 10 / 34 Equitable Coloring Flowers, colors, and bins

bin1

bin2

bin3 y bin4

0 1 2 3 4

aj = 2, k = 4 k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins

bin1

bin2

bin3 y bin4

0 1 2 3 4

aj = 2, k = 4 k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins

bin1

bin2

bin3 y bin4

0 1 2 3 4

aj = 2, k = 4 k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins

bin1

bin2

bin3 y bin4

0 1 2 3 4

aj = 2, k = 4 k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins

bin1

bin2

bin3 y bin4

0 1 2 3 4

aj = 2, k = 4 k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins

bin1

bin2

bin3 y bin4

0 1 2 3 4

aj = 2, k = 4 k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 Equitable Coloring Flowers, colors, and bins

bin1

bin2

bin3 y bin4

0 1 2 3 4

aj = 2, k = 4 k = 4

Grafos e Combinatória - 1a temporada Equitable Coloring 11 / 34 X |V (G)| = ((aj + 1)(k − 1) + 1) j∈[n]   X = (k − 1) n + aj  + n j∈[n] = (k − 1)(n + kB) + n = kn + k2B − n − kB + n = k(kB − B + n)

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Construct a graph G as the disjoint union of ﬂowers Fj = F (aj , k) and try to equitably k-color it.

Grafos e Combinatória - 1a temporada Equitable Coloring 12 / 34   X = (k − 1) n + aj  + n j∈[n] = (k − 1)(n + kB) + n = kn + k2B − n − kB + n = k(kB − B + n)

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Construct a graph G as the disjoint union of ﬂowers Fj = F (aj , k) and try to equitably k-color it. X |V (G)| = ((aj + 1)(k − 1) + 1) j∈[n]

Grafos e Combinatória - 1a temporada Equitable Coloring 12 / 34 = (k − 1)(n + kB) + n = kn + k2B − n − kB + n = k(kB − B + n)

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Construct a graph G as the disjoint union of ﬂowers Fj = F (aj , k) and try to equitably k-color it. X |V (G)| = ((aj + 1)(k − 1) + 1) j∈[n]   X = (k − 1) n + aj  + n j∈[n]

Grafos e Combinatória - 1a temporada Equitable Coloring 12 / 34 = kn + k2B − n − kB + n = k(kB − B + n)

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Construct a graph G as the disjoint union of ﬂowers Fj = F (aj , k) and try to equitably k-color it. X |V (G)| = ((aj + 1)(k − 1) + 1) j∈[n]   X = (k − 1) n + aj  + n j∈[n] = (k − 1)(n + kB) + n

Grafos e Combinatória - 1a temporada Equitable Coloring 12 / 34 = k(kB − B + n)

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Construct a graph G as the disjoint union of ﬂowers Fj = F (aj , k) and try to equitably k-color it. X |V (G)| = ((aj + 1)(k − 1) + 1) j∈[n]   X = (k − 1) n + aj  + n j∈[n] = (k − 1)(n + kB) + n = kn + k2B − n − kB + n

Grafos e Combinatória - 1a temporada Equitable Coloring 12 / 34 Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Grafos e Combinatória - 1a temporada Equitable Coloring 12 / 34 X |ψi | = |ϕi | + (aj + 1)

j|yj ∈/ψi X X = |ϕi | + (aj + 1) − (aj + 1)

j∈[n] j|yj ∈ψi

= |ϕi | + n + kB − B − |ϕi | |V (G)| = kB − B + n = k

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ϕ to bin-packing, ψ(yj ) = i if aj ∈ ϕi .

Grafos e Combinatória - 1a temporada Equitable Coloring 13 / 34 X X = |ϕi | + (aj + 1) − (aj + 1)

j∈[n] j|yj ∈ψi

= |ϕi | + n + kB − B − |ϕi | |V (G)| = kB − B + n = k

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ϕ to bin-packing, ψ(yj ) = i if aj ∈ ϕi . X |ψi | = |ϕi | + (aj + 1)

j|yj ∈/ψi

Grafos e Combinatória - 1a temporada Equitable Coloring 13 / 34 = |ϕi | + n + kB − B − |ϕi | |V (G)| = kB − B + n = k

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ϕ to bin-packing, ψ(yj ) = i if aj ∈ ϕi . X |ψi | = |ϕi | + (aj + 1)

j|yj ∈/ψi X X = |ϕi | + (aj + 1) − (aj + 1)

j∈[n] j|yj ∈ψi

Grafos e Combinatória - 1a temporada Equitable Coloring 13 / 34 |V (G)| = kB − B + n = k

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ϕ to bin-packing, ψ(yj ) = i if aj ∈ ϕi . X |ψi | = |ϕi | + (aj + 1)

j|yj ∈/ψi X X = |ϕi | + (aj + 1) − (aj + 1)

j∈[n] j|yj ∈ψi

= |ϕi | + n + kB − B − |ϕi |

Grafos e Combinatória - 1a temporada Equitable Coloring 13 / 34 Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ϕ to bin-packing, ψ(yj ) = i if aj ∈ ϕi . X |ψi | = |ϕi | + (aj + 1)

j|yj ∈/ψi X X = |ϕi | + (aj + 1) − (aj + 1)

j∈[n] j|yj ∈ψi

= |ϕi | + n + kB − B − |ϕi | |V (G)| = kB − B + n = k

Grafos e Combinatória - 1a temporada Equitable Coloring 13 / 34 kB − B + n = |ψi | X X = 1 + (aj + 1)

j|yj ∈ψi j|yj ∈/ψi X X X = 1 + (aj + 1) − (aj + 1)

j|yj ∈ψi j∈[n] j|yj ∈ψi X = kB + n − aj

j|yj ∈ψi X B = aj

j|yj ∈ψi

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ψ to equitable coloring, put aj in ϕi if ψ(yj ) = i.

Grafos e Combinatória - 1a temporada Equitable Coloring 14 / 34 X X = 1 + (aj + 1)

j|yj ∈ψi j|yj ∈/ψi X X X = 1 + (aj + 1) − (aj + 1)

j|yj ∈ψi j∈[n] j|yj ∈ψi X = kB + n − aj

j|yj ∈ψi X B = aj

j|yj ∈ψi

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ψ to equitable coloring, put aj in ϕi if ψ(yj ) = i.

kB − B + n = |ψi |

Grafos e Combinatória - 1a temporada Equitable Coloring 14 / 34 X X X = 1 + (aj + 1) − (aj + 1)

j|yj ∈ψi j∈[n] j|yj ∈ψi X = kB + n − aj

j|yj ∈ψi X B = aj

j|yj ∈ψi

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ψ to equitable coloring, put aj in ϕi if ψ(yj ) = i.

kB − B + n = |ψi | X X = 1 + (aj + 1)

j|yj ∈ψi j|yj ∈/ψi

Grafos e Combinatória - 1a temporada Equitable Coloring 14 / 34 X = kB + n − aj

j|yj ∈ψi X B = aj

j|yj ∈ψi

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ψ to equitable coloring, put aj in ϕi if ψ(yj ) = i.

kB − B + n = |ψi | X X = 1 + (aj + 1)

j|yj ∈ψi j|yj ∈/ψi X X X = 1 + (aj + 1) − (aj + 1)

j|yj ∈ψi j∈[n] j|yj ∈ψi

Grafos e Combinatória - 1a temporada Equitable Coloring 14 / 34 X B = aj

j|yj ∈ψi

Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ψ to equitable coloring, put aj in ϕi if ψ(yj ) = i.

kB − B + n = |ψi | X X = 1 + (aj + 1)

j|yj ∈ψi j|yj ∈/ψi X X X = 1 + (aj + 1) − (aj + 1)

j|yj ∈ψi j∈[n] j|yj ∈ψi X = kB + n − aj

j|yj ∈ψi

Grafos e Combinatória - 1a temporada Equitable Coloring 14 / 34 Equitable Coloring Block Graphs

Theorem equitable coloring of block graphs is NP-complete.

Given a solution ψ to equitable coloring, put aj in ϕi if ψ(yj ) = i.

kB − B + n = |ψi | X X = 1 + (aj + 1)

j|yj ∈ψi j|yj ∈/ψi X X X = 1 + (aj + 1) − (aj + 1)

j|yj ∈ψi j∈[n] j|yj ∈ψi X = kB + n − aj

j|yj ∈ψi X B = aj

j|yj ∈ψi

Grafos e Combinatória - 1a temporada Equitable Coloring 14 / 34 P |V (G)| = aj ∈A(k−1)+(aj +1) = k(n + B). Try equitably k-color it.

Each aj becomes a split graph with k −1 vertices in the clique and aj + 1 vertices in the independent set (key vertices).

Equitable Coloring Disjoint union of split graphs (complete k-partite)

Theorem Equitable Coloring of disjoint union of split graphs is NP-complete.

Grafos e Combinatória - 1a temporada Equitable Coloring 15 / 34 P |V (G)| = aj ∈A(k−1)+(aj +1) = k(n + B). Try equitably k-color it.

Equitable Coloring Disjoint union of split graphs (complete k-partite)

Theorem Equitable Coloring of disjoint union of split graphs is NP-complete.

Each aj becomes a split graph with k −1 vertices in the clique and aj + 1 vertices in the independent set (key vertices).

Grafos e Combinatória - 1a temporada Equitable Coloring 15 / 34 Try equitably k-color it.

Equitable Coloring Disjoint union of split graphs (complete k-partite)

Theorem Equitable Coloring of disjoint union of split graphs is NP-complete.

Each aj becomes a split graph with k −1 vertices in the clique and aj + 1 vertices in the independent set (key vertices). P |V (G)| = aj ∈A(k−1)+(aj +1) = k(n + B).

Grafos e Combinatória - 1a temporada Equitable Coloring 15 / 34 Equitable Coloring Disjoint union of split graphs (complete k-partite)

Theorem Equitable Coloring of disjoint union of split graphs is NP-complete.

Each aj becomes a split graph with k −1 vertices in the clique and aj + 1 vertices in the independent set (key vertices). P |V (G)| = aj ∈A(k−1)+(aj +1) = k(n + B). Try equitably k-color it.

Grafos e Combinatória - 1a temporada Equitable Coloring 15 / 34 P |V (G)| = aj ∈A aj (k − 1) + aj k = k(2kB − B). Again, try to equitably k-color G.

Each aj becomes a sequence of aj cliques of size k − 1. Add one universal vertex to each pair of con- secutive cliques. Said vertex also has an extra clique of size k − 1 attached to it.

k = 3 aj = 2

Equitable Coloring

K1,r -free interval graphs

Theorem

Equitable Coloring of K1,r -free interval graphs is NP-complete if r ≥ 4, otherwise it is solvable in polynomial time (consequence of de Werra’85).

Grafos e Combinatória - 1a temporada Equitable Coloring 16 / 34 P |V (G)| = aj ∈A aj (k − 1) + aj k = k(2kB − B). Again, try to equitably k-color G.

Equitable Coloring

K1,r -free interval graphs

Theorem

Equitable Coloring of K1,r -free interval graphs is NP-complete if r ≥ 4, otherwise it is solvable in polynomial time (consequence of de Werra’85).

Each aj becomes a sequence of aj cliques of size k − 1. Add one universal vertex to each pair of con- secutive cliques. Said vertex also has an extra clique of size k − 1 attached to it.

k = 3 aj = 2

Grafos e Combinatória - 1a temporada Equitable Coloring 16 / 34 Again, try to equitably k-color G.

Equitable Coloring

K1,r -free interval graphs

Theorem

Equitable Coloring of K1,r -free interval graphs is NP-complete if r ≥ 4, otherwise it is solvable in polynomial time (consequence of de Werra’85).

Each aj becomes a sequence of aj cliques of size k − 1. Add one universal vertex to each pair of con- secutive cliques. Said vertex also has an extra clique of size k − 1 attached to it. |V (G)| = P a (k − 1) + a k = aj ∈A j j k = 3 a = 2 k(2kB − B). j

Grafos e Combinatória - 1a temporada Equitable Coloring 16 / 34 Equitable Coloring

K1,r -free interval graphs

Theorem

Equitable Coloring of K1,r -free interval graphs is NP-complete if r ≥ 4, otherwise it is solvable in polynomial time (consequence of de Werra’85).

Grafos e Combinatória - 1a temporada Equitable Coloring 16 / 34 v3 v4

v1 v2

v5 v6

Equitable Coloring Unipolar graphs

A graph G is unipolar if it has a clique Q such that G − Q is a disjoint union of cliques.

Grafos e Combinatória - 1a temporada Equitable Coloring 17 / 34 Equitable Coloring Unipolar graphs

A graph G is unipolar if it has a clique Q such that G − Q is a disjoint union of cliques.

v3 v4

v1 v2

v5 v6

Grafos e Combinatória - 1a temporada Equitable Coloring 17 / 34 Equitable Coloring Unipolar graphs

A graph G is unipolar if it has a clique Q such that G − Q is a disjoint union of cliques.

v3 v4

v1 v2

v5 v6

Grafos e Combinatória - 1a temporada Equitable Coloring 17 / 34 w11 v3

c1 w12 v4

c2 w21 v5

w22 v6

Source s, sink t, for each color i, ci , for each color i and clique j, wij , for vertex v` ∈/ Q, v`.

s t

Vertices

Each ﬂow unit gives the color of one vertex. Solid arcs have unit capacity.

Equitable Coloring A max-ﬂow based algorithm

v3 v4

v1 v2

v5 v6

Grafos e Combinatória - 1a temporada Equitable Coloring 18 / 34 w11 v3

c1 w12 v4

c2 w21 v5

w22 v6

Source s, sink t, for each color i, ci , for each color i and clique j, wij , for vertex v` ∈/ Q, v`.

s t

Vertices

Each ﬂow unit gives the color of one vertex. Solid arcs have unit capacity.

Equitable Coloring A max-ﬂow based algorithm

v3 v4

v1 v2

v5 v6

Grafos e Combinatória - 1a temporada Equitable Coloring 18 / 34 w11 v3

c1 w12 v4

c2 w21 v5

w22 v6

for each color i, ci , for each color i and clique j, wij , for vertex v` ∈/ Q, v`.

Each ﬂow unit gives the color of one vertex. Solid arcs have unit capacity.

Equitable Coloring A max-ﬂow based algorithm

v3 v4

v1 v2 s t

v5 v6 Vertices Source s, sink t,

Grafos e Combinatória - 1a temporada Equitable Coloring 18 / 34 w11 v3

w12 v4

w21 v5

w22 v6

for each color i and clique j, wij , for vertex v` ∈/ Q, v`.

Each ﬂow unit gives the color of one vertex. Solid arcs have unit capacity.

Equitable Coloring A max-ﬂow based algorithm

v3 v4

v1 v2 s t

v5 v6 Vertices

Source s, sink t, for each color i, ci ,

Grafos e Combinatória - 1a temporada Equitable Coloring 18 / 34 v3

for vertex v` ∈/ Q, v`.

Each ﬂow unit gives the color of one vertex. Solid arcs have unit capacity.

Equitable Coloring A max-ﬂow based algorithm

v3 v4 w11

c1 w12

v1 v2 s t

c2 w21

v5 v6 w22 Vertices

Source s, sink t, for each color i, ci , for each color i and clique j, wij ,

Grafos e Combinatória - 1a temporada Equitable Coloring 18 / 34 Each ﬂow unit gives the color of one vertex. Solid arcs have unit capacity.

Equitable Coloring A max-ﬂow based algorithm

v3 v4 w11 v3

c1 w12 v4

v1 v2 s t

c2 w21 v5

v5 v6 w22 v6 Vertices

Source s, sink t, for each color i, ci , for each color i and clique j, wij , for vertex v` ∈/ Q, v`.

Grafos e Combinatória - 1a temporada Equitable Coloring 18 / 34 Equitable Coloring A max-ﬂow based algorithm

v3 v4 w11 v3

c1 w12 v4

v1 v2 s t

c2 w21 v5

v5 v6 w22 v6 Vertices

Source s, sink t, for each color i, ci , for each color i and clique j, wij , for vertex v` ∈/ Q, v`.

Each ﬂow unit gives the color of one vertex. Solid arcs have unit capacity.

Grafos e Combinatória - 1a temporada Equitable Coloring 18 / 34 Parameterized Complexity

Parameterized Complexity

Grafos e Combinatória - 1a temporada Equitable Coloring 19 / 34 Very useful when k << |x|.

Instances are now a pair (x, k) where x is the input to the problem and the parameter k describes some property of x.

FPT Problem can be solved in f (k)nO(1) time. Example: Vertex Cover parameterized by size of the solution. Kernelization x can be compressed into a kernel x 0 so that |x 0| ≤ g(k). There exists an FPT algorithm if and only if there is a kernel. W[1]-hard No f (k)nO(1) time algorithm unless FPT = W[1]. Example: Clique parameterized by size of the solution.

Parameterized Complexity A primer on parameterized complexity

If a problem is NP-hard (we believe that) there is no exact algorithm that solves all instances in time polynomial on the size input.

Grafos e Combinatória - 1a temporada Equitable Coloring 20 / 34 Very useful when k << |x|.

Instances are now a pair (x, k) where x is the input to the problem and the parameter k describes some property of x.

Parameterized Complexity A primer on parameterized complexity

If a problem is NP-hard (we believe that) there is no exact algorithm that solves all instances in time polynomial on the size input.

Grafos e Combinatória - 1a temporada Equitable Coloring 20 / 34 Very useful when k << |x|.

Parameterized Complexity A primer on parameterized complexity

If a problem is NP-hard (we believe that) there is no exact algorithm that solves all instances in time polynomial on the size input.

Instances are now a pair (x, k) where x is the input to the problem and the parameter k describes some property of x.

Grafos e Combinatória - 1a temporada Equitable Coloring 20 / 34 FPT Problem can be solved in f (k)nO(1) time. Example: Vertex Cover parameterized by size of the solution. Kernelization x can be compressed into a kernel x 0 so that |x 0| ≤ g(k). There exists an FPT algorithm if and only if there is a kernel. W[1]-hard No f (k)nO(1) time algorithm unless FPT = W[1]. Example: Clique parameterized by size of the solution.

Parameterized Complexity A primer on parameterized complexity

If a problem is NP-hard (we believe that) there is no exact algorithm that solves all instances in time polynomial on the size input.

Instances are now a pair (x, k) where x is the input to the problem and the parameter k describes some property of x. Very useful when k << |x|.

Grafos e Combinatória - 1a temporada Equitable Coloring 20 / 34 Kernelization x can be compressed into a kernel x 0 so that |x 0| ≤ g(k). There exists an FPT algorithm if and only if there is a kernel. W[1]-hard No f (k)nO(1) time algorithm unless FPT = W[1]. Example: Clique parameterized by size of the solution.

Parameterized Complexity A primer on parameterized complexity

If a problem is NP-hard (we believe that) there is no exact algorithm that solves all instances in time polynomial on the size input.

Instances are now a pair (x, k) where x is the input to the problem and the parameter k describes some property of x. Very useful when k << |x|.

FPT Problem can be solved in f (k)nO(1) time. Example: Vertex Cover parameterized by size of the solution.

Grafos e Combinatória - 1a temporada Equitable Coloring 20 / 34 W[1]-hard No f (k)nO(1) time algorithm unless FPT = W[1]. Example: Clique parameterized by size of the solution.

Parameterized Complexity A primer on parameterized complexity

If a problem is NP-hard (we believe that) there is no exact algorithm that solves all instances in time polynomial on the size input.

Instances are now a pair (x, k) where x is the input to the problem and the parameter k describes some property of x. Very useful when k << |x|.

Grafos e Combinatória - 1a temporada Equitable Coloring 20 / 34 Parameterized Complexity A primer on parameterized complexity

If a problem is NP-hard (we believe that) there is no exact algorithm that solves all instances in time polynomial on the size input.

Instances are now a pair (x, k) where x is the input to the problem and the parameter k describes some property of x. Very useful when k << |x|.

Grafos e Combinatória - 1a temporada Equitable Coloring 20 / 34 Parameterized Complexity The parameterized story so far...

Class Parameterized Complexity Bipartite W[1]-hard parameterized by #colors Cographs W[1]-hard parameterized by #colors Chordal W[1]-hard parameterized by #colors Block ? Disjoint union of Split ? interval ? Independent set +kv FPT param. by k Split +kv W[1]-hard parameterized by k Cluster +kv ? Co-cluster +kv ? Forest +kv W[1]-hard parameterized by k+ #colors Path +kv ? Max leaf ` FPT parameterized by `

Grafos e Combinatória - 1a temporada Equitable Coloring 21 / 34 Parameterized Complexity After this talk

Class Parameterized Complexity Bipartite W[1]-hard parameterized by #colors Cographs W[1]-hard param. by #colors Chordal W[1]-hard param. by #colors Block W[1]-hard param. by #colors + treedepth Disjoint union of Split W[1]-hard param. by #colors + tw Interval W[1]-hard param. by #colors + bandwidth Independent set +kv FPT param. by k Split +kv W[1]-hard param. by k Cluster +kv FPT param. by k Co-cluster +kv FPT param. by k Forest +kv W[1]-hard param. by k + #colors Path +kv W[1]-hard param. by k + #colors Max leaf ` Cubic kernel parameterized by `

Grafos e Combinatória - 1a temporada Equitable Coloring 22 / 34 U

G is a cluster +kv graph if there is a set U ⊂ V (G) of size k such that G − U is a cluster graph, with clusters {C1,..., C`}.

Parameterized Complexity FPT algorithms Cluster +kv

G is a cluster graph if each of its connected components is a clique (cluster).

Grafos e Combinatória - 1a temporada Equitable Coloring 23 / 34 U

G is a cluster +kv graph if there is a set U ⊂ V (G) of size k such that G − U is a cluster graph, with clusters {C1,..., C`}.

Parameterized Complexity FPT algorithms Cluster +kv

G is a cluster graph if each of its connected components is a clique (cluster).

Grafos e Combinatória - 1a temporada Equitable Coloring 23 / 34 Parameterized Complexity FPT algorithms Cluster +kv

G is a cluster graph if each of its connected components is a clique (cluster).

G is a cluster +kv graph if there is a set U ⊂ V (G) of size k such that G − U is a cluster graph, with clusters {C1,..., C`}.

Grafos e Combinatória - 1a temporada Equitable Coloring 23 / 34 U

Parameterized Complexity FPT algorithms Cluster +kv

G is a cluster graph if each of its connected components is a clique (cluster).

G is a cluster +kv graph if there is a set U ⊂ V (G) of size k such that G − U is a cluster graph, with clusters {C1,..., C`}.

Grafos e Combinatória - 1a temporada Equitable Coloring 23 / 34 w11 v3

c1 w12 v4

c2 w21 v5

w22 v6

For each of the kk colorings of U, construct the auxiliary graph. Take into account the #times color i was used in U on the capacity of the (s, ci ) arcs.

s t

Algorithm

Parameterized Complexity FPT algorithms Cluster +kv: max-ﬂow again

v3 v4

v1 v2 v7

v5 v6

Grafos e Combinatória - 1a temporada Equitable Coloring 24 / 34 w11 v3

c1 w12 v4

c2 w21 v5

w22 v6

construct the auxiliary graph. Take into account the #times color i was used in U on the capacity of the (s, ci ) arcs.

s t

Algorithm For each of the kk colorings of U,

Parameterized Complexity FPT algorithms Cluster +kv: max-ﬂow again

v3 v4

v1 v2 v7

v5 v6

Grafos e Combinatória - 1a temporada Equitable Coloring 24 / 34 Take into account the #times color i was used in U on the capacity of the (s, ci ) arcs.

Parameterized Complexity FPT algorithms Cluster +kv: max-ﬂow again

v3 v4 w11 v3

c1 w12 v4

v1 v2 s t v7

c2 w21 v5

v5 v6 w22 v6

Algorithm For each of the kk colorings of U, construct the auxiliary graph.

Grafos e Combinatória - 1a temporada Equitable Coloring 24 / 34 Parameterized Complexity FPT algorithms Cluster +kv: max-ﬂow again

v3 v4 w11 v3

c1 w12 v4

v1 v2 s t v7

c2 w21 v5

v5 v6 w22 v6

Algorithm For each of the kk colorings of U, construct the auxiliary graph. Take into account the #times color i was used in U on the capacity of the (s, ci ) arcs.

Grafos e Combinatória - 1a temporada Equitable Coloring 24 / 34 Theorem (Estivill-Castro et al. (2005)) G has max leaf number number ` if and only if it is the subdivision of a graph H on 4` vertices.

Theorem (Cappelle, G., dos Santos (LAGOS2021?)) G \ H is a collection of at most 5.5` paths and contains most vertices of G.

Parameterized Complexity Kernelization Max leaf number

The max leaf number ` of a graph G is the maximum number of leaves on a spanning tree of G.

Grafos e Combinatória - 1a temporada Equitable Coloring 25 / 34 Theorem (Estivill-Castro et al. (2005)) G has max leaf number number ` if and only if it is the subdivision of a graph H on 4` vertices.

Theorem (Cappelle, G., dos Santos (LAGOS2021?)) G \ H is a collection of at most 5.5` paths and contains most vertices of G.

Parameterized Complexity Kernelization Max leaf number

The max leaf number ` of a graph G is the maximum number of leaves on a spanning tree of G.

Theorem (Cappelle, G., dos Santos (LAGOS2021?)) G \ H is a collection of at most 5.5` paths and contains most vertices of G.

Parameterized Complexity Kernelization Max leaf number

The max leaf number ` of a graph G is the maximum number of leaves on a spanning tree of G.

Theorem (Cappelle, G., dos Santos (LAGOS2021?)) G \ H is a collection of at most 5.5` paths and contains most vertices of G.

Parameterized Complexity Kernelization Max leaf number

The max leaf number ` of a graph G is the maximum number of leaves on a spanning tree of G.

Grafos e Combinatória - 1a temporada Equitable Coloring 25 / 34 Theorem (Cappelle, G., dos Santos (LAGOS2021?)) G \ H is a collection of at most 5.5` paths and contains most vertices of G.

Parameterized Complexity Kernelization Max leaf number

The max leaf number ` of a graph G is the maximum number of leaves on a spanning tree of G.

Theorem (Estivill-Castro et al. (2005)) G has max leaf number number ` if and only if it is the subdivision of a graph H on 4` vertices.

Grafos e Combinatória - 1a temporada Equitable Coloring 25 / 34 Parameterized Complexity Kernelization Max leaf number

The max leaf number ` of a graph G is the maximum number of leaves on a spanning tree of G.

Theorem (Estivill-Castro et al. (2005)) G has max leaf number number ` if and only if it is the subdivision of a graph H on 4` vertices.

Theorem (Cappelle, G., dos Santos (LAGOS2021?)) G \ H is a collection of at most 5.5` paths and contains most vertices of G.

Grafos e Combinatória - 1a temporada Equitable Coloring 25 / 34 P If |P| ≥ k + 2 + x(k + 1), we can extend ϕ to a coloring that is α−x or 1-balanced.

V (H) If |P| ≥ k + 2, we can extend ϕ u to H ∪ P while maintaining it α- balanced or making it 1-balanced if α = 0.

Parameterized Complexity Kernelization Improving the balance of the force/coloring

Let ϕ be a k-coloring of H and α = |ϕk | − |ϕ1|.

Grafos e Combinatória - 1a temporada Equitable Coloring 26 / 34 If |P| ≥ k + 2 + x(k + 1), we can extend ϕ to a coloring that is α−x or 1-balanced.

If |P| ≥ k + 2, we can extend ϕ to H ∪ P while maintaining it α- balanced or making it 1-balanced if α = 0. P

Parameterized Complexity Kernelization Improving the balance of the force/coloring

Let ϕ be a k-coloring of H and α = |ϕk | − |ϕ1|.

V (H)

Grafos e Combinatória - 1a temporada Equitable Coloring 26 / 34 If |P| ≥ k + 2 + x(k + 1), we can extend ϕ to a coloring that is α−x or 1-balanced.

If |P| ≥ k + 2, we can extend ϕ to H ∪ P while maintaining it α- balanced or making it 1-balanced if α = 0.

Parameterized Complexity Kernelization Improving the balance of the force/coloring

Let ϕ be a k-coloring of H and α = |ϕk | − |ϕ1|.

V (H)

Grafos e Combinatória - 1a temporada Equitable Coloring 26 / 34 If |P| ≥ k + 2 + x(k + 1), we can extend ϕ to a coloring that is α−x or 1-balanced.

Parameterized Complexity Kernelization Improving the balance of the force/coloring

Let ϕ be a k-coloring of H and α = |ϕk | − |ϕ1|.

V (H) If |P| ≥ k + 2, we can extend ϕ u to H ∪ P while maintaining it α- balanced or making it 1-balanced if α = 0. P

Grafos e Combinatória - 1a temporada Equitable Coloring 26 / 34 Parameterized Complexity Kernelization Improving the balance of the force/coloring

Let ϕ be a k-coloring of H and α = |ϕk | − |ϕ1|.

V (H) If |P| ≥ k + 2, we can extend ϕ u to H ∪ P while maintaining it α- balanced or making it 1-balanced if α = 0. P If |P| ≥ k + 2 + x(k + 1), we can extend ϕ to a coloring that is α−x v or 1-balanced.

Grafos e Combinatória - 1a temporada Equitable Coloring 26 / 34 Observation 1 Q has O (k`) vertices: Q is a supergraph of H and at most 5.5` paths with at most 2k + 2 vertices each remain in Q.

Observation 2 If |V (G)| ≤ |Q|(k + 2) + 11`(k + 1), G is already a kernel, so we are done.

Observation 3 P Otherwise, the paths in G \ Q satisfy Pi ∈G\Q xi (k + 1) ≥ |Q|(k + 1), so we can bring any coloring of Q to G and make it equitable.

Parameterized Complexity Kernelization A kernel parameterized by k and `

Let Q be the subgraph of G obtained by removing all paths with ≥ 2k + 3 vertices.

Grafos e Combinatória - 1a temporada Equitable Coloring 27 / 34 Observation 2 If |V (G)| ≤ |Q|(k + 2) + 11`(k + 1), G is already a kernel, so we are done.

Observation 3 P Otherwise, the paths in G \ Q satisfy Pi ∈G\Q xi (k + 1) ≥ |Q|(k + 1), so we can bring any coloring of Q to G and make it equitable.

Parameterized Complexity Kernelization A kernel parameterized by k and `

Let Q be the subgraph of G obtained by removing all paths with ≥ 2k + 3 vertices.

Observation 1 Q has O (k`) vertices: Q is a supergraph of H and at most 5.5` paths with at most 2k + 2 vertices each remain in Q.

Grafos e Combinatória - 1a temporada Equitable Coloring 27 / 34 Observation 3 P Otherwise, the paths in G \ Q satisfy Pi ∈G\Q xi (k + 1) ≥ |Q|(k + 1), so we can bring any coloring of Q to G and make it equitable.

Parameterized Complexity Kernelization A kernel parameterized by k and `

Let Q be the subgraph of G obtained by removing all paths with ≥ 2k + 3 vertices.

Observation 1 Q has O (k`) vertices: Q is a supergraph of H and at most 5.5` paths with at most 2k + 2 vertices each remain in Q.

Observation 2 If |V (G)| ≤ |Q|(k + 2) + 11`(k + 1), G is already a kernel, so we are done.

Grafos e Combinatória - 1a temporada Equitable Coloring 27 / 34 Parameterized Complexity Kernelization A kernel parameterized by k and `

Let Q be the subgraph of G obtained by removing all paths with ≥ 2k + 3 vertices.

Observation 1 Q has O (k`) vertices: Q is a supergraph of H and at most 5.5` paths with at most 2k + 2 vertices each remain in Q.

Observation 2 If |V (G)| ≤ |Q|(k + 2) + 11`(k + 1), G is already a kernel, so we are done.

Observation 3 P Otherwise, the paths in G \ Q satisfy Pi ∈G\Q xi (k + 1) ≥ |Q|(k + 1), so we can bring any coloring of Q to G and make it equitable.

Grafos e Combinatória - 1a temporada Equitable Coloring 27 / 34 Using the Brooks-like Hajnal-Szemerédi theorem, any interesting instance asks for a coloring with at most ∆ colors.

Replacing on Observation 2: G has at most 176`3 + O `2 vertices.

Parameterized Complexity Kernelization But you promised a kernel on `...

Since H has 4` vertices, the maximum degree of G is 4` − 1.

Grafos e Combinatória - 1a temporada Equitable Coloring 28 / 34 Replacing on Observation 2: G has at most 176`3 + O `2 vertices.

Parameterized Complexity Kernelization But you promised a kernel on `...

Since H has 4` vertices, the maximum degree of G is 4` − 1.

Using the Brooks-like Hajnal-Szemerédi theorem, any interesting instance asks for a coloring with at most ∆ colors.

Grafos e Combinatória - 1a temporada Equitable Coloring 28 / 34 Parameterized Complexity Kernelization But you promised a kernel on `...

Since H has 4` vertices, the maximum degree of G is 4` − 1.

Using the Brooks-like Hajnal-Szemerédi theorem, any interesting instance asks for a coloring with at most ∆ colors.

Replacing on Observation 2: G has at most 176`3 + O `2 vertices.

Grafos e Combinatória - 1a temporada Equitable Coloring 28 / 34 Theorem Equitable Coloring does not admit a polynomial kernel when parameterized by vertex cover and number of colors, unless NP ⊆ coNP/poly.

The proof is by a (way too technical) OR-cross-composition from Multicolored Clique: given H and a partition V of V (H), ﬁnd a clique with one vertex in each part of V.

But lets try to get some very high level intuition.

Parameterized Complexity Kernelization Back to the future: Vertex Cover

Known to yield an FPT algorithm for Equitable Coloring and a poly kernel for Vertex Coloring.

Grafos e Combinatória - 1a temporada Equitable Coloring 29 / 34 The proof is by a (way too technical) OR-cross-composition from Multicolored Clique: given H and a partition V of V (H), ﬁnd a clique with one vertex in each part of V.

But lets try to get some very high level intuition.

Parameterized Complexity Kernelization Back to the future: Vertex Cover

Known to yield an FPT algorithm for Equitable Coloring and a poly kernel for Vertex Coloring.

Theorem Equitable Coloring does not admit a polynomial kernel when parameterized by vertex cover and number of colors, unless NP ⊆ coNP/poly.

Grafos e Combinatória - 1a temporada Equitable Coloring 29 / 34 But lets try to get some very high level intuition.

Parameterized Complexity Kernelization Back to the future: Vertex Cover

Known to yield an FPT algorithm for Equitable Coloring and a poly kernel for Vertex Coloring.

Theorem Equitable Coloring does not admit a polynomial kernel when parameterized by vertex cover and number of colors, unless NP ⊆ coNP/poly.

The proof is by a (way too technical) OR-cross-composition from Multicolored Clique: given H and a partition V of V (H), ﬁnd a clique with one vertex in each part of V.

Grafos e Combinatória - 1a temporada Equitable Coloring 29 / 34 Parameterized Complexity Kernelization Back to the future: Vertex Cover

Known to yield an FPT algorithm for Equitable Coloring and a poly kernel for Vertex Coloring.

Theorem Equitable Coloring does not admit a polynomial kernel when parameterized by vertex cover and number of colors, unless NP ⊆ coNP/poly.

The proof is by a (way too technical) OR-cross-composition from Multicolored Clique: given H and a partition V of V (H), ﬁnd a clique with one vertex in each part of V.

But lets try to get some very high level intuition.

Grafos e Combinatória - 1a temporada Equitable Coloring 29 / 34 Parameterized Complexity Kernelization OR-cross-composition

H1, V1 (G, k) is YES iﬀ some (Hi , Vi ) is YES H2, V2 t instances of Multicolored Clique (G, k) V (Hi ) = [n] for every Hi ... Parameters must be bounded by poly(n + log t)

Ht , Vt

Grafos e Combinatória - 1a temporada Equitable Coloring 30 / 34 H1, V1 H2, V2

(G, k) I

Hi , Vi

Key idea 2: Use an instance selector gadget I to isolate the gadgets of G \I that encode the instance with the solution.

Parameterized Complexity Kernelization OR-cross-composition

Key idea 1: Overlay the gadgets for each (Hi , Vi ): 1 gadget belongs to many instances.

Grafos e Combinatória - 1a temporada Equitable Coloring 31 / 34 H1, V1 H2, V2

Hi , Vi

Key idea 2: Use an instance selector gadget I to isolate the gadgets of G \I that encode the instance with the solution.

Parameterized Complexity Kernelization OR-cross-composition

Key idea 1: Overlay the gadgets for each (Hi , Vi ): 1 gadget belongs to many instances.

(G, k)

Grafos e Combinatória - 1a temporada Equitable Coloring 31 / 34 H2, V2

Hi , Vi

Key idea 2: Use an instance selector gadget I to isolate the gadgets of G \I that encode the instance with the solution.

Parameterized Complexity Kernelization OR-cross-composition

Key idea 1: Overlay the gadgets for each (Hi , Vi ): 1 gadget belongs to many instances.

H1, V1

(G, k)

Grafos e Combinatória - 1a temporada Equitable Coloring 31 / 34 I

Hi , Vi

Key idea 2: Use an instance selector gadget I to isolate the gadgets of G \I that encode the instance with the solution.

Parameterized Complexity Kernelization OR-cross-composition

Key idea 1: Overlay the gadgets for each (Hi , Vi ): 1 gadget belongs to many instances.

H1, V1 H2, V2

(G, k)

Grafos e Combinatória - 1a temporada Equitable Coloring 31 / 34 I

Key idea 2: Use an instance selector gadget I to isolate the gadgets of G \I that encode the instance with the solution.

Parameterized Complexity Kernelization OR-cross-composition

Key idea 1: Overlay the gadgets for each (Hi , Vi ): 1 gadget belongs to many instances.

H1, V1 H2, V2

(G, k)

Hi , Vi

Grafos e Combinatória - 1a temporada Equitable Coloring 31 / 34 Parameterized Complexity Kernelization OR-cross-composition

Key idea 1: Overlay the gadgets for each (Hi , Vi ): 1 gadget belongs to many instances.

H1, V1 H2, V2

(G, k) I

Hi , Vi

Key idea 2: Use an instance selector gadget I to isolate the gadgets of G \I that encode the instance with the solution.

Grafos e Combinatória - 1a temporada Equitable Coloring 31 / 34 Parameterized Complexity Kernelization OR-cross-composition

Key idea 1: Overlay the gadgets for each (Hi , Vi ): 1 gadget belongs to many instances.

H1, V1 H2, V2

(G, k) I

Hi , Vi

Key idea 2: Use an instance selector gadget I to isolate the gadgets of G \I that encode the instance with the solution.

Grafos e Combinatória - 1a temporada Equitable Coloring 31 / 34 Parameterized Complexity Kernelization Parameterized landscape

Vertex Max Leaf Cover Number*

Distance to Clique

D. to Bounded FVS + Disjoint Paths* Max Degree

Minimum Distance Distance Distance Distance to Feedback Treedepth Bandwidth Clique Cover to Co-cluster* to Cluster* to Threshold Disjoint Paths* Edge Set

Maximum Distance Distance Distance Feedback Pathwidth Independent Set to Cograph to Interval to Split Vertex Set

Minimum Distance Distance Distance Treewidth Dominating Set to Chordal to Unipolar to Bipartite

Grafos e Combinatória - 1a temporada Equitable Coloring 32 / 34 Parameterized Complexity Kernelization Open Problems

• Can we ﬁnd a subcubic kernel when parameterizing by the max leaf number? • Using ETH, can we prove that currently known algorithms have optimal running time? • Is Equitable Coloring FPT when parameterized by feedback edge set?

Grafos e Combinatória - 1a temporada Equitable Coloring 33 / 34 Thank you!

Thank you!

Grafos e Combinatória - 1a temporada Equitable Coloring 34 / 34