1. Computing the Chromatic Number Using Graph Decompositions via Matrix Rank
Bart M. P. Jansen Jesper Nederlof
Dagstuhl Seminar New Horizons in Param. Complexity
January 21st 2018, Dagstuhl, Germany

2. Graph 𝑞-Coloring
Input: An undirected graph 𝐺 Question: Can we assign each vertex a color 𝑓 𝑣 ∈ 1,…, 𝑞 , such that 𝑓 𝑢 ≠𝑓(𝑣) for all edges 𝑢,𝑣 ∈𝐸(𝐺) ?
NP-complete for 𝑞≥3

3. Coloring algorithms using vertex separators
Parameterize the problem by measures of graph complexity
Can 𝑞-Coloring be solved efficiently on graphs that are structurally simple?
Parameterized by the treewidth 𝑡𝑤 of the graph, for 𝑞≥3:
𝑞-Coloring can be solved in time 𝑂 ∗ ( 𝑞 𝑡𝑤 ) 𝑞-Coloring can not be solved in time 𝑂 ∗ (𝑞−𝜀 𝑡𝑤 ) (under SETH) [Lokshtanov, Marx & TALG’18] – 𝑂 ∗ suppresses 𝑝𝑜𝑙𝑦(𝑛) factors
Solid algorithmic understanding for parameterizations based on vertex separators
Treewidth lower-bound extends to pathwidth & feedback vertex number. So complexity is understood quite well for parameterizations based on the width/size of vertex separators of the graph.

4. Coloring algorithms using edge separators
Parameterize the problem by measures of graph complexity
Can 𝑞-Coloring be solved efficiently on graphs that are structurally simple?
What about graphs decomposable by small edge separators?
If the input graph has maximum degree 𝑑, is given with a path decomposition of width 𝑝𝑤, and has at most 𝑠 proper 𝑞-colorings, then with high probability:
SWAT ’16]
Maximum degree 𝑑, pathwidth 𝑝𝑤 implies cutwidth ≤ 𝑑⋅𝑝𝑤 [Thilikos, Serna, J. Algorithms ‘05]
𝑞-Coloring can be solved in randomized time 𝑂 ∗ 𝑑/2 +1 𝑝𝑤 ⋅𝑠
Setting of bounded-degree graphs of bounded pathwidth is related to the graph width parameter cutwidth.

5. Decomposing graphs by small edge-cuts
The cutwidth of a linear ordering of 𝑉(𝐺) is: max. # edges intersected by a vertical line between vertices The cutwidth of a graph is min. cutwidth of any ordering of 𝑉(𝐺)
For any graph 𝐺: 𝒕𝒓𝒆𝒆𝑤𝑖𝑑𝑡ℎ 𝐺 ≤𝒑𝒂𝒕𝒉𝑤𝑖𝑑𝑡ℎ 𝐺 ≤𝒄𝒖𝒕𝑤𝑖𝑑𝑡ℎ(𝐺)
Cutwidth gives an upper-bound on the size of edge cuts that decompose the graph linearly, similarly as how pathwidth upper-bounds the size of vertex separators that decompose the graph linearly. So: how fast can q-Coloring be solved parameterized by cutwidth?
cutwidth 3

6. Results for cutwidth
If the input graph is given with a layout of cutwidth 𝑐𝑡𝑤, then:
𝜔≈ is the matrix multiplication constant; key ingredient is a new rank bound
one-sided error caused by Isolation Lemma; key ingredient is communication protocol
for any 𝜀>0, unless SETH fails
Note: running time is independent of 𝑞 using cutwidth running time 𝑂 ∗ 𝑞 𝑡𝑤 is SETH-optimal for treewidth
𝑞-Coloring can be solved deterministically in time 𝑂 ∗ ( 2 𝜔⋅𝑐𝑡𝑤 )
𝑞-Coloring can be solved in time 𝑂 ∗ ( 2 𝑐𝑡𝑤 ) w. high probability
Planar 3-Coloring cannot be solved in time 𝑂 ∗ 2−𝜀 𝑐𝑡𝑤
SETH lower bound holds in the same setting, where a layout is given in the input. Isolation lemma is by Mulmuley, Vazirani and Vazirani [Combinatorica ’87].

7. Results for bounded degree and pathwidth
If the input graph has maximum degree 𝑑, and is given with a path decomposition of width 𝑝𝑤, then:
Removes dependence on solution space from Björklund’s algorithm [SWAT ’16]
Let 𝑑≥5 be odd, let 𝑞 𝑑 ≔ 𝑑/2 +1. Then:
for any 𝜀>0, unless SETH fails
𝑞-Coloring can be solved in time 𝑂 ∗ 𝑑/2 +1 𝑝𝑤 w.h.p.
𝑞 𝑑 -Coloring cannot be solved in time 𝑂 ∗ 𝑑/2 +1−𝜀 𝑝𝑤 on graphs of maximum degree 𝑑

8. Deterministic ideas: a rank bound
Uses rank-based approach for dynamic programming [Bodlaender,Cygan,Kratsch & N ’15; Cygan,Kratsch & N ’13; Fomin,Lokshtanov,Panolan & Saurabh ’16]
Nr. of partial solutions needed in dynamic-programming table, can be reduced using rank bounds for ‘compatibility matrices’
Our high-level insight:
Compatibility matrix for partial 𝑞-colorings on two sides of a
size-𝑘 vertex separator can have rank ≈ 𝑞 𝑘
size-𝑘 edge separator has rank ≤ 2 𝑘
Yields 𝑂 ∗ ( 2 𝜔⋅𝑐𝑢𝑡𝑤 ) algorithm using rank-based approach
Bodlaender Cygan Kratsch Nederlof; Cygan Kratsch Nederlof; Fomin Lokhstanov Panolan Saurabh.

9. A compatibility matrix for 𝑞-colorings
Consider a bipartite graph 𝐻 on 𝑘 edges with bipartition 𝑋∪𝑌 Corresponds to the edges crossing a cut in the linear layout
Fix an integer 𝑞 and define a matrix 𝑀 𝐻 as follows:
For each 𝑞-coloring 𝒙∈ 𝑞 𝑋 of 𝑋, there is a row in 𝑀 𝐻
For each 𝑞-coloring 𝒚∈ 𝑞 𝑌 of 𝑌, there is a column in 𝑀 𝐻
𝑀 𝐻 [𝒙,𝒚]= 1 0
𝑀 𝐻 is useful to prune partial solutions, but has too large rank
if 𝒙∪𝒚 is a proper 𝑞-coloring of 𝐻 otherwise
⬤ ⬤ …
⬤ ⬤ ⬤ 1
... 𝑋
Intuitively, this is the compatibility matrix for partial solutions to q-coloring on two sides of a small edge separator. 𝑌

10. A related integer-valued matrix
Consider a bipartite graph 𝐻 on 𝑘 edges with bipartition 𝑋∪𝑌 Orient edges from 𝑋 to 𝑌
Fix an integer 𝑞 and define a matrix 𝑀 𝐻 ′ as follows:
For each 𝑞-coloring 𝒙∈ 𝑞 𝑋 of 𝑋, there is a row in 𝑀 𝐻 ′
For each 𝑞-coloring 𝒚∈ 𝑞 𝑌 of 𝑌, there is a column in 𝑀 𝐻 ′
𝑀 𝐻 ′ [𝒙,𝒚]= (𝑖,𝑗)∈𝐸 𝐻 𝑥 𝑖 − 𝑦 𝑗
𝑀 𝐻 ′ 𝒙,𝒚 ≠0 ⇔ (𝒙∪𝒚) is a proper 𝑞-coloring of 𝐻
For all 𝑞: 𝑟𝑘 𝑀 𝐻 ′ ≤ 2 𝑘 for bipartite graphs 𝐻 on 𝑘 edges
3 2
1 3
3 3 …
1 2 3 2
1 2 2 -2
3 1 1
... 𝑋 𝑌
= 1−3 ⋅ 2−3 ⋅ 3−2 = −2 −1 1 =2
= 1−3 ⋅ 2−3 ⋅ 2−2 = −2 −1 0 =0
= 3−3 ⋅ 1−3 ⋅ 1−2 = 0 −2 −1 =0

11. Exploiting the rank bound algorithmically
Dynamic programming over a given linear layout From left to right, compute proper 𝑞-colorings of first 𝑖 vertices Can safely restrict to a row-basis of the partial colorings At most 𝑟𝑘( 𝑀′ 𝐻 )≤ 2 𝑘 rows
𝑞-Coloring can be solved deterministically in time 𝑂 ∗ ( 2 𝜔⋅𝑐𝑡𝑤 )
𝑞-Coloring can be solved deterministically in time 𝑂 ∗ ( 2 𝜔⋅𝑐𝑡𝑤 )
e f …
b d ≠0
...
Running time dominated by the maximum number of colorings stored simultaneously at any point in time.

12. Intermezzo: The lower bounds
I’ve heard Daniel Marx say that every talk should have a picture of a reduction no-one understands, and since he’s one of the organizers I feel obliged to comply. Reduction from CNF-SAT with n variables to Planar 3-Coloring instance of cutwidth n + O(1).

13. Randomized ideas: the graph polynomial
Let 𝐺 be an undirected graph on vertex set 𝑛
The 𝑛-variate graph polynomial 𝑓 𝐺 is defined as follows:
𝑓 𝐺 𝑥 1 ,…, 𝑥 𝑛 ≔ {𝑖,𝑗}∈𝐸(𝐺) 𝑖<𝑗 𝑥 𝑖 − 𝑥 𝑗
Extensively studied, i.e. in context of Alon-Tarsi Theorem [Combinatorica ‘92]
For any 𝒙= 𝑥 1 ,…, 𝑥 𝑛 ∈ 𝑞 𝑛 , we have:
We analyze 𝑞-colorability of 𝐺 by summing the graph polynomial If 𝒙∈ 𝑞 𝑛 𝑓 𝐺 (𝒙) ≠0, then 𝐺 is 𝑞-colorable; but converse may fail
𝒙 is a proper 𝑞-coloring of 𝐺⇔ 𝑓 𝐺 𝒙 ≠0
If 𝑥 is proper, then for each edge the endpoints have different colors, so 𝑝 𝐺 (𝒙) is a product of nonzero terms, so nonzero. If 𝑥 is not proper, then there is an edge whose endpoints have the same color, which yields a 0-term in the product.

14. Isolating a coloring
Pick random weight ∈[2𝑛𝑞] for each (𝑣∈𝑉(𝐺),𝑐∈ 𝑞 ) pair Algorithmic implication: To get a Monte Carlo algorithm for 𝑞-Coloring, pick random weight function and test whether ∃𝑧∈ℕ such that: 𝑃 𝐺 𝑧 ≔ 𝒙∈ 𝑞 𝑛 weight 𝒙 =𝑧 𝑓 𝐺 (𝒙) ≠0 If there is a unique weight-𝑧 proper 𝑞-coloring, then 𝑃 𝐺 𝑧 ≠0 If 𝑃 𝐺 𝑧 ≠0, then there is a proper 𝑞-coloring of weight 𝑧
Isolation Lemma [Mulmuley, Vazirani, and Vazirani] If there is a proper 𝑞-coloring, then with probability ≥ 1 2 there is a unique proper 𝑞-coloring of minimum total weight

15. Rewriting the sum
𝑃 𝐺 𝑧 ≔ 𝒙∈ 𝑞 𝑛 weight 𝒙 =𝑧 𝑓 𝐺 𝒙 = 𝒙∈ 𝑞 𝑛 weight 𝒙 =𝑧 𝑖,𝑗 ∈𝐸 𝐺 𝑖<𝑗 𝑥 𝑖 − 𝑥 𝑗 = 𝒙∈ 𝑞 𝑛 weight 𝒙 =𝑧 𝑊⊆𝐸(𝐺) 𝑖,𝑗 ∈𝑊 𝑖<𝑗 𝑥 𝑖 𝑖,𝑗 ∈𝐸 𝐺 ∖𝑊 𝑖<𝑗 − 𝑥 𝑗
For fixed 𝑊, degree of 𝑥 𝑘 in (∏ 𝑥 𝑖 )( − 𝑥 𝑗 ) equals sum of:
number of edges ∈𝑊 to neighbors 𝑗>𝑘
number of edges ∉𝑊 to neighbors 𝑖<𝑘
Let’s look at the graph polynomial, summed over all colorings of weight z. For each way of choosing the left or right endpoint of an edge, you get a term in the expansion of the product. Degree consideration hints at connection between degree of (partial evaluations of) this polynomial, and cutwidth of the graph.

16. Evaluate the sum using a linear layout
𝑃 𝐺 𝑧 = 𝒙∈ 𝑞 𝑛 weight 𝒙 =𝑧 𝑊⊆𝐸(𝐺) 𝑖,𝑗 ∈𝑊 𝑖<𝑗 𝑥 𝑖 𝑖,𝑗 ∈𝐸 𝐺 ∖𝑊 𝑖<𝑗 − 𝑥 𝑗 To solve 𝑞-Coloring on a graph with a bounded-cutwidth layout: Left-to-right dynamic programming over the layout, computing partial sums over colorings of the first 𝑖 vertices Leave some terms un-evaluated to allow partial sum for prefix of length 𝑖+1 to be computed from partial sums for prefixes of length 𝑖 Small cutwidth layout ⇒ degree of un-evaluated part is small

17. Main idea for the dynamic program
Communication protocol for evaluating polynomial sums Table entries stored in DP correspond to message that allows other player to find the answer Alice has univariate polynomial 𝑃 𝐴 (𝑥) of degree 𝑑 𝐴 Bob has univariate polynomial 𝑃 𝐵 𝑥 of degree 𝑑 𝐵 Both parties know 𝑑 𝐴 , 𝑑 𝐵 , and 𝑞 Goal: Using little communication from Alice to Bob, enable Bob to output 𝑥∈ 𝑞 𝑃 𝐴 𝑥 ⋅ 𝑃 𝐵 𝑥 Straight-forward communication strategy: Consider coefficients of 𝑃 𝐴 𝑥 = 𝑐 0 𝑥 0 + 𝑐 1 𝑥 1 +…+ 𝑐 𝑑 𝐴 𝑥 𝑑 𝐴 Alice sends her 𝑑 𝐴 +1 coefficients ( 𝑐 0 , 𝑐 1 ,…, 𝑐 𝑑 𝐴 ) to Bob
Summing evaluations of a polynomial that is split into two bounded-degree factors (one corresponding to edges already encountered in the linear layout, other corresponding to edges to-be encountered in the future).

18. Alternative communication strategy
Goal: Using little communication from Alice to Bob, enable Bob to output 𝑥∈ 𝑞 𝑃 𝐴 𝑥 ⋅ 𝑃 𝐵 𝑥 Straight-forward communication strategy: Alice sends the 𝑑 𝐴 +1 coefficients ( 𝑐 0 , 𝑐 1 ,…, 𝑐 𝑑 𝐴 ) of 𝑃 𝐴 to Bob Alternative communication strategy: Consider coefficients of 𝑃 𝐵 𝑥 = 𝑐 0 ′ 𝑥 0 + 𝑐 1 ′ 𝑥 1 +…+ 𝑐 𝑑 𝐵 ′ 𝑥 𝑑 𝐵 Desired output can be rewritten as: 𝑥∈ 𝑞 𝑃 𝐴 𝑥 𝑐 0 ′ 𝑥 0 + 𝑐 1 ′ 𝑥 1 +…+ 𝑐 𝑑 𝐵 ′ 𝑥 𝑑 𝐵 = 𝑐 0 ′ 𝑥∈ 𝑞 𝑃 𝐴 𝑥 𝑥 0 + 𝑐 1 ′ 𝑥∈ 𝑞 𝑃 𝐴 𝑥 𝑥 1 +⋯+ 𝑐 𝑑 𝐵 ′ 𝑥∈ 𝑞 𝑃 𝐴 𝑥 𝑥 𝑑 𝐵
First split the single sum into 𝑑 𝐵 +1 sums, one for every power; within one such sum, we can move the coefficient in front.
These 𝑑 𝐵 +1 partial evaluations allow Bob to compute the output!

19. Alternative communication strategy
Goal: Using little communication from Alice to Bob, enable Bob to output 𝑥∈ 𝑞 𝑃 𝐴 𝑥 ⋅ 𝑃 𝐵 𝑥 Straight-forward communication strategy: Alice sends the 𝑑 𝐴 +1 coefficients ( 𝑐 0 , 𝑐 1 ,…, 𝑐 𝑑 𝐴 ) of 𝑃 𝐴 to Bob Alternative communication strategy: Alice sends 𝑑 𝐵 +1 evaluations 𝑥∈ 𝑞 𝑃 𝐴 𝑥 𝑥 𝑖 for 0≤𝑖≤ 𝑑 𝐵 Best protocol depends on which polynomial has larger degree Extends to multivariate polynomials for Alice and Bob Mix the two strategies, choosing most efficient one for each variable
First split the single sum into 𝑑 𝐵 +1 sums, one for every power; within one such sum, we can move the coefficient in front.

20. From communication protocol to 𝑞-Coloring
From left to right in the linear layout, compute partial evaluations of 𝑃 𝐺 (𝑧) about the first 𝑖 vertices 𝑉 𝑖 Un-evaluated variables in the polynomial for 𝑉 𝑖 correspond to: vertices incident to an edge in the cut after 𝑖 Independently pick communication strategy for each vertex 𝑣: does the majority of the neighbors occur at positions ≤𝑖, or at positions >𝑖?
𝑞-Coloring can be solved in time 𝑂 ∗ ( 2 𝑐𝑡𝑤 ) w. high probability
𝑞-Coloring can be solved in time 𝑂 ∗ 𝑑/2 +1 𝑝𝑤 w.h.p.

21. THANK YOU! Conclusion PhD & Postdoc
SETH-optimal algorithms for 𝑞-Coloring, using several parameterizations concerning edge separators
Uniform approach works for both cutwidth / degree+pathwidth
Running times for cutwidth parameterization independent of 𝑞
Running time 𝑂 ∗ ( 2 𝑐𝑡𝑤 ) is already optimal for Planar 3-Coloring
Open problems:
Derandomize the algorithms
Experiment with the cut-based reduction of partial solutions
PhD & Postdoc
THANK YOU!