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Tight lower bound for the Channel Assignment problem Arkadiusz Socaa University of Warsaw Arkadiusz Socaa Tight lower bound for the Channel Assignment problem Graph Coloring Recall the Graph Coloring problem. 2 1 3 3 1 2 1 coloring

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Tight lower bound for the Channel Assignment problem

Arkadiusz Socała

University of Warsaw

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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Graph Coloring

Recall the Graph Coloring problem. 1 2 3 1 2 1 3 coloring c : V → {1, . . . , s} c(u) = c(v) for every edge uv ∈ E

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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Graph Coloring

Recall the Graph Coloring problem. 1 1 1 1 1 1 1 1 1 2 3 1 2 1 1 1 3 coloring c : V → {1, . . . , s} |c(u) − c(v)| ≥ 1 for every edge uv ∈ E

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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Channel Assignment

Weights on edges mean minimum allowed color differences. 4 2 5 3 2 5 4 1 1 9 3 1 6 7 3 2 4 coloring c : V → {1, . . . , s} |c(u) − c(v)| ≥ w(uv) for every edge uv ∈ E

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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The Channel Assignment problem.

Input graph G(V , E) weight function w : E → N+. Definitions An assignment c : V → {1, . . . , s} is called proper when ∀uv∈E|c(u) − c(v)| ≥ w(uv). The number s is called the span of c. Problem Find a proper assignment of minimum span.

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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The Channel Assignment problem.

Motivation Assignment of the radio frequencies: n radio emitters they interfere each

ther

minimize the range

f the used

frequencies (span) Introduced by Hale in 1980.

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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Algorithms for Channel Assignment

General version O∗(n!)-time (McDiarmid, 2003) Our result: There is no 2o(n log n)-time algorithm (under ETH) ℓ-bounded version Assume ∀uv∈E w(uv) ≤ ℓ. O∗((2ℓ + 1)n)-time (McDiarmid, 2003) O∗((ℓ + 2)n)-time (Kral, 2005) O∗((ℓ + 1)n)-time (Cygan and Kowalik, 2011) O∗((2 √ ℓ + 1)n)-time (Kowalik and S, 2014) These are all dynamic programming. All algorithms above can be modified to count proper colorings.

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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CSP Hierarchy

(s, 2)-CSP Generalized T-Coloring Channel Assignment T-Coloring Graph Coloring

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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T-Coloring

List of forbidden differences. 1 7 4 1 7 1 4 T = {0, 1, 2, 5} coloring c : V → {1, . . . , s} |c(u) − c(v)| ∈ T for every edge uv ∈ E

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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Generalized T-Coloring

Lists of forbidden differences. 0, 1, 2 0, 2 0, 1, 2, 3 0, 1, 7 2, 4 0, 3, 5 0, 1, 3, 4 0, 1, 3 1 7 6 6 2 4 0, 2, 5 0, 1, 2 3 coloring c : V → {1, . . . , s} |c(u) − c(v)| ∈ T(uv) for every edge uv ∈ E

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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(s, 2)-CSP (Constraint Satisfaction Problem)

Lists of forbidden pairs of colors. (1, 1), (1, 2) (3, 1), (3, 3) (3, 3) (1, 2), (2, 1) (2, 4) (1, 1), (2, 2) (2, 1), (2, 3) (1, 2), (1, 4) 1 3 2 2 1 5 (1, 4), (2, 2) (1, 1) 4 coloring c : V → {1, . . . , s} (c(u), c(v)) ∈ C(uv) for every edge uv ∈ E

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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CSP Hierarchy

(s, 2)-CSP Generalized T-Coloring Channel Assignment T-Coloring Graph Coloring no 2O(n) under ETH (Traxler) O∗(n!) (McDiarmid) O∗(2n) (Björklund et al.) no 22o(√n) under ETH (Kowalik and S)

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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CSP Hierarchy

(s, 2)-CSP Generalized T-Coloring Channel Assignment T-Coloring Graph Coloring no 2O(n) under ETH (Traxler) no 22o(√n) under ETH (Kowalik and S) O∗(n!) (McDiarmid) O∗(2n) (Björklund et al.) Our Result: no 2o(n log n) under ETH

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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Summary of our results

Channel Assignment cannot be solved in 2o(n log n) under ETH (tight!) nor in 2n·o(log log ℓ)

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From 3-CNF-SAT to Family Intersection From 3-CNF-SAT to Family Intersection

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From 3-CNF-SAT to Family Intersection

Family Intersection: Input: We are given two matrices. We have to pick exactly one element from every row. Question: Is it possible to pick elements in such a way that both sums are equal? Choice 1: 1 14 1 Choice 2: 7 6 10 Choice 3: 6 4 7 Choice 4: 2 14 Choice 1: 5 7 5 4 Choice 2: 2 8 7 7 Choice 3: 3 8 8 9

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From 3-CNF-SAT to Family Intersection

Family Intersection: Input: We are given two matrices. We have to pick exactly one element from every row. Question: Is it possible to pick elements in such a way that both sums are equal? Choice 1: 1 14 1 Choice 2: 7 6 10 Choice 3: 6 4 7 Choice 4: 2 14 Choice 1: 5 7 5 4 Choice 2: 2 8 7 7 Choice 3: 3 8 8 9 e.g. 1 + 7 + 7 + 0 = 5 + 7 + 3

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From 3-CNF-SAT to Family Intersection

An example for a 2-CNF-SAT formula ϕ = (α ∨ β) ∧ (¬α ∨ γ): We number all the occurrences (α1 ∨ β2) ∧ (¬α3 ∨ γ4) We interpret 4-bit numbers as the valuations of the

ccurrences

False True α: 00002 10102 β: 00002 01002 γ: 00002 00012 1st 2nd 1st and 2nd α1 ∨ β2: 10002 01002 11002 ¬α3 ∨ γ4: 00002 00112 00012

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From 3-CNF-SAT to Family Intersection

An example for a 2-CNF-SAT formula ϕ = (α ∨ β) ∧ (¬α ∨ γ): We number all the occurrences (α1 ∨ β2) ∧ (¬α3 ∨ γ4) We interpret 4-bit numbers as the valuations of the

ccurrences

False True α: 00002 10102 β: 00002 01002 γ: 00002 00012 1st 2nd 1st and 2nd α1 ∨ β2: 10002 01002 11002 ¬α3 ∨ γ4: 00002 00112 00012 Left matrix represents consistent valuations of the occurrences. Right matrix represents valuations of the occurrences such that every clause is satisfied.

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From Family Intersection to Common Matching Weight From Family Intersection to Common Matching Weight

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From Family Intersection to Common Matching Weight

For a matrix with n rows and c columns we can build a weighted full bipartite graph such that: set of weights of all perfect matchings = set of all possible sums of choices number of the vertices is O

log n

(for fixed c)

Common Matching Weight: Input: We are given two weighted bipartite graphs. Question: Is it possible to pick two perfect matchings in such a way that both sum of weights are equal?

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From Family Intersection to Common Matching Weight

How do we compress? A nice recursive idea behind: Assume that we already have matchings that represents ”something”. By merging them we can additionally represent a function ρ on the left vertices of the first matching. ˆ 1, ˆ 1 ˆ 1, ˆ 2 ˆ 1, ˆ 3 ˆ 2, ˆ 1 ˆ 2, ˆ 2 ˆ 2, ˆ 3 ˆ 3, ˆ 1 ˆ 3, ˆ 2 ˆ 3, ˆ 3 1, 1 1, 2 1, 3 (ρ(ˆ 2) = 1) 2, 1 2, 2 2, 3 3, 1 (ρ(ˆ 3) = 3) 3, 2 (ρ(ˆ 1) = 3) 3, 3 ˆ 1 ˆ 2 ˆ 3 ˆ 1 ˆ 2 ˆ 3 ˆ 1 ˆ 2 ˆ 3 1 2 3 1 2 3 1 2 3 (ρ)

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From Family Intersection to Common Matching Weight

We can use this idea to build our weighted graph recursively. When merging bipartite graphs: We copy the weights that for every vertex of the left side there is no difference to which of the merged parts it will be matched. Then for every vertex of the first part we can add to the weights of its edges a function depending only on the part to which it is connected.

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From Family Intersection to Common Matching Weight

For the matrix with 4 rows and 2 columns we get:

A1,1

A1,2 A2,1 A2,2

   A1,1 + A3,1 A1,2 + A3,1 A1,1 + A3,2 A1,2 + A3,2 A4,1 A4,1 A4,2 A4,2 A2,1 A2,2 A2,1 A1,2    

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From Common Matching Weight to Channel Assignment From Common Matching Weight to Channel Assignment

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From Common Matching Weight to Channel Assignment

We encode a bipartite graph and its matchings into: v1 v2 v3 v4 v5 v6 v7 v8 w1 w2 w3 vM = w2 aπ(1) aπ(2) bπ(1) bπ(2) 2M 2M 2M 2M 2M 2M 2M M M M M M M M M M M vL = v1 vR = v8 8M 8M The position (color) of the middle vertex represent the weight of the matching corresponding to the permutation π.

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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From Common Matching Weight to Channel Assignment

We can encode two bipartite graphs with the common middle

vertex. Its position (color) needs to represent the common weight
f both matchings.

w(1)

L

w(2)

L

w(1)

R

w(2)

R

vM v(1)

L

v(2)

L

v(1)

R

v(2)

R

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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Reductions

3-CNF- SAT O(n) → Family In- tersection O(n) → Common Matching Weight O

log n

Channel Assignment O

log n

Arkadiusz Socała

Tight lower bound for the Channel Assignment problem

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Summary

Our results Channel Assignment cannot be solved in 2o(n log n) under ETH (tight!) nor in 2n·o(log log ℓ) Further research Faster algorithm for ℓ-bounded version? e.g. O(log ℓ)n?

Arkadiusz Socała Tight lower bound for the Channel Assignment problem

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The end.

Arkadiusz Socała Tight lower bound for the Channel Assignment problem