Lower Bounds for Dynamic Programming on Planar Graphs

Lower Bounds for Dynamic Programming on Planar Graphs of Bounded Cutwidth Bas A. M. van Geffen Bart M. P. Jansen Arnoud A. W. M. de Kroon Rolf Morel NWO-JSPS joint seminar: Computations on Networks with a Tree-Structure September 14 th 2018, Eindhoven, Netherlands

Lower bounds for treewidth-based algorithms •

Motivating questions Do faster algorithms exist … 1. when the bounded-width graph is also planar? [open problem from Lokshtanov, Marx & Saurabh] 2. when parameterizing by more restrictive width measures? Width measures considered earlier: 3 Boolean width h NLC width MIM width t wid Cl width lar width th d i w ique Tree Rank width du DAG h dt i w h Pat Mo Kelly width

Motivating questions Do faster algorithms exist … 1. when the bounded-width graph is also planar? [open problem from Lokshtanov, Marx & Saurabh] 2. when parameterizing by more restrictive width measures? More restrictive width measures considered earlier: • Pathwidth • Feedback vertex number • Vertex cover number 4

Decomposing graphs by small edge-cuts • cutwidth 3 5

Motivating questions Do faster algorithms exist … 1. when the bounded-width graph is also planar? [open problem from Lokshtanov, Marx & Saurabh] 2. when parameterizing by more restrictive width measures such as cutwidth? The answers to these questions are related! Lower bounds against cutwidth parameterizations, transfer to planar graphs by suitable crossover gadgets 6

Main results • 7

Conceptual contribution • 8

Planarization by vertices preserves cutwidth Replacing crossings by degree-4 vertices preserves cutwidth 9

Gadget planarization does not blow up cutwidth 10

Proof strategy for lower bounds • 11

Technical contribution Following NP-completeness, several crossover gadgets were developed [Garey, Johnson & Stockmeyer, TCS ‘ 76] [Lichtenstein, SICOMP ‘ 82] VERTEX COVER INDEPENDENT SET 3 -COLORABILITY SATISFIABILITY Existing INDEPENDENT SET gadget directly gives first lower bound We design a novel crossover gadget for DOMINATING SET 12

The DOMINATING SET problem • 13

Replacing single edges 14

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Replacing single edges 19

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Eliminating crossing triangles 21

Complete planarization operation • 22

Behavior of the Vertex Cover gadget 23

Conclusion • 24