Colored cut games

Nils Morawietz, Niels Grüttemeier, Christian Komusiewicz, Frank Sommer

Abstract: In a graph G=(V,E)<math><mi is="true">G</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mo stretchy="false" is="true">(</mo><mi is="true">V</mi><mo is="true">,</mo><mi is="true">E</mi><mo stretchy="false" is="true">)</mo></math> with an edge coloring ℓ:E→C<math><mi is="true">ℓ</mi><mo is="true">:</mo><mi is="true">E</mi><mo stretchy="false" is="true">→</mo><mi is="true">C</mi></math> and two distinguished vertices s and t, a colored (s,t)<math><mo stretchy="false" is="true">(</mo><mi is="true">s</mi><mo is="true">,</mo><mi is="true">t</mi><mo stretchy="false" is="true">)</mo></math>-cut is a set C˜⊆C<math><mover accent="true" is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mo stretchy="false" is="true">˜</mo></mrow></mover><mo is="true">⊆</mo><mi is="true">C</mi></math> such that deleting all edges with some color c∈C˜<math><mi is="true">c</mi><mo is="true">∈</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mo stretchy="false" is="true">˜</mo></mrow></mover></math> from G disconnects s and t. Motivated by applications in the design of robust networks, we introduce colored cut games. In these games, an attacker and a defender choose colors to delete and to protect, respectively, in an alternating fashion. The attacker wants to achieve a colored (s,t)<math><mo stretchy="false" is="true">(</mo><mi is="true">s</mi><mo is="true">,</mo><mi is="true">t</mi><mo stretchy="false" is="true">)</mo></math>-cut and the defender wants to prevent this. First, we show that for an unbounded number of alternations, colored cut games are PSPACE-complete even on subcubic graphs. We then show that, even on subcubic graphs, colored cut games with i alternations are complete for classes in the polynomial hierarchy whose level depends on i. To complete the dichotomy, we show that all colored cut games are polynomial-time solvable on graphs with maximum degree at most 2.Next, we show that all colored cut games admit a polynomial kernel for the parameter k+κr<math><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><msub is="true"><mrow is="true"><mi is="true">κ</mi></mrow><mrow is="true"><mi is="true">r</mi></mrow></msub></math> where k denotes the total attacker budget and, for any constant r, κr<math><msub is="true"><mrow is="true"><mi is="true">κ</mi></mrow><mrow is="true"><mi is="true">r</mi></mrow></msub></math> is the number of vertex deletions that are necessary to transform G into a graph where the longest path has length at most r. For κ1<math><msub is="true"><mrow is="true"><mi is="true">κ</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub></math>, which is the vertex cover number vc of the input graph, the kernel has size O(vc2k2)<math><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><msup is="true"><mrow is="true"><mi mathvariant="normal" is="true">vc</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><msup is="true"><mrow is="true"><mi is="true">k</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mo stretchy="false" is="true">)</mo></math>. Moreover, we introduce an algorithm solving the most basic colored cut game, Colored (s,t)<math><mo stretchy="false" is="true">(</mo><mi is="true">s</mi><mo is="true">,</mo><mi is="true">t</mi><mo stretchy="false" is="true">)</mo></math>-Cut, in 2vc+knO(1)<math><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mrow is="true"><mi mathvariant="normal" is="true">vc</mi></mrow><mo linebreak="badbreak" linebreakstyle="after" is="true">+</mo><mi is="true">k</mi></mrow></msup><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi mathvariant="script" is="true">O</mi><mo stretchy="false" is="true">(</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></mrow></msup></math> time.