Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2 - epsilon)(n)m(O(1)) time, we show that for any epsilon > 0; INDEPENDENT SET cannot be solved in time (2 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)), DOMINATING SET cannot be solved in time (3 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)), MAX CUT cannot be solved in time (2 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)), ODD CYCLE TRANSVERSAL cannot be solved in time (3 - epsilon)(tw(G))vertical bar V(G)(vertical bar)(O(1)), For any q >= 3, q-COLORING cannot be solved in time (q - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)), PARTITION INTO TRIANGLES cannot be solved in time (2 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)). Our lower bounds match the running times for the best known algorithms for the problems, up to the epsilon in the base.