Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal (1007.5450v1)

Abstract: 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 e > 0; {\sc Independent Set} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Dominating Set} cannot be solved in (3-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Max Cut} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Odd Cycle Transversal} cannot be solved in (3-e)^{tw(G)}|V(G)|^{O(1)} time, For any $q \geq 3$, $q$-{\sc Coloring} cannot be solved in (q-e)^{tw(G)}|V(G)|^{O(1)} time, {\sc Partition Into Triangles} cannot be solved in (2-e)^{tw(G)}|V(G)|^{O(1)} time. Our lower bounds match the running times for the best known algorithms for the problems, up to the e in the base.

• The paper demonstrates that current algorithms for NP-hard problems on bounded treewidth graphs are optimal up to sub-exponential factors under the Strong Exponential Time Hypothesis.
• It employs precise many-one reductions that carefully control treewidth to derive matching lower bounds for problems such as INDEPENDENT SET and DOMINATING SET.
• The results suggest that efforts to design faster algorithms for these problems should instead focus on challenging SETH or exploring alternative graph parameters.

Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

This paper by Lokshtanov, Marx, and Saurabh addresses a fundamental question in the field of graph algorithms: the efficiency of solving problems on graphs with bounded treewidth. The authors establish robust lower bounds on the running time of algorithms for several NP-hard graph problems when constrained to graphs of bounded treewidth. These results are premised on the Strong Exponential Time Hypothesis (SETH) proposed by Impagliazzo and Paturi. The implications of this assertion are significant for the theoretical limits of algorithmic graph theory. The authors emphasize that if any algorithm surpasses these lower bounds, it would challenge the SETH assumption, suggesting that existing known algorithms are theoretically optimal.

Key Results

The paper focuses on the following graph problems:

• INDEPENDENT SET: No algorithm can solve this problem in time $O*((2 - \varepsilon)^{tw(G)})$.
• DOMINATING SET: No algorithm can achieve $O*((3 - \varepsilon)^{tw(G)})$ time complexity.
• MAX CUT: Time bound established at $O*((2 - \varepsilon)^{tw(G)})$.
• ODD CYCLE TRANSVERSAL: Establishes the bound $O*((3 - \varepsilon)^{tw(G)})$.
• q-COLORING: For $q \geq 3$, bounded by $O*((q - \varepsilon)^{tw(G)})$.
• PARTITION INTO TRIANGLES: Time complexity bounded at $O*((2 - \varepsilon)^{tw(G)})$.

These bounds match the running times for the best-known algorithms for these problems, modulo the $\varepsilon$ in their bases.

Technical Approach

The authors employ polynomial time many-one reductions to derive the lower bounds, managing the treewidth carefully through their constructions. Standard reductions are enhanced by meticulously controlling the graph's treewidth, ensuring that proofs are aligned tightly with the hypothesis. For example, in the case of DOMINATING SET, they construct instances whose treewidth grows minimally with each added variable, illustrating the subtlety in achieving these results.

SETH posits that the satisfiability problem (SAT) cannot be solved in sub-exponential time in generality. The authors argue that solving the aforementioned graph problems faster than the noted bounds would consequentially lead to contradicting SETH by yielding a faster SAT algorithm.

Implications and Future Work

The implications of these findings are both practical and theoretical. From a practical perspective, researchers in algorithm design can focus their efforts away from seeking faster algorithms for bounded treewidth graphs for these specific problems. Instead, efforts could be better invested in direct challenges to SETH or in other domains like planar graphs where such tight bounds have not yet been established.

The authors propose several directions for future research:

• Investigating whether similar lower bounds hold for graphs with restrictions like planarity.
• Exploring bounds based on parameters other than treewidth, such as feedback vertex set size.
• Extending these methods to additional graph problems that align with known exponential time parameter traits.

This work delineates an important boundary in algorithmic graph theory, prompting researchers to refine focus and explore untapped edges of computational complexity beyond known territory.