Parameterized Traveling Salesman Problem: Beating the Average

The fundamental theorem of Turán from Extremal Graph Theory determines the exact bound on the number of edges tr(n) in an n-vertex graph that does not contain a clique of size r + 1. We establish an interesting link between Extremal Graph Theory and Algorithms by providing a simple compression algorithm that in linear time reduces the problem of finding a clique of size ℓ in an n-vertex graph G with m ≥ tr(n) − k edges, where ℓ ≤ r + 1, to the problem of finding a maximum clique in a graph on at most 5k vertices. This also gives us an algorithm deciding in time 2.49 k • (n + m) whether G has a clique of size ℓ. As a byproduct of the new compression algorithm, we give an algorithm that in time 2 O(td 2 ) • n 2 decides whether a graph contains an independent set of size at least n/(d + 1) + t. Here d is the average vertex degree of the graph G. The multivariate complexity analysis based on ETH indicates that the asymptotical dependence on several parameters in the running times of our algorithms is tight.

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“…This approach was successfully applied to various problems, see e.g. [1,5,14,15,16,17,18,24,26,21,12].…”

Section: Related Workmentioning

confidence: 99%

Algorithmic Extensions of Dirac's Theorem

Fomin¹,

Golovach²,

Sagunov³

2022

Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

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“…A natural question extending Vizing's question is as follows: for each fixed k, is there a polynomial-time algorithm which, given w, determines if there exists a Hamilton cycle H * satisfying w(H * ) ≤ dn − k? Gutin and Patel [30] answered this question affirmatively: The proof of Theorem 14 is quite involved, and heavily uses the fact that K n is an undirected graph. This motivates the following problem: Open Problem 10.…”

Section: Traveling Salesperson Problemmentioning

confidence: 99%

“…Is there a fixed-parameter algorithm, which given a complete directed graph D whose every arc has an integer weight, decides whether D has a Hamilton cycle of weight not larger that the average weight of a Hamilton cycle minus k, where k is the parameter? Gutin and Patel [30] believe that the answer to Open Problem 10 is positive.…”

Section: Traveling Salesperson Problemmentioning

confidence: 99%

A Survey on Graph Problems Parameterized Above and Below Guaranteed Values

Gutin¹,

Mnich²

2022

Preprint

Self Cite

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“…Both Longest Detour and Longest Path above Diameter fit into the research subarea of parameterized complexity called "above guarantee" parameterization [44,1,15,31,32,33,34,35,43,45]. Besides the work of Bezáková et al [6], several papers study parameterization of longest paths and cycles above different guarantees.…”

Section: Related Workmentioning

confidence: 99%

Detours in Directed Graphs

Fomin¹,

Golovach²,

Lochet³

et al. 2022

Preprint

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We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s, t)-path of length at least dist G (s, t) + k (where dist G (s, t) denotes the length of a shortest path from s to t). Bezáková et al. [7] proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2 O(k) • n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2 O(k) • n O(1) time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs.In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k (diam(G) denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path above Diameter is NP-complete even for k = 1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path above Diameter is solvable in polynomial time for each k ∈ {1, . . . , 4} and is NP-complete for every k ≥ 5. The parameterized complexity of Longest Detour on general directed graphs remains an interesting open problem.

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Parameterized Traveling Salesman Problem: Beating the Average

Cited by 11 publications

References 6 publications

Algorithmic Extensions of Dirac's Theorem

Algorithmic Extensions of Dirac's Theorem

A Survey on Graph Problems Parameterized Above and Below Guaranteed Values

Detours in Directed Graphs

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