The salesman’s improved tours for fundamental classes

Abstract

Finding the exact integrality gap α for the LP relaxation of the metric Travelling Salesman Problem (TSP) has been an open problem for over thirty years, with little progress made. It is known that 4/3 ≤ α ≤ 3/2, and a famous conjecture states α = 4/3. It has also been conjectured that there exist halfinteger basic solutions of the linear program for which the highest integrality gap is reached. For this problem, essentially two "fundamental" classes of instances have been proposed. This fundamental property means that in order to show that the integrality gap is at most ρ for all instances of the metric TSP, it is sufficient to show it only for the instances in the fundamental class. However, despite the importance and the simplicity of such classes, no apparent effort has been deployed for improving the integrality gap bounds for them. In this paper we take a natural first step in this endeavour, and consider the 1/2-integer points of one such class. We successfully improve the upper bound for the integrality gap from 3/2 to 10/7 for a superclass of these points for which a lower bound of 4/3 is proved.A key role in the proof of this result is played by finding Hamiltonian cycles whose existence is equivalent to Kotzig's result on "compatible Eulerian tours", and which lead us to delta-matroids for developing the related algorithms. Our arguments also involve other innovative tools from combinatorial optimization with the potential of a broader use.

Shorter tours and longer detours: uniform covers and a bit beyond

“…Sebő [SBS14] asked if this bound can be improved: Does a 3-edge-connected, cubic graph have a (1 − )-uniform cover (for some small constant )? For the special class of 3-edgeconnected, cubic graphs that are also Hamiltonian, Boyd and Sebő show that the everywhere 6 7 vector for G dominates a convex combination of tours [BS17]. We give an affirmative answer to Sebő's question and improve this factor from 1 to 18 19 for all 3-edge-connected, cubic graphs 1 .…”

“…Based on a polyhedral analysis of Christofides' algorithm, we know that 3 2 x dominates a convex combination of tours in G x [Wol80,SW90]; so far we cannot replace 3 2 with any smaller constant. Following the terminology of Boyd and Sebő [BS17], for a graph G = (V, E) on n vertices, let the everywhere r vector for G, be the vector in R ( V 2 ) that is r in all coordinates corresponding to edges of G and 0 in all other coordinates. Conjecture 1 is closely related to the problem of uniform covers, which we now formally define.…”

“…Since the everywhere 2 3 vector for a 3-edge-connected, cubic graph G is in Subtour = (G), Sebő pointed out that the four-thirds conjecture implies that for a 3-edge-connected, cubic graph, the everywhere 8 9 vector dominates a convex combination of tours [SBS14]. The following is therefore a relaxed version of Conjecture 1 [SBS14,BS17].…”

Efficient constructions of convex combinations for 2-edge-connected subgraphs on fundamental classes

“…(This specific square point is discussed in Section 4.4.) Furthermore, half-integer square points also demonstrate the lower bound of 4 3 for the integrality gap of TSP with respect to the Held-Karp relaxation [BS17].…”

“…for special cases there has been some progress towards validating it. An important such class is half-integer points, which are conjectured to exhibit the largest gap for TSP (e.g., see [SWvZ12,BS17]). Carr and Ravi proved that α LP 2EC ≤ 4 3 if the optimal solution to min x∈LP(G) wx is half-integer [CR98].…”