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

Motivated by the well known four-thirds conjecture for the traveling salesman problem (TSP), we study the problem of {\em uniform covers}. A graph $G=(V,E)$ has an $\alpha$-uniform cover for TSP (2EC, respectively) if the everywhere $\alpha$ vector (i.e. $\{\alpha\}^{E}$) dominates a convex combination of incidence vectors of tours (2-edge-connected spanning multigraphs, respectively). The polyhedral analysis of Christofides' algorithm directly implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP. Seb\H{o} asked if such graphs have $(1-\epsilon)$-uniform covers for TSP for some $\epsilon > 0$. Indeed, the four-thirds conjecture implies that such graphs have 8/9-uniform covers. We show that these graphs have 18/19-uniform covers for TSP. We also study uniform covers for 2EC and show that the everywhere 15/17 vector can be efficiently written as a convex combination of 2-edge-connected spanning multigraphs. For a weighted, 3-edge-connected, cubic graph, our results show that if the everywhere 2/3 vector is an optimal solution for the subtour linear programming relaxation, then a tour with weight at most 27/19 times that of an optimal tour can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall into this category. In this special case, we can apply our tools to obtain an even better approximation guarantee. To extend our approach to input graphs that are 2-edge-connected, we present a procedure to decompose an optimal solution for the subtour relaxation for TSP into spanning, connected multigraphs that cover each 2-edge cut an even number of times. Using this decomposition, we obtain a 17/12-approximation algorithm for minimum weight 2-edge-connected spanning subgraphs on subcubic, node-weighted graphs

Towards Improving Christofides Algorithm on Fundamental Classes by Gluing Convex Combinations of Tours

We present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a $\theta$-cyclic point: $x_e \in \{0,\theta, 1-\theta, 1\}$, where the support graph is subcubic and each vertex is incident to at least one edge with $x$-value 1. Such points are sufficient to resolve TSP in general. For these points, we construct a convex combination of tours in which we can reduce the usage of edges with $x$-value 1 from the $\frac{3}{2}$ of Christofides algorithm to $\frac{3}{2}-\frac{\theta}{10}$ while keeping the usage of edges with fractional $x$-value the same as Christofides algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP: In the case when the objective function of the subtour elimination relaxation is minimized by a $\frac{2}{3}$-uniform point: $x_e \in \{0, \frac{2}{3}\}$, we give a $\frac{17}{12}$-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios of $\frac{3}{2}$ of Christofides algorithm and $\frac{4}{3}$ implied by the famous "four-thirds conjecture"

Finding a Second Hamiltonian cycle in Barnette Graphs

We study the following two problems: (1) finding a second room-partitioning of an oik, and (2) finding a second Hamiltonian cycle in cubic graphs. The existence of solution for both problems is guaranteed by a parity argument. For the first problem we prove that deciding whether a 2-oik has a room-partitioning is NP-hard, even if the 2-oik corresponds to a planar triangulation. For the problem of finding a second Hamiltonian cycle, we state the following conjecture: for every cubic planar bipartite graph finding a second Hamiltonian cycle can be found in time linear in the number of vertices via a standard pivoting algorithm. We fail to settle the conjecture, but we prove it for cubic planar bipartite WH(6)-minor free graphs

Approximation Algorithms for Flexible Graph Connectivity

We present approximation algorithms for several network design problems in the model of Flexible Graph Connectivity (Adjiashvili, Hommelsheim and M\"uhlenthaler, "Flexible Graph Connectivity", Math. Program. pp. 1-33 (2021), and IPCO 2020: pp. 13-26). Let $k\geq 1$, $p\geq 1$ and $q\geq 0$ be integers. In an instance of the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, we have an undirected connected graph $G = (V,E)$, a partition of $E$ into a set of safe edges $S$ and a set of unsafe edges $U$, and nonnegative costs $c: E\to\Re$ on the edges. A subset $F \subseteq E$ of edges is feasible for the $(p,q)$-FGC problem if for any subset $F'$ of unsafe edges with $|F'|\leq q$, the subgraph $(V, F \setminus F')$ is $p$-edge connected. The algorithmic goal is to find a feasible solution $F$ that minimizes $c(F) = \sum_{e \in F} c_e$. We present a simple $2$-approximation algorithm for the $(1,1)$-FGC problem via a reduction to the minimum-cost rooted $2$-arborescence problem. This improves on the $2.527$-approximation algorithm of Adjiashvili et al. Our $2$-approximation algorithm for the $(1,1)$-FGC problem extends to a $(k+1)$-approximation algorithm for the $(1,k)$-FGC problem. We present a $4$-approximation algorithm for the $(p,1)$-FGC problem, and an $O(q\log|V|)$-approximation algorithm for the $(p,q)$-FGC problem. Finally, we improve on the result of Adjiashvili et al. for the unweighted $(1,1)$-FGC problem by presenting a $16/11$-approximation algorithm. The $(p,q)$-FGC problem is related to the well-known Capacitated $k$-Connected Subgraph problem (denoted Cap-k-ECSS) that arises in the area of Capacitated Network Design. We give a $\min(k,2 u_{max})$-approximation algorithm for the Cap-k-ECSS problem, where $u_{max}$ denotes the maximum capacity of an edge.Comment: 23 pages, 1 figure, preliminary version in the Proceedings of the 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021), December 15-17, (LIPIcs, Volume 213, Article No. 9, pp. 9:1-9:14), see https://doi.org/10.4230/LIPIcs.FSTTCS.2021.9. Related manuscript: arXiv:2102.0330

Towards improving Christofides algorithm for half-integer TSP

International audienc

New Bounds on Integrality Gaps by Constructing Convex Combinations

This dissertation studies the integrality gap of linear programming relaxations of integer programs. The integrality gap of a continuous relaxation of the sets of latticepoints corresponding to integer feasible solutions is the worst case ratio between the cost of an integer feasible solution and the optimal value of the continuous relaxation.The main focus in the ?first part of the thesis is on the Traveling Salesperson Problem (TSP) and the 2-edge-connected multigraph problem (2ECM). In TSP and 2ECM we are given n vertices with costs on pairs of vertices. We consider cost functions obeying triangle inequality. In TSP the goal is to fi?nd the minimum cost Hamiltonian cycle andin the 2ECM the goal is to ?find the minimum cost 2-edge-connected subgraph. Both problems can be formulated via a linear programming relaxation known as the subtour elimination relaxation. The most general case for TSP and 2ECM has resisted approximation algorithms (and upper bounds on the integrality gap with the subtourelimination relaxation) better than 3/2 for decades.In Chapter 3 we consider TSP and 2ECM on node-weighted graphs. These are instances where the cost on the pairs of vertices arise from a shortest path between the pair in a node-weighted graph, a graph with edge weights arising from adding the costs of its endpoints. First we show that for 3-edge-connected cubic graphs, there is a 7/5 approximation algorithm for the node-weighted TSP and a 13/10-approximation for the node-weighted 2ECM. The main tool for both algorithms is the fact that 3-edgeconnectedcubic graphs contain 2-factors covering all their small edge cuts. We extend this result to subcubic graphs by providing a decomposition of a point of the subtour elimination relaxation into a convex combination of connected multigraphs, each covering 2-edge cuts an even number of times. An application of this decomposition leads to a17/12 -approximation algorithm for node-weighted 2ECM on subcubic graphs. Chapter 4 focuses on the Uniform Cover Problem for TSP and 2ECM. We establish this framework as a way to approach the most general case of TSP and 2ECM. As a ?first result, we give the ?first positive answer to Seb}o et al. [SBS14] regarding the uniform cover problem for TSP by showing that for a 3-edge-connected cubic graph, the incidence vector of G multiplied by 18=19 can be decomposed into a convex combination of solutions for the TSP: this is equivalent to a 27/19-approximation for TSP on such instances. We also provide a 45/34 -approximation for 2ECM on such instances. This is the first bound below 43 that can be proved via an efficient rounding algorithm. Improving this factor further requires a technique commonly known as \gluing". We show how gluing on 3-edge cuts reduces our problems to more structured instances. For such structured instances we use a novel application of a rainbow 1-tree decomposition that serves a top-down coloring algorithm in order to improve the factor of 45/34 ≈ 1:323 to 123/94 ≈ 1:308. In Chapter 5 our focus is on half-integer points of the subtour elimination relaxationmotivated by the conjecture of Schalekamp, Williamson, van Zuylen [SWvZ13] that the largest integrality gap is achieved for instances where the optimal solution of the subtourelimination relaxation is half-integer. Our focus is on fundamental classes that are a class of interesting yet highly structured points in the subtour elimination relaxation.In particular, we study half-square points and half-triangle points. For half-squarepoints we provide a 9/7 -approximation for 2ECM and for half-triangle points we show a ( 6/5 + 1/120 )-approximation for 2ECM. In Chapter 6 we investigate the possibility of gluing the solutions for T SP over 3-edge cuts. Gluing over 3-edge cuts has proven to be successful for 2-edge-connected subgraphs but there is not much known in this direction for gluing connected multigraphs. We introduce a novel approach of gluing solutions to the TSP based on different parts ofa tour: (i) the connected skeleton of a solution which is a connected subgraph and (ii) the parity correction part of the solution that augments the connected skeleton into anEulerian connected multigraph. Using this approach we show that for a half-integer point x of the subtour elimination relaxation, we can reduce the usage of edges with x-value 1 from the 3/2 of Christo?des' algorithm to 3/2 - 1/20 while keeping the usage of edges with x-value of 1 2 the same as Christo?des' algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP, where we show that for a 3-edge-connected cubic graph, the incidence vector of G multiplied by 17/18 can be decomposed into a convex combination of solutions for the TSP: In this way we improvethe 27/19 -approximation algorithm in Chapter 4 to a 17/12 -approximation algorithm for TSP on these instances.In the fi?nal chapter of this thesis, we focus on general binary integer programs (binary IPs) and show an efficient algorithm, called the Fractional Decomposition Tree Algorithm (FDT), that provides an upper bound on the integrality gap of an instance of a binary IP with its linear programming relaxation. As a stepping stone, we design an efficient algorithm for ?finding a feasible integer solution to binary IPs with bounded integrality gap which may be of independent interest. We extend FDT to ?find convex combinations of 2-edge-connected multigraphs which is a non-binary problem. We run experiments and compare upper bounds provided by FDT with that of polyhedral version of Christo?des' algorithm.</div

Towards Improving Christofides Algorithm for Half-Integer TSP

International audienceWe study the traveling salesman problem (TSP) in the case when the objective function of the subtour linear programming relaxation is minimized by a half-cycle point: xe ∈ {0, 1 2 , 1} where the half-edges form a 2-factor and the 1-edges form a perfect matching. Such points are sufficient to resolve half-integer TSP in general and they have been conjectured to demonstrate the largest integrality gap for the subtour relaxation. For half-cycle points, the best-known approximation guarantee is 3 2 due to Christofides' famous algorithm. Proving an integrality gap of α for the subtour relaxation is equivalent to showing that αx can be written as a convex combination of tours, where x is any feasible solution for this relaxation. To beat Christofides' bound, our goal is to show that (3 2 −)x can be written as a convex combination of tours for some positive constant. Let ye = 3 2 − when xe = 1 and ye = 3 4 when xe = 1 2. As a first step towards this goal, our main result is to show that y can be written as a convex combination of tours. In other words, we show that we can save on 1-edges, which has several applications. Among them, it gives an alternative algorithm for the recently studied uniform cover problem. Our main new technique is a procedure to glue tours over proper 3-edge cuts that are tight with respect to x, thus reducing the problem to a base case in which such cuts do not occur

Towards Improving Christofides Algorithm on Fundamental Classes by Gluing Convex Combinations of Tours

International audienceWe present a new approach for gluing tours over certain tight, 3-edge cuts. Gluing over 3-edge cuts has been used in algorithms for finding Hamilton cycles in special graph classes and in proving bounds for 2-edge-connected subgraph problem, but not much was known in this direction for gluing connected multigraphs. We apply this approach to the traveling salesman problem (TSP) in the case when the objective function of the subtour elimination relaxation is minimized by a θ-cyclic point: x e ∈ {0, θ, 1 − θ, 1}, where the support graph is subcubic and each vertex is incident to at least one edge with x-value 1. Such points are sufficient to resolve TSP in general. For these points, we construct a convex combination of tours in which we can reduce the usage of edges with x-value 1 from the 3 2 of Christofides algorithm to 3 2 − θ 10 while keeping the usage of edges with fractional x-value the same as Christofides algorithm. A direct consequence of this result is for the Uniform Cover Problem for TSP: In the case when the objective function of the subtour elimination relaxation is minimized by a 2 3-uniform point: x e ∈ {0, 2 3 }, we give a 17 12-approximation algorithm for TSP. For such points, this lands us halfway between the approximation ratios of 3 2 of Christofides algorithm and 4 3 implied by the famous "four-thirds conjecture"