András Sebő

Shorter tours by nicer ears: 7/5-Approximation for the graph-TSP, 3/2 for the path version, and 4/3 for two-edge-connected subgraphs

We prove new results for approximating the graph-TSP and some related problems. We obtain polynomial-time algorithms with improved approximation guarantees.For the graph-TSP itself, we improve the approximation ratio to 7=5. For a generalization, the minimum T-tour problem, we obtain the first nontrivial approximation algorithm, with ratio 3=2. This contains the s-t-path graph-TSP as a special case. Our approximation guarantee for finding a smallest 2-edge-connected spanning subgraph is 4=3.The key new ingredient of all our algorithms is a special kind of ear-decomposition optimized using forest representations of hypergraphs. The same methods also provide the lower bounds (arising from LP relaxations) that we use to deduce the approximation ratios.

Flows, View Obstructions, and the Lonely Runner

We prove the following result: LetGbe an undirected graph. IfGhas a nowhere zero flow with at mostkdifferent values, then it also has one with values from the set {1, ?, k}. Whenk?5, this is a trivial consequence of Seymours “six-flow theorem”. Whenk?4 our proof is based on a lovely number theoretic problem which we call the “Lonely Runner Conjecture:” Supposekrunners having nonzero constant speeds run laps on a unit-length circular track. Then there is a time at which all runners are at least 1/(k+1) from their common starting point. This conjecture appears to have been formulated by J. Wills (Monatsch. Math.71, 1967) and independently by T. Cusick (Aequationes Math.9, 1973). This conjecture has been verified fork?4 by Cusick and Pomerance (J. Number Theory19, 1984) in a complicated argument involving exponential sums and electronic case checking. A major part of this paper is an elementary selfcontained proof of the casek=4 of the Lonely Runner Conjecture.

Eight-Fifth approximation for the path TSP

We prove the approximation ratio 8/5 for the metric {s, t}-path-TSP problem, and more generally for shortest connected T -joins. The algorithm that achieves this ratio is the simple “Best of Many” version of Christofides’ algorithm (1976), suggested by An, Kleinberg and Shmoys (2012), which consists in determining the best Christofides {s, t}-tour out of those constructed from a family F>0 of trees having a convex combination dominated by an optimal solution x of the fractional relaxation. They give the approximation guarantee √ 5+1 2 for such an {s, t}-tour, which is the first improvement after the 5/3 guarantee of Hoogeveen’s Christofides type algorithm (1991). Cheriyan, Friggstad and Gao (2012) extended this result to a 13/8-approximation of shortest connected T -joins, for |T | ≥ 4. The ratio 8/5 is proved by simplifying and improving the approach of An, Kleinberg and Shmoys that consists in completing x/2 in order to dominate the cost of “parity correction” for spanning trees. We partition the edge-set of each spanning tree in F>0 into an {s, t}-path (or more generally, into a T -join) and its complement, which induces a decomposition of x. This decomposition can be refined and then efficiently used to complete x/2 without using linear programming or particular properties of T , but by adding to each cut deficient for x/2 an individually tailored explicitly given vector, inherent in x. A simple example shows that the Best of Many Christofides algorithm may not find a shorter {s, t}-tour than 3/2 times the incidentally common optima of the problem and of its fractional relaxation. keywords: traveling salesman problem, path TSP, approximation algorithm, T -join, polyhedron

Coloring precolored perfect graphs

We consider the question of the computational complexity of coloring perfect graphs with some precolored vertices. It is well known that a perfect graph can be colored optimally in polynomial time. Our results give a sharp border between the polynomial and NP-complete instances, when precolored vertices occur. The key result on the polynomially solvable cases includes a good characterization theorem on the existence of an optimal coloring of a perfect graph where a given stable set is precolored with only one color. The key negative result states that the 3-colorability of a graph whose odd circuits go through two fixed vertices is NP-complete. The polynomial algorithms use Grotschel, Lovasz and Schrijvers algorithm for finding a maximum clique in a graph, but are otherwise purely combinatorial. c

An Improved Approximation Algorithm for Minimum Size 2-Edge Connected Spanning Subgraphs

We give a \( \frac{{17}} {{12}} \) -approximation algorithm for the following NP- hard problem: Given a simple undirected graph, find a 2-edge connected span- ning subgraph that has the minimum number of edges. The best previous approximation guarantee was \( \frac{3} {2} \) . If the well known TSP \( \frac{4} {3} \) conjecture holds, then there is a \( \frac{4} {3} \) -approximation algorithm. Thus our main result gets half-way to this target.

The connectivity of minimal imperfect graphs

fr ABSTRACT We prove that partitionable graphs are 2w - 2-connected, that this bound is sharp, and prove some structural properties of cutsets of cardinality 2w - 2. The proof of the con- nectivity result is a simple linear algebraic proof. 0 1996 John Wiley & Sons, Inc.

Multiflow Feasibility: An Annotated Tableau

We provide a tableau of 189 entries and some annotations presenting the computational complexity of integer multiflow feasibility problems; 21 entries remain open. The tableau is followed by an introduction to the field, providing more problems, reproving some results with new insights, simple proofs, or slight sharpenings motivated by the tableau, paying particular attention to planar (di)graphs with terminals on the boundary. Last, the key-theorems and key-problems of the tableau are listed.

Minmax relations for cyclically ordered digraphs

We prove a range of minmax theorems about cycle packing and covering in digraphs whose vertices are cyclically ordered, a notion promoted by Bessy and Thomassé in their beautiful proof of the following conjecture of Gallai: the vertices of a strongly connected digraph can be covered by at most as many cycles as the stability number. The results presented here provide relations between cycle packing and covering and various objects in graphs such as stable sets, their unions, or feedback vertexand arc-sets. They contain the results of Bessy and Thomassé with simple algorithmic proofs, including polynomial algorithms for weighted variants, classical results on posets extending Greene and Kleitman’s theorem (that in turn contains Dilworth’s theorem), and a common generalization of these. The most general minmax results concern the maximum number of vertices of a k-chromatic subgraph (k ∈ N)—as a consequence, this number is greater than or equal to the minimum of |X| + k|C|, running on subsets of vertices X and families of cycles C covering all vertices not in X, in strongly connected digraphs. This is the “circuit cover” version of a conjecture of Linial (like Gallai’s conjecture is the circuit cover version of the Gallai– Milgram theorem, meaning that path partitions are replaced by circuit covers under strong connectivity); we also deduce the circuit cover version of a conjecture of Berge on path partitions; these conjectures remain open, but the proven statements also bound the maximum size of a k-chromatic subgraph, contain Gallai’s conjecture, and Bessy and Thomassé’s theorems. All presented minmax relations are proved using cyclic orders and a unique elementary argument based on network flows—algorithmically only shortest paths and potentials in conservative digraphs—varying the parameters of the network flow model. In this way antiblocking and blocking relations can be established, leading to a general polyhedral phenomenon—a combination of “integer decomposition, integer rounding” and “dual integrality”—that also contains the matroid partition theorem and Dilworth’s theorem. We provide ✩ This research is part of the ADONET network’s program (a Marie Curie training network of the European community), “milestone 3.1.” It was completed when the author was a visitor at the Caesarea Edmond Benjamin de Rothschild Foundation Institute for Interdisciplinary Applications of Computer Science at the University of Haifa. The support and the exceptional hospitality of the Institute were a decisive factor in the successful termination of the work. E-mail address: andras.sebo@imag.fr. 0095-8956/

Imperfect and Nonideal Clutters: A Common Approach

– see front matter

Cyclic orders: Equivalence and duality

We prove three theorems. First, Lovász’s theorem about minimal imperfect clutters, including also Padberg’s corollaries. Second, Lehman’s result on minimal nonideal clutters. Third, a common generalization of these two. The endeavor of working out a ‘common denominator’ for Lovász’s and Lehman’s theorems leads, besides the common generalization, to a better understanding and simple polyhedral proofs of both.