Sophie Toulouse

The path partition problem and related problems in bipartite graphs

We prove that it is NP-complete to decide whether a bipartite graph of maximum degree three on nk vertices can be partitioned into n paths of length k. Finally, we propose some approximation and inapproximation results for several related problems.

Approximation algorithms for the traveling salesman problem

Abstract. We first prove that the minimum and maximum traveling salesman problems, their metric versions as well as some versions defined on parameterized triangle inequalities (called sharpened and relaxed metric traveling salesman) are all equi-approximable under an approximation measure, called differential-approximation ratio, that measures how the value of an approximate solution is placed in the interval between the worst- and the best-value solutions of an instance. We next show that the 2OPT, one of the most-known traveling salesman algorithms, approximately solves all these problems within differential-approximation ratio bounded above by 1/2. We analyze the approximation behavior of 2OPT when used to approximately solve traveling salesman problem in bipartite graphs and prove that it achieves differential-approximation ratio bounded above by 1/2 also in this case. We also prove that, for any ε>0, it is NP-hard to differentially approximate metric traveling salesman within better than 649/650 + ε and traveling salesman with distances 1 and 2 within better than 741/742 + ε. Finally, we study the standard approximation of the maximum sharpened and relaxed metric traveling salesman problems. These are versions of maximum metric traveling salesman defined on parameterized triangle inequalities and, to our knowledge, they have not been studied until now.

Differential approximation results for the traveling salesman problem with distances 1 and 2

Abstract We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min_TSP12 and max_TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard-approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any ϵ>0, it is NP-hard to approximate both problems better than within 741/742+ϵ. The same results hold when dealing with a generalization of min_ and max_TSP12, where instead of 1 and 2, edges are valued by a and b.

The Pk Partition Problem and Related Problems in Bipartite Graphs

In this paper, we continue the investigation proposed in [15] about the approximability of P k partition problems, but focusing here on their complexity. More precisely, we prove that the problem consisting of deciding if a graph of nkvertices has nvertex disjoint simple paths {P 1 , i¾? ,P n } such that each path P i has kvertices is NP -complete, even in bipartite graphs of maximum degree 3. Note that this result also holds when each path P i is chordless in G[V(P i )]. Then, we prove that the optimization version of these problems, denoted by Max P 3 Packing and MaxInduced P 3 Packing , are not in PTAS in bipartite graphs of maximum degree 3. Finally, we propose a 3/2-approximation for Min 3 -PathPartition in general graphs within O(nm+ n2logn) time and a 1/3 (resp., 1/2)-approximation for MaxW P 3 Packing in general (resp., bipartite) graphs of maximum degree 3 within O(i¾?(n,3n/2)n) (resp., O(n2logn)) time, where i¾?is the inverse Ackermans function and n= |V|, m= |E|.

Approximability of the Multiple Stack TSP

STSP seeks a pair of pickup and delivery tours in two distinct networks, where the two tours are related by LIFO contraints. We address here the problem approximability. We notably establish that asymmetric MaxSTSP and MinSTSP12 are APX, and propose a heuristic that yields to a 1/2, 3/4 and 3/2 standard approximation for respectively Max2STSP, Max2STSP12 and Min2STSP12.

Differential Approximation Results for the Traveling Salesman Problem with Distances 1 and 2

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any Ɛ > 0, it is NP-hard to approximate both problems within better than 5379/5380 + Ɛ.

Approximation results for the weighted P 4 partition problems

We present several new standard and differential approximation results for P4-partition problem by using the algorithm proposed in Hassin and Rubinstein (Information Processing Letters, 63: 63-67, 1997), for both minimization and maximization versions of the problem. However, the main point of this paper is the robustness of this algorithm, since it provides good solutions, whatever version of the problem we deal with, whatever the approximation framework within which we estimate its approximate solutions.

How Far From a Worst Solution a Random Solution of a \(k\,\)CSP Instance Can Be?

Given an instance I of an optimization constraint satisfaction problem (CSP), finding solutions with value at least the expected value of a random solution is easy. We wonder how good such solutions can be. Namely, we initiate the study of ratio \(\rho _E(I) =(\mathrm {E}_X[v(I, X)] -\mathrm {wor}(I))/(\mathrm {opt}(I) -\mathrm {wor}(I))\) where \(\mathrm {opt}(I)\), \(\mathrm {wor}(I)\) and \(\mathrm {E}_X[v(I, X)]\) refer to respectively the optimal, the worst, and the average solution values on I. We here focus on the case when the variables have a domain of size \(q \ge 2\) and the constraint arity is at most \(k \ge 2\), where k, q are two constant integers. Connecting this ratio to the highest frequency in orthogonal arrays with specified parameters, we prove that it is \(\varOmega (1/n^{k/2})\) if \(q =2\), \(\varOmega (1/n^{k -1 -\lfloor \log _{p^\kappa } (k -1)\rfloor })\) where \(p^\kappa \) is the smallest prime power such that \(p^\kappa \ge q\) otherwise, and \(\varOmega (1/q^k)\) in \((\max \{q, k\} +1\})\)-partite instances.

2 CSPs All Are Approximable Within a Constant Differential Factor

Only a few facts are known regarding the approximability of optimization CSPs with respect to the differential approximation measure, which compares the gain of a given solution over the worst solution value to the instance diameter. Notably, the question whether \(\mathsf {k\,CSP\!-\!q}\) is approximable within any constant factor is open in case when \(q \ge 3\) or \(k\ge 4\). Using a family of combinatorial designs we introduce for our purpose, we show that, given any three constant integers \(k\ge 2\), \(p\ge k\) and \(q >p\), \(\mathsf {k\,CSP\!-\!q}\) reduces to \(\mathsf {k\,CSP\!-\!p}\) with an expansion of \(1/(q~-~p~+~k/2)^k\) on the approximation guarantee. When \(p =k =2\), this implies together with the result of Nesterov as regards \(\mathsf {2\,CSP\!-\!2}\) [1] that for all constant integers \(q\ge 2\), \(\mathsf {2\,CSP\!-\!q}\) is approximable within factor \((2~-~\pi /2)/(q~-~1)^2\).

A new effective unified model for solving the Pre-marshalling and Block Relocation Problems

Abstract Container terminals are exchange hubs that interconnect many transportation modes and facilitate the flow of containers. Among other elements, terminals include a yard which serves as temporary storage space. In the yard, containers are piled up by cranes to form blocks of stacks. During the shipment process, containers are carried from the stacks to ships following a given sequence. Hence, if a high priority container is placed below low priority ones, such obstructing containers have to be moved (relocated) to other stacks. Given a set of stacks and a retrieval sequence, the aim in the Pre-marshalling Problem ( pmp ) is to sort the initial configuration according to the retrieval sequence using a minimum number of relocations, so that no new relocations are needed during the shipment. The objective in the Block Relocation Problem ( brp ) is to retrieve all the containers according to the retrieval sequence by using a minimum number of relocations. This paper presents a new unified integer programming model for solving the pmp , the brp , and the Restricted brp ( r-brp ) variant. The new formulations are compared with existing mathematical models for these problems, as well as with other exact methods that combines combinatorial lower bounds and the branch-and-bound (B&B) framework, by using a large set of instances available in the literature. The numerical experiments show that the proposed models are able to outperform the approaches based on mathematical programming. Nevertheless, the B&B algorithms achieve the best results both in terms of computation time and number of instances solved to optimality.