In this paper we analyze the complexity of algorithms for two problems that arise in automatic test path generation for programs: the problem of building a path through a specified set of program statements and the problem of building a path which satisfies impossible-pairs restrictions on statement pairs. These problems are both reduced to graph traversal problems. We give an efficient algorithm for the first, and show that the second is NP-complete.

On Two Problems in the Generation of Program Test Paths ; CU-CS-081-75

Small-scale DNA physical mapping (such as the Double Digest Problem or DDP) is an important and difficult problem in computational molecular biology. When enzyme sites are modeled by a random process, the number of solutions to DDP is known to increase exponentially as the length of DNA increases. However, the overwhelming majority of solutions are very similar and can be transformed into each other by simple transformations. Recently, Schmitt and Waterman [SW] introduced equivalence classes on the set of DDP solutions and raised an open problem to completely characterize equivalent physical maps. We study the combinatorics of multiple solutions and the cassette transformations of Schmitt and Waterman. We demonstrate that the solutions to DDP are closely associated with alternating Eulerian cycles in colored graphs and study order transformations of alternating cycles. We prove that every two alternating Eulerian cycles in a bicolored graph can be transformed into each other by means of order transformations. Using this result we obtain a complete characterization of equivalent physical maps in the Schmitt-Waterman problem. It also allows us to prove Ukkonen's conjecture on word transformations preservingq-gram composition.

DNA physical mapping and alternating Eulerian cycles in colored graphs

An efficient fixed-parameter algorithm for 3-hitting set

https://www.lri.fr/~yannis/dm-leandro.pdf

Proper connection of graphs

We survey approximation algorithms for some well-known and very natural combinatorial optimization problems, the minimum set covering, the minimum vertex covering, the maximum set packing, and maximum independent set problems; we discuss their approximation performance and their complexity. For already known results, any time we have conceived simpler proofs than those already published, we give these proofs, and, for the rest, we cite the simpler published ones. Finally, we discuss how one can relate the approximability behavior (from both a positive and a negative point of view) of vertex covering to the approximability behavior of a restricted class of independent set problems.

A survey of approximately optimal solutions to some covering and packing problems

Paths and trails in edge-colored graphs

On the Complexity of Some Hamiltonian and Eulerian Problems in Edge-Colored Complete Graphs

In an edge-colored, we say that a path (cycle) is alternating if it has length at least 2 (3) and if any 2 adjacent edges of this path (cycle) have different colors. We give efficient algorithms for finding alternating factors with a minimum number of cycles and then, by using this result, we obtain polynomial algorithms for finding alternating Hamiltonian cycles and paths in 2-edge-colored complete graphs. We then show that some extensions of these results to k-edge-colored complete graphs, k ≥ 3, are NP-complete. related problems are proposed. Finally, we give a polynomial characterization of the existence of alternating Eulerian cycles in edge-colored graphs. Our proof is algorithmic and uses a procedure that finds a perfect matching in a complete k-partite graph.

/pdf/hamiltonian-problems-in-edge-colored-complete-graphs-and-57mbpglnk4.pdf

Hamiltonian problems in edge-colored complete graphs and Eulerian cycles in edge-colored graphs: some complexity results

This paper deals with the existence and search of Properly Edge-Colored paths/trails between two, not necessarily distinct, vertices s and t in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s - t path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest Properly Edge-Colored path/trail between s and t for some particular graphs and characterize edge-colored graphs without Properly Edge-Colored closed trails. Next, we prove that deciding whether there exist k pairwise vertex/edge disjoint Properly Edge-Colored s - t paths/trails in a c-edge-colored graph Gc is NP-complete even for k = 2 and c = Ω(n2), where n denotes the number of vertices in Gc. Moreover, we prove that these problems remain NP-complete for c-colored graphs containing no Properly Edge-Colored cycles and c = Ω(n). We obtain some approximation results for those maximization problems together with polynomial results for some particulars classes of edge-colored graphs.

https://www.lri.fr/~yannis/pathstrails.pdf

Sufficient degree conditions for the existence of properly edge-colored cycles and paths in edge-colored graphs, multigraphs and random graphs are investigated. In particular, we prove that an edge-colored multigraph of order n on at least three colors and with minimum colored degree greater than or equal to ⌈(n+1)-2⌉ has properly edge-colored cycles of all possible lengths, including hamiltonian cycles. Longest properly edge-colored paths and hamiltonian paths between given vertices are considered as well. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 63–86, 2010

https://www.lri.fr/~yannis/cyclespaths.pdf

R. Saad

Papers

Paths and trails in edge-colored graphs

On the Complexity of Some Hamiltonian and Eulerian Problems in Edge-Colored Complete Graphs

Hamiltonian problems in edge-colored complete graphs and Eulerian cycles in edge-colored graphs: some complexity results

Paths and trails in edge-colored graphs

Cycles and paths in edge-colored graphs with given degrees