Graph classes and the complexity of the graph orientation minimizing the maximum weighted outdegree

Given an undirected graph with edge weights, we are asked to find an orientation, that is, an assignment of a direction to each edge, so as to minimize the weighted maximum outdegree in the resulted directed graph. The problem is called MMO, and is a restricted variant of the well-known minimum makespan problem. As in previous studies, it is shown that MMO is in P for trees, weak NP-hard for planar bipartite graphs, and strong NP-hard for general graphs. There are still gaps between those graph classes. The objective of this paper is to show tighter thresholds of complexity: We show that MMO is (i) in P for cactus graphs, (ii) weakly NP-hard for outerplanar graphs, and also (iii) strongly NP-hard for graphs which are both planar and bipartite. This implies the NP-hardness for P₄-bipartite, diamond-free or house-free graphs, each of which is a superclass of cactus. We also show (iv) the NP-hardness for series-parallel graphs and multi-outerplanar graphs, and (v) present a pseudo-polynomial time algorithm for graphs with bounded treewidth.     

On the complexity of the multicut problem in bounded tree-width graphs and digraphs

Given an edge- or vertex-weighted graph or digraph and a list of source-sink pairs, the minimum multicut problem consists in selecting a minimum weight set of edges or vertices whose removal leaves no path from each source to the corresponding sink. This is a classical NP-hard problem, and we show that the edge version becomes tractable in bounded tree-width graphs if the number of source-sink pairs is fixed, but remains NP-hard in directed acyclic graphs and APX-hard in bounded tree-width and bounded degree unweighted digraphs. The vertex version, although tractable in trees, is proved to be NP-hard in unweighted cacti of bounded degree and bounded path-width.     

Maximum transversal in partial Latin squares and rainbow matchings

In a partial Latin square P a set of distinct entries, such that no two of which are in the same row or column is called a transversal. By the size of a transversal T, we mean the number of its entries. We define a duplex to be a partial Latin square of order n containing 2n entries such that exactly two entries lie in each row and column and each of n symbols occurs exactly twice. We show that determining the maximum size of a transversal in a given duplex is an NP-complete problem. This problem relates to independent sets in certain subfamilies of cubic graphs. Generalizing the concept of transversals in edge coloring of graphs we are led to introduce the concept of rainbow matching. We show that if each color appears at most twice then it is a polynomial time problem to know whether there exists a rainbow matching of size at least ⌊n/2⌋-t for each fixed t, where n is the order of the graph. As an application we show that for any fixed t, there is a polynomial time algorithm which decides whether α(G)?n-t, for any graph G on 2n vertices containing a perfect matching. At the end we mention some other applications of rainbow matching.     

Some complexity problems on single input double output controllers

We investigate the time complexity of constructing single input double output state feedback controller structures, given the directed structure graph G of a system. Such a controller structure defines a restricted type of P₃-partition of the graph G. A necessary condition (∗) is described and some classes of graphs are identified where the search problem of finding a feasible P₃-partition is polynomially solvable and, in addition, (∗) is not only necessary but also sufficient for the existence of a P₃-partition. It is also proved that the decision problem on two particular graph classes — defined in terms of forbidden subgraphs — remains NP-complete, but is polynomially solvable on the intersection of those two classes. The polynomial-time solvability of some further related problems is shown, too.     

On distance-3 matchings and induced matchings

For a finite undirected graph G=(V,E) and positive integer k≥1, an edge set M⊆E is a distance-k matching if the pairwise distance of edges in M is at least k in G. For k=1, this gives the usual notion of matching in graphs, and for general k≥1, distance-k matchings were called k-separated matchings by Stockmeyer and Vazirani. The special case k=2 has been studied under the names induced matching (i.e., a matching which forms an induced subgraph in G) by Cameron and strong matching by Golumbic and Laskar in various papers.Finding a maximum induced matching is NP-complete even on very restricted bipartite graphs and on claw-free graphs but it can be done efficiently on various classes of graphs such as chordal graphs, based on the fact that an induced matching in G corresponds to an independent vertex set in the square L(G)² of the line graph L(G) of G which, by a result of Cameron, is chordal for any chordal graph G.We show that, unlike for k=2, for a chordal graph G, L(G)³ is not necessarily chordal, and finding a maximum distance-3 matching, and more generally, finding a maximum distance-(2k+1) matching for k≥1, remains NP-complete on chordal graphs. For strongly chordal graphs and interval graphs, however, the maximum distance-k matching problem can be solved in polynomial time for every k≥1. Moreover, we obtain various new results for maximum induced matchings on subclasses of claw-free graphs.     

Least and most colored bases

Consider a matroid M=(E,B), where B denotes the family of bases of M, and assign a color c(e) to every element e∈E (the same color can go to more than one element). The palette of a subset F of E, denoted by c(F), is the image of F under c. Assume also that colors have prices (in the form of a function π(?), where ? is the label of a color), and define the chromatic price as: π(F)=∑_?∈c(F)π(?). We consider the following problem: find a base B∈B such that π(B) is minimum. We show that the greedy algorithm delivers a lnr(M)-approximation of the unknown optimal value, where r(M) is the rank of matroid M. By means of a reduction from SETCOVER, we prove that the lnr(M) ratio cannot be further improved, even in the special case of partition matroids, unless . The results apply to the special case where M is a graphic matroid and where the prices π(?) are restricted to be all equal. This special case was previously known as the minimum label spanning tree (MLST) problem. For the MLST, our results improve over the ln(n-1)+1 ratio achieved by Wan, Chen and Xu in 2002. Inspired by the generality of our results, we study the approximability of coloring problems with different objective function π(F), where F is a common independent set on matroids M₁,…,M_k and, more generally, to independent systems characterized by the k-for-1 property.     

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.     

Finding monotone paths in edge-ordered graphs

An edge-ordering of a graph G=(V,E) is a one-to-one function f from E to a subset of the set of positive integers. A path P in G is called an f-ascent if f increases along the edge sequence of P. The heighth(f) of f is the maximum length of an f-ascent in G.In this paper we deal with computational problems concerning finding ascents in graphs. We prove that for a given edge-ordering f of a graph G the problem of determining the value of h(f) is NP-hard. In particular, the problem of deciding whether there is an f-ascent containing all the vertices of G is NP-complete. We also study several variants of this problem, discuss randomized and deterministic approaches and provide an algorithm for the finding of ascents of order at least k in graphs of order n in running time O(4^kn^O(1)).     

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S⊆V, such that, for each u∈V, |N[u]∩S|?|N[u]|/2 in a given graph G=(V,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k?5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k-1)-regular graphs, where k?3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.     

Solution techniques for the Large Set Covering Problem

Given a finite set E and a family F={E₁,…,E_m} of subsets of E such that F covers E, the famous unicost set covering problem (USCP) is to determine the smallest possible subset of F that also covers E. We study in this paper a variant, called the Large Set Covering Problem (LSCP), which differs from the USCP in that E and the subsets E_i are not given in extension because they are very large sets that are possibly infinite. We propose three exact algorithms for solving the LSCP. Two of them determine minimal covers, while the third one produces minimum covers. Heuristic versions of these algorithms are also proposed and analysed. We then give several procedures for the computation of a lower bound on the minimum size of a cover. We finally present algorithms for finding the largest possible subset of F that does not cover E. We also show that a particular case of the LSCP is to determine irreducible infeasible sets in inconsistent constraint satisfaction problems. All concepts presented in the paper are illustrated on the k-colouring problem which is formulated as a constraint satisfaction problem.     

On the two-dimensional orthogonal drawing of series-parallel graphs

It has been known that every planar 4-graph has a 2-bend 2-D orthogonal drawing, with the only exception being the octahedron, every planar 3-graph has a 1-bend 2-D orthogonal drawing with the only exception being K₄, and every outerplanar 3-graph with no triangles has a 0-bend 2-D orthogonal drawing. We show in this paper that every series-parallel 4-graph has a 1-bend 2-D orthogonal drawing.     

On disconnected cuts and separators

For a connected graph G=(V,E), a subset U⊆V is called a disconnected cut if U disconnects the graph, and the subgraph induced by U is disconnected as well. A natural condition is to impose that for any u∈U, the subgraph induced by (V?U)∪{u} is connected. In that case, U is called a minimal disconnected cut. We show that the problem of testing whether a graph has a minimal disconnected cut is NP-complete. We also show that the problem of testing whether a graph has a disconnected cut separating two specified vertices, s and t, is NP-complete.     

Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, Part II

We continue our work (Y. Li, C. Zhao, Locating the peaks of least-energy solutions to a quasilinear elliptic Neumann problem, J. Math. Anal. Appl. 336 (2007) 1368-1383) to study the shape of least-energy solutions to the quasilinear problem ε^mΔ_mu−u^m−1+f(u)=0 with homogeneous Neumann boundary condition. In this paper we focus on the case 1<m<2 as a complement to our previous work on the case m≥2. We use an intrinsic variation method to show that as the case m≥2, when ε→0⁺, the global maximum point P_ε of least-energy solutions goes to a point on the boundary ∂Ω at a rate of o(ε) and this point on the boundary approaches a global maximum point of mean curvature of ∂Ω.     

On extension of isometries and approximate isometries between unit spheres

In this paper, we will study the isometric extension problem for L¹-spaces and prove that every surjective isometry from the unit sphere of L¹(μ) onto that of a Banach space E can be extended to a linear surjective isometry from L¹(μ) onto E. Moreover, we introduce the approximate isometric extension problem and show that, if E and F are Banach spaces and E satisfies the property (m) (special cases are L^∞(Γ), C₀(Ω) and L^∞(μ)), then every bijective ?-isometry between the unit spheres of E and F can be extended to a bijective 5?-isometry between their closed unit balls. At last, we will give an example to show that the surjectivity assumption cannot be omitted. Using this, we solve the non-surjective isometric extension problem in the negative.     

On the complexity of crossings in permutations

We investigate crossing minimization problems for a set of permutations, where a crossing expresses a disarrangement between elements. The goal is a common permutation π^∗ which minimizes the number of crossings. In voting and social science theory this is known as the Kemeny optimal aggregation problem minimizing the Kendall-τ distance. This rank aggregation problem can be phrased as a one-sided two-layer crossing minimization problem for a series of bipartite graphs or for an edge coloured bipartite graph, where crossings are counted only for monochromatic edges. We contribute the max version of the crossing minimization problem, which attempts to minimize the discrimination against any permutation. As our results, we correct the construction from [C. Dwork, R. Kumar, M. Noar, D. Sivakumar, Rank aggregation methods for the Web, Proc. WWW10 (2001) 613-622] and prove the NP-hardness of the common crossing minimization problem for k=4 permutations. Then we establish a 2−2/k-approximation, improving the previous factor of 2. The max version is shown NP-hard for every k≥4, and there is a 2-approximation. Both approximations are optimal, if the common permutation is selected from the given ones. For two permutations crossing minimization is solved by inspecting the drawings, whereas it remains open for three permutations.     

Tree 3-spanners in 2-sep chordal graphs: Characterization and algorithms

A spanning tree T of a graph G is said to be a treet-spanner if the distance between any two vertices in T is at most t times their distance in G. A graph that has a tree t-spanner is called a treet-spanner admissible graph. The problem of deciding whether a graph is tree t-spanner admissible is NP-complete for any fixed t≥4 and is linearly solvable for t≤2. The case t=3 still remains open. A chordal graph is called a 2-sep chordal graph if all of its minimal a−b vertex separators for every pair of non-adjacent vertices a and b are of size two. It is known that not all 2-sep chordal graphs admit tree 3-spanners. This paper presents a structural characterization and a linear time recognition algorithm of tree 3-spanner admissible 2-sep chordal graphs. Finally, a linear time algorithm to construct a tree 3-spanner of a tree 3-spanner admissible 2-sep chordal graph is proposed.     

An improved randomized approximation algorithm for maximum triangle packing

This paper deals with the maximum triangle packing problem. For this problem, Hassin and Rubinstein gave a randomized polynomial-time approximation algorithm that achieves an expected ratio of for any constant ?>0. By modifying their algorithm, we obtain a new randomized polynomial-time approximation algorithm for the problem which achieves an expected ratio of 0.5257(1−?) for any constant ?>0.     

A survey of stratified domination in graphs

A graph G is 2-stratified if its vertex set is partitioned into two nonempty classes (each of which is a stratum or a color class). We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph with one fixed blue vertex v specified. We say that F is rooted at v. The F-domination number of a graph G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that for every blue vertex v of G, there is a copy of F in G rooted at v. In this paper, we survey recent results on the F-domination number for various 2-stratified graphs F.     

Homomorphisms of 2-edge-colored graphs

In this paper, we study homomorphisms of 2-edge-colored graphs, that is graphs with edges colored with two colors. We consider various graph classes (outerplanar graphs, partial 2-trees, partial 3-trees, planar graphs) and the problem is to find, for each class, the smallest number of vertices of a 2-edge-colored graph H such that each graph of the considered class admits a homomorphism to H.