DAG‐Width and Circumference of Digraphs

We prove that every digraph of circumference l has DAG‐width at most l. This is best possible and solves a recent conjecture from S. Kintali (ArXiv:1401.2662v1 [math.CO], January 2014).¹ As a consequence of this result we deduce that the k‐linkage problem is polynomially solvable for every fixed k in the class of digraphs with bounded circumference. This answers a question posed in J. Bang‐Jensen, F. Havet, and A. K. Maia (Theor Comput Sci 562 (2014), 283–303). We also prove that the weak k‐linkage problem (where we ask for arc‐disjoint paths) is polynomially solvable for every fixed k in the class of digraphs with circumference 2 as well as for digraphs with a bounded number of disjoint cycles each of length at least 3. The case of bounded circumference digraphs is still open. Finally, we prove that the minimum spanning strong subdigraph problem is NP‐hard on digraphs of DAG‐width at most 5.     

The Klein bottle and multicommodity flows

LetG be an eulerian graph embedded on the Klein bottle. Then the maximum number of pairwise edge-disjoint orientation-reversing circuits inG is equal to the minimum number of edges intersecting all orientation-reversing circuits. This generalizes a theorem of Lins for the projective plane. As consequences we derive results on disjoint paths in planar graphs, including theorems of Okamura and of Okamura and Seymour.     

On Toughness and Hamiltonicity of 2K2‐Free Graphs

The toughness of a (noncomplete) graph G is the minimum value of t for which there is a vertex cut A whose removal yields components. Determining toughness is an NP‐hard problem for general input graphs. The toughness conjecture of Chvátal, which states that there exists a constant t such that every graph on at least three vertices with toughness at least t is hamiltonian, is still open for general graphs. We extend some known toughness results for split graphs to the more general class of 2K₂‐free graphs, that is, graphs that do not contain two vertex‐disjoint edges as an induced subgraph. We prove that the problem of determining toughness is polynomially solvable and that Chvátal's toughness conjecture is true for 2K₂‐free graphs.     

A note on packing paths in planar graphs

Seymour (1981) proved that the cut criterion is necessary and sufficient for the solvability of the edge-disjoint paths problem when the union of the supply graph and the demand graph is planar and Eulerian. When only planarity is required, Middendorf and Pfeiffer (1993) proved the problem to be NP-complete. For this case, Korach and Penn (1992) proved that the cut criterion is sufficient for the existence of a near-complete packing of paths. Here we generalize this result by showing how a natural strengthening of the cut criterion yields better packings of paths.Analogously to Seymour's approach, we actually prove a theorem on packing cuts in an arbitrary graph and then the planar edge-disjoint paths case is obtained by planar dualization. The main result is derived from a theorem of Seb (1990) on the structure of ±1 weightings of a bipartite graph with no negative circuits.Research partially supported by the Hungarian National Foundation for Scientific Research Grants OTKA 2118 and 4271.     

Induced disjoint paths problem in a planar digraph

As an extension of the disjoint paths problem, we introduce a new problem which we call the induced disjoint paths problem. In this problem we are given a graph G and a collection of vertex pairs {(s₁,t₁),…,(s_k,t_k)}. The objective is to find k paths P₁,…,P_k such that P_i is a path from s_i to t_i and P_i and P_j have neither common vertices nor adjacent vertices for any distinct i,j.The induced disjoint paths problem has several variants depending on whether k is a fixed constant or a part of the input, whether the graph is directed or undirected, and whether the graph is planar or not. We investigate the computational complexity of several variants of the induced disjoint paths problem. We show that the induced disjoint paths problem is (i) solvable in polynomial time when k is fixed and G is a directed (or undirected) planar graph, (ii) NP-hard when k=2 and G is an acyclic directed graph, (iii) NP-hard when k=2 and G is an undirected general graph.As an application of our first result, we show that we can find in polynomial time certain structures called a “hole” and a “theta” in a planar graph.     

Robust discrete optimization and network flows

We propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution, and is computationally tractable both practically and theoretically. In particular, when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty, we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation. When only the cost coefficients are subject to uncertainty and the problem is a 0–1 discrete optimization problem on n variables, then we solve the robust counterpart by solving at most n+1 instances of the original problem. Thus, the robust counterpart of a polynomially solvable 0–1 discrete optimization problem remains polynomially solvable. In particular, robust matching, spanning tree, shortest path, matroid intersection, etc. are polynomially solvable. We also show that the robust counterpart of an NP-hard -approximable 0–1 discrete optimization problem, remains -approximable. Finally, we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network. The research of the author was partially supported by the Singapore-MIT alliance.The research of the author is supported by a graduate scholarship from the National University of Singapore.Mathematics Subject Classification (2000): 90C10, 90C15     

Estimation of the Absolute Error and Polynomial Solvability for a Classical <Emphasis Type="Italic">NP</Emphasis>-Hard Scheduling Problem

A method for finding an approximate solution for NP-hard scheduling problems is proposed. The example of the classical NP-hard in the strong sense problem of minimizing the maximum lateness of job processing with a single machine shows how a metric introduced on the instance space of the problem and polynomially solvable areas can be used to find an approximate solution with a guaranteed absolute error. The method is evaluated theoretically and experimentally and is compared with the ED-heuristic. Additionally, for the problem under consideration, we propose a numerical characteristic of polynomial unsolvability, namely, an upper bound for the guaranteed absolute error for each equivalence class of the instance space.     

Two-machine flowshop scheduling to minimize the number of tardy jobs

We consider the two-machine flowshop problem with the objective of minimizing the total number of tardy jobs. Since this problem is known to be strongly NP-hard, algorithms are described for four polynomially solvable special cases. In addition, several heuristic algorithms are developed to find optimal or near optimal schedules. Results of computational tests in solving problems up to 60 jobs are reported and directions for future research are provided.     

On the disjoint paths problem

Using flow and matching algorithms to solve the problem of finding disjoint paths through a given node, and with a technique of Chekuri and Khanna, we give an approximation for the edge-disjoint paths problem in undirected graphs, directed acyclic graphs and directed graphs with edge capacity at least 2.     

Query complexity,or why is it difficult to separateNP A ∩coNP A fromP A by random oraclesA?

By thequery-time complexity of a relativized algorithm we mean the total length of oracle queries made; thequery-space complexity is the maximum length of the queries made. With respect to these cost measures one can define polynomially time- or space-bounded deterministic, nondeterministic, alternating, etc. Turing machines and the corresponding complexity classes. It turns out that all known relativized separation results operate essentially with this cost measure. Therefore, if certain classes do not separate in the query complexity model, this can be taken as an indication that their relativized separation in the classical cost model will require entirely new principles.A notable unresolved question in relativized complexity theory is the separation of NP^A co NP^A fromP ^A under random oraclesA. We conjecture that the analogues of these classes actually coincide in the query complexity model, thus indicating an answer to the question in the title. As a first step in the direction of establishing the conjecture, we prove the following result, where polynomial bounds refer to query complexity.If two polynomially query-time-bounded nondeterministic oracle Turing machines accept precisely complementary (oracle dependent) languages L^A and {0, 1}^*L^A under every oracle A then there exists a deterministic polynomially query-time-bounded oracle Turing machine that accept L^A. The proof involves a sort of greedy strategy to selecting deterministically, from the large set of prospective queries of the two nondeterministic machines, a small subset that suffices to perform an accepting computation in one of the nondeterministic machines. We describe additional algorithmic strategies that may resolve the same problem when the condition holds for a (1–) fraction of the oracles A, a step that would bring us to a non-uniform version of the conjecture. Thereby we reduce the question to a combinatorial problem on certain pairs of sets of partial functions on finite sets.     

Constructing disjoint paths on expander graphs

In a typical parallel or distributed computation model processors are connected by a spars interconnection network. To establish open-line communication between pairs of processors that wish to communicate interactively, a set of disjoint paths has to be constructed on the network. Since communication needs vary in time, paths have to be dynamically constructed and destroyed.We study the complexity of constructing disjoint paths between given pairs of vertices on expander interconnection graphs. These graphs have been shown before to possess desirable properties for other communication tasks.We present a sufficient condition for the existence ofKn ^Q edge-disjoint paths connecting any set ofK pairs of vertices on an expander graph, wheren is the number of vertices and<1 is some constant. We then show that the computational problem of constructing these paths lies in the classes Deterministic-P and Random-P C.Furthermore, we show that the set of paths can be constructed in probabilistic polylog time in the parallel-distributed model of computation, in which then participating processors reside in the nodes of the communication graph and all communication is done through edges of the graph. Thus, the disjoint paths are constructed in the very computation model that uses them.Finally, we show how to apply variants of our parallel algorithms to find sets ofvertex-disjoint paths when certain conditions are satisfied.Supported in part by a Weizmann fellowship and by contract ONR N00014-85-C-0731.     

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.     

Complexity of the satisfiability problem for multilinear forms over a finite field

Multilinear forms over finite fields are considered. Multilinear forms over a field are products in which each factor is the sum of variables or elements of this field. Each multilinear form defines a function over this field. A multilinear form is called satisfiable if it represents a nonzero function. We show the N P-completeness of the satisfiability recognition problem for multilinear forms over each finite field of q elements for q ≥ 3. A theorem is proved that distinguishes cases of polynomiality and NP-completeness of the satisfiability recognition problem for multilinear fields for each possible q ≥ 3.     

Vertex-disjoint paths and edge-disjoint branchings in directed graphs

A theorem of J. Edmonds states that a directed graph has k edge-disjoint branchings rooted at a vertex r if and only if every vertex has k edge-disjoint paths to r. We conjecture an extension of this theorem to vertex-disjoint paths and give a constructive proof of the conjecture in the case k = 2.     

Independent trees in graphs

IfG is a finite undirected graph ands is a vertex ofG, then two spanning treesT ₁ andT ₂ inG are calleds — independent if for each vertexx inG the paths fromx tos inT ₁ andT ₂ are openly disjoint. It is known that the following statement is true fork3: IfG isk-connected, then there arek pairwises — independent spanning, trees inG. As a main result we show that this statement is also true fork=4 if we restrict ourselves to planar graphs. Moreover we consider similar statements for weaklys — independent spanning trees (i.e., the tree paths from a vertex tos are edge disjoint) and for directed graphs.     

The gallai-younger conjecture for planar graphs

Younger conjectured that for everyk there is ag(k) such that any digraphG withoutk vertex disjoint cycles contains a setX of at mostg(k) vertices such thatG–X has no directed cycles. Gallai had previously conjectured this result fork=1. We prove this conjecture for planar digraphs. Specifically, we show that ifG is a planar digraph withoutk vertex disjoint directed cycles, thenG contains a set of at mostO(klog(k)log(log(k))) vertices whose removal leaves an acyclic digraph. The work also suggests a conjecture concerning an extension of Vizing's Theorem for planar graphs.     

Partitioning 2‐Edge‐Colored Ore‐Type Graphs by Monochromatic Cycles

Consider a graph G on n vertices satisfying the following Ore‐type condition: for any two nonadjacent vertices x and y of G, we have . We conjecture that if we color the edges of G with two colors then the vertex set of G can be partitioned to two vertex disjoint monochromatic cycles of distinct colors. In this article, we prove an asymptotic version of this conjecture.     

Analysis of the impact of the number of edges in connected graphs on the computational complexity of the independent set problem

Under study is the complexity status of the independent set problem in a class of connected graphs that are defined by functional constraints on the number of edges depending on the number of vertices. For every natural number C, this problem is shown to be polynomially solvable in the class of graphs

On Central Endomorphisms of a Group with Operators

Let G be a group with a set of operators such that Z(G) is -admissible. Central -automorphisms occur in the Krull-Remak-Schmidt Theorem. We discuss the existence of a central -endomorphism of G that is not an automorphism.2000 Mathematics Subject Classification: 20E36