Approximation algorithms for the bi-criteria weighted MAX-CUT problem

We consider a generalization of the classical MAX-CUT problem where two objective functions are simultaneously considered. We derive some theorems on the existence and the non-existence of feasible cuts that are at the same time near optimal for both criteria. Furthermore, two approximation algorithms with performance guarantee are presented. The first one is deterministic while the second one is randomized. A generalization of these results is given for the bi-criteria MAX-k-CUT problem.     

Improved induced matchings in sparse graphs

An induced matching in a graph G=(V,E) is a matching M such that (V,M) is an induced subgraph of G. Clearly, among two vertices with the same neighbourhood (called twins) at most one is matched in any induced matching, and if one of them is matched then there is another matching of the same size that matches the other vertex. Motivated by this, Kanj et al. [10] studied induced matchings in twinless graphs. They showed that any twinless planar graph contains an induced matching of size at least and that there are twinless planar graphs that do not contain an induced matching of size greater than . We improve both these bounds to , which is tight up to an additive constant. This implies that the problem of deciding whether a planar graph has an induced matching of size k has a kernel of size at most 28k. We also show for the first time that this problem is fixed parameter tractable for graphs of bounded arboricity.Kanj et al. also presented an algorithm which decides in -time whether an n-vertex planar graph contains an induced matching of size k. Our results improve the time complexity analysis of their algorithm. However, we also show a more efficient -time algorithm. Its main ingredient is a new, O^∗(4^l)-time algorithm for finding a maximum induced matching in a graph of branch width at most l.     

Differential approximation of min sat, max sat and related problems

We present differential approximation results (both positive and negative) for optimal satisfiability, optimal constraint satisfaction, and some of the most popular restrictive versions of them. As an important corollary, we exhibit an interesting structural difference between the landscapes of approximability classes in standard and differential paradigms.     

A sparse counterpart of Reichel and Gragg’s package QRUP

We describe how to maintain the triangular factor of a sparse QR factorization when columns are added and deleted and Q cannot be stored for sparsity reasons. The updating procedures could be thought of as a sparse counterpart of Reichel and Gragg’s package QRUP. They allow us to solve a sequence of sparse linear least squares subproblems in which each matrix B_k is an independent subset of the columns of a fixed matrix A, and B_k+1 is obtained by adding or deleting one column. Like Coleman and Hulbert [T. Coleman, L. Hulbert, A direct active set algorithm for large sparse quadratic programs with simple bounds, Math. Program. 45 (1989) 373-406], we adapt the sparse direct methodology of Björck and Oreborn of the late 1980s, but without forming A^TA, which may be only positive semidefinite. Our Matlab 5 implementation works with a suitable row and column numbering within a static triangular sparsity pattern that is computed in advance by symbolic factorization of A^TA and preserved with placeholders.     

clever or smart: Strategies for the online target date assignment problem

In this paper, we consider the Online Target Date Assignment Problem (OnlineTDAP) for general downstream problems, where the downstream cost are nonnegative, additive and satisfy the triangle inequality.We analyze algorithm smart, which was introduced by Angelelli et al. [3] and give its exact competitive ratio depending on the number of requests. Since the obtained competitive ratio is at most we answer the question posed in Angelelli et al. [4] if smart has a competitive ratio strictly less than 2.Moreover, we introduce a new algorithm called clever and show that this strategy has a competitive ratio of 3/2. We show that this is asymptotically optimal by proving that no online algorithm can perform better than 3/2−ε.     

Approximation of max independent set, min vertex cover and related problems by moderately exponential algorithms

Using ideas and results from polynomial time approximation and exact computation we design approximation algorithms for several NP-hard combinatorial problems achieving ratios that cannot be achieved in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. We study in particular two paradigmatic problems, max independent set and min vertex cover.     

On a generalization of Chen's iterated integrals

Text

In this paper, Chen's iterated integrals are generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated integrals satisfy both an additive iterative property and comultiplication formula. In a particular example, a (non-classical) multiplicative iterative property is also shown to hold. After developing this theory in the first part of the paper we discuss various applications, including the expression of certain zeta functions as complex iterated integrals (from which an obstruction to the existence of a contour integration proof of the functional equation for the Dedekind zeta function emerges); a way of thinking about complex iterated derivatives arising from a reformulation of a result of Gel'fand and Shilov in the theory of distributions; and a direct topological proof of the monodromy of polylogarithms.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=dsVvo7s8BYU.     

Eliciting Electre Tri category limits for a group of decision makers

Multiple criteria sorting aims at assigning alternatives evaluated on several criteria to predefined ordered categories. In this paper, we consider a well known multiple criteria sorting method, Electre Tri, which involves three types of preference parameters: (1) category limits defining the frontiers between consecutive categories, (2) weights and majority level specifying which coalitions form a majority, and (3) veto thresholds characterizing discordance effects. We propose an elicitation procedure to infer category limits from assignment examples provided by multiple decision makers. The procedure computes a set of category limits and vetoes common to all decision makers, with variable weights for each decision maker. Hence, the method helps reaching a consensus among decision makers on the category limits and veto thresholds, whereas finding a consensus on weights is left aside. The inference procedure is based on mixed integer linear programming and performs well even for datasets corresponding to real-world decision problems. We provide an illustrative example of the use of the method and analyze the performance of the proposed algorithms.     

Combinatorial 5/6-approximation of Max Cut in graphs of maximum degree 3

The best approximation algorithm for Max Cut in graphs of maximum degree 3 uses semidefinite programming, has approximation ratio 0.9326, and its running time is Θ(n^3.5logn); but the best combinatorial algorithms have approximation ratio 4/5 only, achieved in O(n²) time [J.A. Bondy, S.C. Locke, J. Graph Theory 10 (1986) 477–504; E. Halperin, et al., J. Algorithms 53 (2004) 169–185]. Here we present an improved combinatorial approximation, which is a 5/6-approximation algorithm that runs in O(n²) time, perhaps improvable even to O(n). Our main tool is a new type of vertex decomposition for graphs of maximum degree 3.     

An almost linear time algorithm for finding Hamilton cycles in sparse random graphs with minimum degree at least three

We describe an algorithm for finding Hamilton cycles in random graphs. Our model is the random graph . In this model G is drawn uniformly from graphs with vertex set [n], m edges and minimum degree at least three. We focus on the case where m = cn for constant c. If c is sufficiently large then our algorithm runs in time and succeeds w.h.p. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 73–98, 2015     

The relationship between k-forcing and k-power domination

Zero forcing and power domination are iterative processes on graphs where an initial set of vertices are observed, and additional vertices become observed based on some rules. In both cases, the goal is to eventually observe the entire graph using the fewest number of initial vertices. The concept of $k$-power domination was introduced by Chang et al. (2012) as a generalization of power domination and standard graph domination. Independently, $k$-forcing was defined by Amos et al. (2015) to generalize zero forcing. In this paper, we combine the study of $k$-forcing and $k$-power domination, providing a new approach to analyze both processes. We give a relationship between the $k$-forcing and the $k$-power domination numbers of a graph that bounds one in terms of the other. We also obtain results using the contraction of subgraphs that allow the parallel computation of $k$-forcing and $k$-power dominating sets.     

An exact algorithm for minimizing resource availability costs in project scheduling

A new exact algorithm that solves the Resource Availability Cost Problem (RACP) in project scheduling is shown to yield a significant improvement over the existing algorithm in the literature. The new algorithm consists of a hybrid method where an initial feasible solution is found heuristically. The branching scheme solves a Resource-Constrained Project Scheduling Problem (RCPSP) at each node where the resources of the RACP are fixed. The knowledge of previously solved RCPSPs is used to produce cuts in the search tree. A worst-case-performance theorem is established for this new algorithm. Experiments are performed on instances adapted from the PSPLIB database. The new algorithm can be used to minimize any resource availability cost problem once a procedure for the underlying resource-constrained problem is available.     

Pointlike sets with respect to R and J

We present an algorithm to compute the pointlike subsets of a finite semigroup with respect to the pseudovariety of all finite R-trivial semigroups. The algorithm is inspired by Henckell’s algorithm for computing the pointlike subsets with respect to the pseudovariety of all finite aperiodic semigroups. We also give an algorithm to compute -pointlike sets, where denotes the pseudovariety of all finite J-trivial semigroups. We finally show that, in contrast with the situation for , the natural adaptation of Henckell’s algorithm to computes pointlike sets, but not all of them.     

Applying relational algebra and RelView to coalition formation

We present an application of relational algebra to coalition formation. This leads to specifications, which can be executed with the help of the RelView tool after a simple translation into the tool’s programming language. As an example we consider a simplification of the situation in Poland after the 2001 elections.     

Computing tournament solutions using relation algebra and RelView

We describe a simple computing technique for the tournament choice problem. It rests upon relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView’s labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.     

An update on non-Hamiltonian 54-tough maximal planar graphs

Studying the shortness of longest cycles in maximal planar graphs, we improve the upper bound on the shortness exponent of the class of $\frac{5}{4}$-tough maximal planar graphs presented by Harant and Owens (1995). In addition, we present two generalizations of a similar result of Tká? who considered $1$-tough maximal planar graphs (Tká?, 1996); we remark that one of these generalizations gives a tight upper bound. We fix a problematic argument used in both mentioned papers.