No abstract

The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponential-time algorithms for problems of that kind. The goal is getting the time complexity still of order O(c n ), but with the constant c as small as possible. In this work we extend this line of research and we investigate whether the constant c can be made even smaller when one allows constant factor approximation.In fact, we describe a kind of approximation schemes -trade-offs between approximation factor and the time complexity. We use general transformations from exponentialtime exact algorithms to approximations that are faster but still exponential-time. For example, we show that for any reduction rate r, one can transform any O * (c n )-time 1 algorithm for Set Cover into a (1 + ln r)-approximation algorithm running in time O * (c n/r ). We believe that results of that kind extend the applicability of exact algorithms for NPhard problems.

We introduce a new algebraic sieving technique to detect constrained multilinear monomials in multivariate polynomial generating functions given by an evaluation oracle. As applications of the technique, we show an O * (2 k )-time polynomial space algorithm for the k-sized Graph Motif problem. We also introduce a new optimization variant of the problem, called Closest Graph Motif and solve it within the same time bound. The Closest Graph Motif problem encompasses several previously studied optimization variants, like Maximum Graph Motif, Min-Substitute Graph Motif, and Min-Add Graph Motif. Finally, we provide a piece of evidence that our result might be essentially tight: the existence of an O * ((2 − ) k )time algorithm for the Graph Motif problem implies an O((2 − ) n )-time algorithm for Set Cover.*A preliminary conference abstract of this work has appeared as A. Björklund, P. Kaski, and L.

Abstract. We deal with the problem of finding such an orientation of a given graph that the largest number of edges leaving a vertex (called the outdegree of the orientation) is small. For any ε ∈ (0, 1) we show anÕ(|E(G)|/ε) time algorithm 3 which finds an orientation of an input graph G with outdegree at most (1 + ε)d * , where d * is the maximum density of a subgraph of G. It is known that the optimal value of orientation outdegree is d * . Our algorithm has applications in constructing labeling schemes, introduced by Kannan et al. in [18] and in approximating such graph density measures as arboricity, pseudoarboricity and maximum density. Our results improve over the previous, 2-approximation algorithms by Aichholzer et al.

We deal with the problem of maintaining a dynamic graph so that queries of the form "is there an edge between u and v?" are processed fast. We consider graphs of bounded arboricity, i.e., graphs with no dense subgraphs, like for example planar graphs. Brodal and Fagerberg [WADS'99] described a very simple linear-size data structure which processes queries in constant worst-case time and performs insertions and deletions in O(1) and O(log n) amortized time, respectively. We show a complementary result that their data structure can be used to get O(log n) worst-case time for query, O(1) amortized time for insertions and O(1) worst-case time for deletions. Moreover, our analysis shows that by combining the data structure of Brodal and Fagerberg with efficient dictionaries one gets O(log log log n) worst-case time bound for queries and deletions and O(log log log n) amortized time for insertions, with size of the data structure still linear. This last result holds even for graphs of arboricity bounded by O(log k n), for some constant k.

The central problem of the total-colorings is the total-coloring conjecture, which asserts that every graph of maximum degree Δ admits a (Δ + 2)-total-coloring. Similar to edgecolorings-with Vizing's edge-coloring conjecture-this bound can be decreased by 1 for plane graphs of higher maximum degree. More precisely, it is known that if Δ ≥ 10, then every plane graph of maximum degree Δ is (Δ + 1)-totally-colorable. On the other hand, such a statement does not hold if Δ ≤ 3. We prove that every plane graph of maximum degree 9 can be 10-totally-colored.

We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E) one can build in O(|V |) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so a shortest path between them is returned. This significantly improves the previous result of D. Eppstein [5] where after a linear preprocessing the queries are answered in O(log |V |) time. Our approach can be applied to compute the girth of a planar graph and a corresponding shortest cycle in O(|V |) time provided that the constant bound on the girth is known.Our results can be easily generalized to other wide classes of graphs -for instance we can take graphs embeddable in a surface of bounded genus or graphs of bounded tree-width.

We study the problem of scheduling equal-length jobs with release times and deadlines, where the objective is to maximize the number of completed jobs. Preemptions are not allowed. In Graham's notation, the problem is described as 1|r j ; p j = p| U j . We give the following results:(1) We show that the often cited algorithm by Carlier from 1981 is not correct. (2) We give an algorithm for this problem with running time O(n 5 ).