It is shown that one can count @math -edge paths in an @math -vertex graph and @math -set @math -packings on an @math -element universe, respectively, in time @math and @math , up to a factor polynomial in @math , @math , and @math ; in polynomial space, the bounds hold if multiplied by @math or @math , respectively. These are implications of a more general result: given two set families on an @math -element universe, one can count the disjoint pairs of sets in the Cartesian product of the two families with @math basic operations, where @math is the number of members in the two families and their subsets.
The presented meet-in-the-middle approach resembles the randomized divide-and-conquer technique by Chen, Lu, Sze, and Zhang @cite_20 and the similar divide-and-color method by Kneis, M " o lle, Richter, and Rossmanith @cite_6 , designed for parameterized decision problems. These can, in turn, be viewed as extensions of the recursive partitioning technique of Gurevich and Shelah @cite_10 for the Hamiltonian Path problem. That said, our contribution is rather in the observation that, in the counting context, the join operation can be done efficiently using the inclusion--exclusion machinery. While our formalization of the problem as the Disjoint Sum problem is new, the solution itself can, in essence, already be found in Kennes @cite_18 , even though in terms of possibility calculus and without the idea of trimming,'' that is, restricting the computations to small subsets. Kennes's results were rediscovered in a dual form and extended to accommodate trimming in the authors' recent works @cite_23 @cite_3 @cite_21 .
@cite_18 @cite_21 @cite_6 @cite_3 @cite_23 @cite_10 @cite_20
2026924428 2409645877 2787728049 2170675255
where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. We show that the Graph Isomorphism (GI) problem and the more general problems of String Isomorphism (SI) andCoset Intersection (CI) can be solved in quasipolynomial(exp((logn)O(1))) time. The best previous bound for GI was exp(O( √n log n)), where n is the number of vertices (Luks, 1983); for the other two problems, the bound was similar, exp(O (√ n)), where n is the size of the permutation domain (Babai, 1983). Following the approach of Luks’s seminal 1980 82 paper, the problem we actually address is SI. This problem takes two strings of length n and a permutation group G of degree n (the “ambient group”) as input (G is given by a list of generators) and asks whether or not one of the strings can be transformed into the other by some element of G. Luks’s divide-and-conquer algorithm for SI proceeds by recursion on the ambient group. We build on Luks’s framework and attack the obstructions to efficient Luks recurrence via an interplay between local and global symmetry. We construct group theoretic “local certificates” to certify the presence or absence of local symmetry, aggregate the negative certificates to canonical k-ary relations where k = O(log n), and employ combinatorial canonical partitioning techniques to split the k-ary relational structure for efficient divide-and- conquer. We show that in a well–defined sense, Johnson graphs are the only obstructions to effective canonical partitioning. The central element of the algorithm is the “local certificates” routine which is based on a new group theoretic result, the “Unaffected stabilizers lemma,” that allows us to construct global automorphisms out of local information. The well-known @math -disjoint path problem ( @math -DPP) asks for pairwise vertex-disjoint paths between @math specified pairs of vertices @math in a given graph, if they exist. The decision version of the shortest @math -DPP asks for the length of the shortest (in terms of total length) such paths. Similarly the search and counting versions ask for one such and the number of such shortest set of paths, respectively. We restrict attention to the shortest @math -DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdiere and Schrijver for the general one-face problem. We do so by providing a randomised @math algorithm along with an @math time randomised sequential algorithm. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals. In addition, the earlier best known sequential algorithms (e.g. ) were randomised while ours are also deterministic. The algorithms are based on a bijection between a shortest @math -tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "good" cycle covers. We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of @math , a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the @math bound.Our main result is a @math -approximation online algorithm for stochastic bipartite matching, breaking this @math barrier. Furthermore, we show that no online algorithm can produce a @math approximation for an arbitrarily small @math for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-) matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.