Set partitioning via inclusion-exclusion

Το παρόν έργο αποτελεί μελέτη του paper των Andreas Bjorklund, Thore Husfeldt και Mikko Koivisto, ”Set partitioning via inclusion-exclusion”. Κύριος στόχος κατά τη συγγραφή ήταν να καταστούν οι έννοιες που παρουσιάζονται όσο το δυνατόν περισσότερο εύληπτες από προπτυχιακούς φοιτητές. Αποδεικνύουμε την αρχή εγκλεισμού-αποκλεισμού και ορίζουμε το z-μετασχηματισμό ενώ δίνουμε και έναν αλγόριθμο που τον υπολογίζει. Δεδομένου ενός συνόλου N, n στοιχείων και μιας οικογένειας F υποσυνόλων του N καθώς και ενός ακεραίου k, παρέχουμε έναν ακριβή αλγόριθμο που υπολογίζει το πλήθος των k-κατατμήσεων σε εκθετικό χρόνο. Επίσης παρέχουμε και άλλους οι οποίοι λύνουν παρόμοια προβλήματα όπως η καταμέτρηση των k-καλυμμάτων, η άθροιση κατατμήσεων με βάρη και η εύρεση της πιο βαριάς κατάτμησης. Στη συνέχεια παρέχουμε παραδείγματα προβλημάτων τα οποία ανάγονται σε αυτά που λύσαμε παραπάνω και για τα οποία οι αναγωγές δεν απαιτούν πολύ χρόνο. Οι προαναφερθέντες αλγόριθμοι στοχεύουν στον ελάχιστο χρόνο, με τη χωρική πολυ- πλοκότητα να είναι εκθετική. Δεδομένου ότι την ευθύνη για αυτό φέρουν αποκλειστικά οι υπολογισμοί του z-μετασχηματισμού, δίνουμε εναλλακτικούς τρόπους επίλυσης των παραπάνω χωρίς τη χρήση του z-μετασχηματισμού σε πολυωνιμικό χώρο. Το μειονέκτημα αυτών είναι ότι χρειάζονται περισσότερο χρόνο. Κλείνουμε με έναν προσεγγιστικό αλγόριθμο πολυωνυμικού χώρου ο οποίος λύνει το Πρόβλημα Χρωματικού Αριθμού Γραφήματος.The present work is a study of the paper by Andreas Bjorklund, Thore Husfeldt and Mikko Koivisto, ”Set partitioning via inclusion-exclusion”. The main aim of the writer was for the ideas presented to be as accessible as possible to undergraduate students. We prove the principle of inclusion-exclusion and define the zeta transform while also giving an algorithm that computes it. Given a n element set N and a family F of subsets of N we provide an exact algorithm that computes the number of k-partitions in time exponential. We also provide others that solve similar problems like k-covers, sum of weighted partitions and max-weighted partition. We then provide examples of problems which are reducible to the ones solved above and for which the reduction does not dominate the time complexity. The aforementioned algorithms are optimized for time with the space complexity being also exponential. Considering that the responsibility for this falls squarely on the calculations for the z-transform, we provide alternate ways of solving the previous problems where we substitute the z-transform by polynomial space tools with the drawback of them being more costly on time. We conclude with an approximation algorithm for the Chromatic Number Problem in polynomial space

Set multi-covering via inclusion-exclusion

Set multi-covering is a generalization of the set covering problem where each element may need to be covered more than once and thus some subset in the given family of subsets may be picked several times for minimizing the number of sets to satisfy the coverage requirement. In this paper, we propose a family of exact algorithms for the set multi-covering problem based on the inclusion-exclusion principle. The presented ESMC (Exact Set Multi-Covering) algorithm takes O* ((2 t)n) time and O* ((t + 1)n) space where t is the maximum value in the coverage requirement set (The O* (f (n)) notation omits a p o l y log (f (n)) factor). We also propose the other three exact algorithms through different tradeoffs of the time and space complexities. To the best of our knowledge, this present paper is the first one to give exact algorithms for the set multi-covering problem with nontrivial time and space complexities. This paper can also be regarded as a generalization of the exact algorithm for the set covering problem given in [A. Björklund, T. Husfeldt, M. Koivisto, Set partitioning via inclusion-exclusion, SIAM Journal on Computing, in: FOCS 2006 (in press, special issue)]. © 2009 Elsevier B.V. All rights reserved.postprin

How proofs are prepared at Camelot

We study a design framework for robust, independently verifiable, and workload-balanced distributed algorithms working on a common input. An algorithm based on the framework is essentially a distributed encoding procedure for a Reed--Solomon code, which enables (a) robustness against byzantine failures with intrinsic error-correction and identification of failed nodes, and (b) independent randomized verification to check the entire computation for correctness, which takes essentially no more resources than each node individually contributes to the computation. The framework builds on recent Merlin--Arthur proofs of batch evaluation of Williams~[{\em Electron.\ Colloq.\ Comput.\ Complexity}, Report TR16-002, January 2016] with the observation that {\em Merlin's magic is not needed} for batch evaluation---mere Knights can prepare the proof, in parallel, and with intrinsic error-correction. The contribution of this paper is to show that in many cases the verifiable batch evaluation framework admits algorithms that match in total resource consumption the best known sequential algorithm for solving the problem. As our main result, we show that the $k$-cliques in an $n$-vertex graph can be counted {\em and} verified in per-node $O(n^{(\omega+\epsilon)k/6})$ time and space on $O(n^{(\omega+\epsilon)k/6})$ compute nodes, for any constant $\epsilon>0$ and positive integer $k$ divisible by $6$, where $2\leq\omega<2.3728639$ is the exponent of matrix multiplication. This matches in total running time the best known sequential algorithm, due to Ne{\v{s}}et{\v{r}}il and Poljak [{\em Comment.~Math.~Univ.~Carolin.}~26 (1985) 415--419], and considerably improves its space usage and parallelizability. Further results include novel algorithms for counting triangles in sparse graphs, computing the chromatic polynomial of a graph, and computing the Tutte polynomial of a graph.Comment: 42 p

Exact Covers via Determinants

Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the k-dimensional matching problem asks whether there is a disjoint collection of the hyperedges which covers all vertices. We show it can be solved by a randomized polynomial space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors polynomial in n and k. When we drop the partition constraint and permit arbitrary hyperedges of cardinality k, we obtain the exact cover by k-sets problem. We show it can be solved by a randomized polynomial space algorithm in time O*(c_k^n), where c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k. Both results substantially improve on the previous best algorithms for these problems, especially for small k, and follow from the new observation that Lovasz' perfect matching detection via determinants (1979) admits an embedding in the recently proposed inclusion-exclusion counting scheme for set covers, despite its inability to count the perfect matchings