Families with infants: a general approach to solve hard partition problems

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with $O^*((2-\varepsilon(d))^n)$ time and exponential space for graphs of average degree $d$. This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in $O^*((2-\varepsilon(d))^n)$ time and polynomial space for graphs of average degree $d$. The previously known results of this kind is a polyspace algorithm by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~$d$ we present an algorithm with running time $O^*((2-\varepsilon(d))^{n/2})$ and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at http://arxiv.org/abs/1410.220

Determining Distributions of Security Means for WSNs based on the Model of a Neighbourhood Watch

Neighbourhood watch is a concept that allows a community to distribute a complex security task in between all members. Members of the community carry out individual security tasks to contribute to the overall security of it. It reduces the workload of a particular individual while securing all members and allowing them to carry out a multitude of security tasks. Wireless sensor networks (WSNs) are composed of resource-constraint independent battery driven computers as nodes communicating wirelessly. Security in WSNs is essential. Without sufficient security, an attacker is able to eavesdrop the communication, tamper monitoring results or deny critical nodes providing their service in a way to cut off larger network parts. The resource-constraint nature of sensor nodes prevents them from running full-fledged security protocols. Instead, it is necessary to assess the most significant security threats and implement specialised protocols. A neighbourhood-watch inspired distributed security scheme for WSNs has been introduced by Langend\"orfer. Its goal is to increase the variety of attacks a WSN can fend off. A framework of such complexity has to be designed in multiple steps. Here, we introduce an approach to determine distributions of security means on large-scale static homogeneous WSNs. Therefore, we model WSNs as undirected graphs in which two nodes connected iff they are in transmission range. The framework aims to partition the graph into $n$ distinct security means resulting in the targeted distribution. The underlying problems turn out to be NP hard and we attempt to solve them using linear programs (LPs). To evaluate the computability of the LPs, we generate large numbers of random {\lambda}-precision unit disk graphs (UDGs) as representation of WSNs. For this purpose, we introduce a novel {\lambda}-precision UDG generator to model WSNs with a minimal distance in between nodes

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

A New Optimality Measure for Distance Dominating Sets

  We study the problem of finding the smallest power of an input graph that has k disjoint dominating sets, where the ith power of an input graph G is constructed by adding edges between pairs of vertices in G at distance i or less, and a subset of vertices in a graph G is a dominating set if and only if every vertex in G is adjacent to a vertex in this subset.   The problem is a different view of the d-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the dth power of the input graph.   This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are k types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance d".   In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the d-domatic number problem. We show that the problem is NP-complete for any fixed k greater than two even when the input graph is restricted to split graphs, 2-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering k as a constant, and for strongly chordal graphs, for any k. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time O(2. 73n). Moreover, we prove that the problem cannot be approximated within ratio smaller than 2 even for split graphs, 2-connected graphs, and planar bipartite graphs of degree four. We propose a greedy 3-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work

Quantum Algorithms for Graph Coloring and other Partitioning, Covering, and Packing Problems

Let U be a universe on n elements, let k be a positive integer, and let F be a family of (implicitly defined) subsets of U. We consider the problems of partitioning U into k sets from F, covering U with k sets from F, and packing k non-intersecting sets from F into U. Classically, these problems can be solved via inclusion-exclusion in O*(2^n) time [BjorklundHK09]. Quantumly, there are faster algorithms for graph coloring with running time O(1.9140^n) [ShimizuM22] and for Set Cover with a small number of sets with running time O(1.7274^n |F|^O(1)) [AmbainisBIKPV19]. In this paper, we give a quantum speedup for Set Partition, Set Cover, and Set Packing whenever there is a classical enumeration algorithm that lends itself to a quadratic quantum speedup, which, for any subinstance on a subset X of U, enumerates at least one member of a k-partition, k-cover, or k-packing (if one exists) restricted to (or projected onto, in the case of k-cover) the set X in O*(c^{|X|}) time with c<2. Our bounded-error quantum algorithm runs in O*((2+c)^(n/2)) for Set Partition, Set Cover, and Set Packing. When c<=1.147899, our algorithm is slightly faster than O*((2+c)^(n/2)); when c approaches 1, it matches the running time of [AmbainisBIKPV19] for Set Cover when |F| is subexponential in n. For Graph Coloring, we further improve the running time to O(1.7956^n) by leveraging faster algorithms for coloring with a small number of colors to better balance our divide-and-conquer steps. For Domatic Number, we obtain a O((2-\epsilon)^n) running time for some \epsilon>0

Separate, measure and conquer: faster polynomial-space algorithms for Max 2-CSP and counting dominating sets

We show a method resulting in the improvement of several polynomial-space, exponential-time algorithms. The method capitalizes on the existence of small balanced separators for sparse graphs, which can be exploited for branching to disconnect an instance into independent components. For this algorithm design paradigm, the challenge to date has been to obtain improvements in worst-case analyses of algorithms, compared with algorithms that are analyzed with advanced methods, such as Measure and Conquer. Our contribution is the design of a general method to integrate the advantage from the separator-branching into Measure and Conquer, for an improved running time analysis. We illustrate the method with improved algorithms for Max (r,2) -CSP and #Dominating Set. For Max (r,2) -CSP instances with domain size r and m constraints, the running time improves from r m/6 to r m/7.5 for cubic instances and from r 0.19⋅m to r 0.18⋅m for general instances, omitting subexponential factors. For #Dominating Set instances with n vertices, the running time improves from 1.4143 n to 1.2458 n for cubic instances and from 1.5673 n to 1.5183 n for general instances. It is likely that other algorithms relying on local transformations can be improved using our method, which exploits a non-local property of graphs