128 research outputs found
Near-linear Time Algorithm for Approximate Minimum Degree Spanning Trees
Given a graph , we wish to compute a spanning tree whose maximum
vertex degree, i.e. tree degree, is as small as possible. Computing the exact
optimal solution is known to be NP-hard, since it generalizes the Hamiltonian
path problem. For the approximation version of this problem, a
time algorithm that computes a spanning tree of degree at most is
previously known [F\"urer \& Raghavachari 1994]; here denotes the
minimum tree degree of all the spanning trees. In this paper we give the first
near-linear time approximation algorithm for this problem. Specifically
speaking, we propose an time algorithm that
computes a spanning tree with tree degree for any constant .
Thus, when , we can achieve approximate solutions with
constant approximate ratio arbitrarily close to 1 in near-linear time.Comment: 17 page
Improved Algorithm for Degree Bounded Survivable Network Design Problem
We consider the Degree-Bounded Survivable Network Design Problem: the
objective is to find a minimum cost subgraph satisfying the given connectivity
requirements as well as the degree bounds on the vertices. If we denote the
upper bound on the degree of a vertex v by b(v), then we present an algorithm
that finds a solution whose cost is at most twice the cost of the optimal
solution while the degree of a degree constrained vertex v is at most 2b(v) +
2. This improves upon the results of Lau and Singh and that of Lau, Naor,
Salavatipour and Singh
Exact Algorithms for Maximum Independent Set
We show that the maximum independent set problem (MIS) on an -vertex graph
can be solved in time and polynomial space, which even is
faster than Robson's -time exponential-space algorithm
published in 1986. We also obtain improved algorithms for MIS in graphs with
maximum degree 6 and 7, which run in time of and
, respectively. Our algorithms are obtained by using fast
algorithms for MIS in low-degree graphs in a hierarchical way and making a
careful analyses on the structure of bounded-degree graphs
Space Saving by Dynamic Algebraization
Dynamic programming is widely used for exact computations based on tree
decompositions of graphs. However, the space complexity is usually exponential
in the treewidth. We study the problem of designing efficient dynamic
programming algorithm based on tree decompositions in polynomial space. We show
how to construct a tree decomposition and extend the algebraic techniques of
Lokshtanov and Nederlof such that the dynamic programming algorithm runs in
time , where is the maximum number of vertices in the union of
bags on the root to leaf paths on a given tree decomposition, which is a
parameter closely related to the tree-depth of a graph. We apply our algorithm
to the problem of counting perfect matchings on grids and show that it
outperforms other polynomial-space solutions. We also apply the algorithm to
other set covering and partitioning problems.Comment: 14 pages, 1 figur
An Efficient Local Search for Partial Latin Square Extension Problem
A partial Latin square (PLS) is a partial assignment of n symbols to an nxn
grid such that, in each row and in each column, each symbol appears at most
once. The partial Latin square extension problem is an NP-hard problem that
asks for a largest extension of a given PLS. In this paper we propose an
efficient local search for this problem. We focus on the local search such that
the neighborhood is defined by (p,q)-swap, i.e., removing exactly p symbols and
then assigning symbols to at most q empty cells. For p in {1,2,3}, our
neighborhood search algorithm finds an improved solution or concludes that no
such solution exists in O(n^{p+1}) time. We also propose a novel swap
operation, Trellis-swap, which is a generalization of (1,q)-swap and
(2,q)-swap. Our Trellis-neighborhood search algorithm takes O(n^{3.5}) time to
do the same thing. Using these neighborhood search algorithms, we design a
prototype iterated local search algorithm and show its effectiveness in
comparison with state-of-the-art optimization solvers such as IBM ILOG CPLEX
and LocalSolver.Comment: 17 pages, 2 figure
How Fast Can We Multiply Large Integers on an Actual Computer?
We provide two complexity measures that can be used to measure the running
time of algorithms to compute multiplications of long integers. The random
access machine with unit or logarithmic cost is not adequate for measuring the
complexity of a task like multiplication of long integers. The Turing machine
is more useful here, but fails to take into account the multiplication
instruction for short integers, which is available on physical computing
devices. An interesting outcome is that the proposed refined complexity
measures do not rank the well known multiplication algorithms the same way as
the Turing machine model.Comment: To appear in the proceedings of Latin 2014. Springer LNCS 839
The number of matchings in random graphs
We study matchings on sparse random graphs by means of the cavity method. We
first show how the method reproduces several known results about maximum and
perfect matchings in regular and Erdos-Renyi random graphs. Our main new result
is the computation of the entropy, i.e. the leading order of the logarithm of
the number of solutions, of matchings with a given size. We derive both an
algorithm to compute this entropy for an arbitrary graph with a girth that
diverges in the large size limit, and an analytic result for the entropy in
regular and Erdos-Renyi random graph ensembles.Comment: 17 pages, 6 figures, to be published in Journal of Statistical
Mechanic
Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space
For many algorithmic problems on graphs of treewidth , a standard dynamic
programming approach gives an algorithm with time and space complexity
. It turns out that when one
considers the more restrictive parameter treedepth, it is often the case that a
variation of this technique can be used to reduce the space complexity to
polynomial, while retaining time complexity of the form
, where is the treedepth. This
transfer of methodology is, however, far from automatic. For instance, for
problems with connectivity constraints, standard dynamic programming techniques
give algorithms with time and space complexity on graphs of treewidth , but it is not clear how to
convert them into time-efficient polynomial space algorithms for graphs of low
treedepth.
Cygan et al. (FOCS'11) introduced the Cut&Count technique and showed that a
certain class of problems with connectivity constraints can be solved in time
and space complexity . Recently,
Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the
Cut&Count technique can be also applied in the setting of treedepth, and it
gives algorithms with running time
and polynomial space usage. However, a number of important problems eluded such
a treatment, with the most prominent examples being Hamiltonian Cycle and
Longest Path.
In this paper we clarify the situation by showing that Hamiltonian Cycle,
Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit
-time and polynomial space algorithms on graphs of
treedepth . The algorithms are randomized Monte Carlo with only false
negatives.Comment: Presented at WG2020. 20 pages, 2 figure
TRAVEL WATTLE OF LANDSCAPES, ANTIQUITY AND SPIRITUAL AURA OF SPACE IN TRAVELOGUES BY ZLATKO TOMIČIĆ
U radu se prvi put revalorizira još posve neistražen književni opus tiskanih
putopisnih zbirka istaknutoga suvremenog hrvatskog književnika
Zlatka Tomičića (Zagreb, 1930. – Zagreb, 2008.). Današnja međunarodna
književna produkcija snažno je usmjerena prema oblikovanjima tzv.
nefikcionalnih žanrova (memoara, putopisa, autobiografija) pa se prema
ulozi reanimiranja misaonoga poklisara realiziraju i Tomičićevi putopisi.
Tomičićeva putopisna proza jednim se dijelom promatra u kontekstu
njegovih književnih suvremenika i političko-društvene sudbine autora.
Međutim rad se u prvome planu usmjerava na književnu analizu i interpretaciju
šesnaest putopisnih knjiga te se unutar autorova oblikotvornoga
postupka promatraju i njegovi književni utjecaji, književne relacije
i refleksivno-filozofski obzori. Bitna konstanta Tomičićeva putopisa jest
težnja za polifonijskim i polihistorijskim modelom doživljavanja svijeta.
Kao obilježje Tomičićeva umjetničkoga postupka izdvaja se interferencijska
realizacija hibridnoga žanra putopisa i stvaranje književnoga pletera,
tj. ispreplitanja kozmopolitizma i rodoljublja te povezivanja realističkih
zapažanja s konstantom kulturno-duhovne aure opisivanoga prostora.The paper valorizes for the first time a completely unexplored literary
opus of published travelogue collections of prominent contemporary Croatian
writer Zlatko Tomičić (Zagreb, 1930. – Zagreb, 2008.). Contemporary
international literary production is strongly directed to forming of
the so called non-fictional genres (memoirs, travelogues, autobiographies),
therefore Tomičić’s travelogues are also realized in accordance with
the role of reanimating contemplative envoy. Tomičić’s travel prose is partly
considered in the context of his literary contemporaries and authors’
political-social destiny. However, the paper is in the first plan directed to
literary analysis and interpretation of sixteen travelogues and author’s
literary influences, literary relations and reflexive-philosophic visions are
considered within his forming procedure. A very important constant of
author’s travelogue is aspiration to polyphonic and historic model of life
experiencing. The characteristic of Tomičić’s artistic procedure is interferential
realization of hybrid travelogue genre and creation of literary
wattle i.e. interweaving of cosmopolitism and patriotism and connecting
of realistic observations with a constant of cultural-spiritual aura of described
space
On the State Complexity of Partial Derivative Automata For Regular Expressions with Intersection
Extended regular expressions (with complement and intersection) are used in many applications due to their succinctness. In particular, regular expressions extended with intersection only (also called semi-extended) can already be exponentially smaller than standard regular expressions or equivalent nondeterministic finite automata (NFA). For practical purposes it is important to study the average behaviour of conversions between these models. In this paper, we focus on the conversion of regular expressions with intersection to nondeterministic finite automata, using partial derivatives and the notion of support. First, we give a tight upper bound of 2O(n) for the worst-case number of states of the resulting partial derivative automaton, where n is the size of the expression. Using the framework of analytic combinatorics, we then establish an upper bound of (1.056 + o(1))n for its asymptotic average-state complexity, which is significantly smaller than the one for the worst case. (c) IFIP International Federation for Information Processing 2016
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