3,843 research outputs found
Faster exponential-time algorithms in graphs of bounded average degree
We first show that the Traveling Salesman Problem in an n-vertex graph with
average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and
exponential space for a constant \eps_d depending only on d, where the
O*-notation suppresses factors polynomial in the input size. Thus, we
generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of
bounded degree.
Then, we move to the problem of counting perfect matchings in a graph. We
first present a simple algorithm for counting perfect matchings in an n-vertex
graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the
complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on
inclusion-exclusion principle instead of algebraic transformations. Building
upon this result, we show that the number of perfect matchings in an n-vertex
graph with average degree bounded by d can be computed in
O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the
constant obtained by us for the Traveling Salesman Problem in graphs of average
degree at most 2d.
Moreover we obtain a simple algorithm that counts the number of perfect
matchings in an n-vertex bipartite graph of average degree at most d in
O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of
Izumi and Wadayama [FOCS 2012].Comment: 10 page
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
time and exponential space for graphs of average
degree . 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
time and polynomial space for graphs of average
degree . 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~ we present an
algorithm with running time 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
A near-optimal approximation algorithm for Asymmetric TSP on embedded graphs
We present a near-optimal polynomial-time approximation algorithm for the
asymmetric traveling salesman problem for graphs of bounded orientable or
non-orientable genus. Our algorithm achieves an approximation factor of O(f(g))
on graphs with genus g, where f(n) is the best approximation factor achievable
in polynomial time on arbitrary n-vertex graphs. In particular, the
O(log(n)/loglog(n))-approximation algorithm for general graphs by Asadpour et
al. [SODA 2010] immediately implies an O(log(g)/loglog(g))-approximation
algorithm for genus-g graphs. Our result improves the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi [SODA
2011], which applies only to graphs with orientable genus g; ours is the first
approximation algorithm for graphs with bounded non-orientable genus.
Moreover, using recent progress on approximating the genus of a graph, our
O(log(g) / loglog(g))-approximation can be implemented even without an
embedding when the input graph has bounded degree. In contrast, the
O(sqrt(g)*log(g))-approximation algorithm of Oveis Gharan and Saberi requires a
genus-g embedding as part of the input.
Finally, our techniques lead to a O(1)-approximation algorithm for ATSP on
graphs of genus g, with running time 2^O(g)*n^O(1)
Fine-Grained Complexity of k-OPT in Bounded-Degree Graphs for Solving TSP
The Traveling Salesman Problem asks to find a minimum-weight Hamiltonian cycle in an edge-weighted complete graph. Local search is a widely-employed strategy for finding good solutions to TSP. A popular neighborhood operator for local search is k-opt, which turns a Hamiltonian cycle C into a new Hamiltonian cycle C\u27 by replacing k edges. We analyze the problem of determining whether the weight of a given cycle can be decreased by a k-opt move. Earlier work has shown that (i) assuming the Exponential Time Hypothesis, there is no algorithm that can detect whether or not a given Hamiltonian cycle C in an n-vertex input can be improved by a k-opt move in time f(k) n^o(k / log k) for any function f, while (ii) it is possible to improve on the brute-force running time of O(n^k) and save linear factors in the exponent. Modern TSP heuristics are very successful at identifying the most promising edges to be used in k-opt moves, and experiments show that very good global solutions can already be reached using only the top-O(1) most promising edges incident to each vertex. This leads to the following question: can improving k-opt moves be found efficiently in graphs of bounded degree? We answer this question in various regimes, presenting new algorithms and conditional lower bounds. We show that the aforementioned ETH lower bound also holds for graphs of maximum degree three, but that in bounded-degree graphs the best improving k-move can be found in time O(n^((23/135+epsilon_k)k)), where lim_{k -> infty} epsilon_k = 0. This improves upon the best-known bounds for general graphs. Due to its practical importance, we devote special attention to the range of k in which improving k-moves in bounded-degree graphs can be found in quasi-linear time. For k <= 7, we give quasi-linear time algorithms for general weights. For k=8 we obtain a quasi-linear time algorithm when the weights are bounded by O(polylog n). On the other hand, based on established fine-grained complexity hypotheses about the impossibility of detecting a triangle in edge-linear time, we prove that the k = 9 case does not admit quasi-linear time algorithms. Hence we fully characterize the values of k for which quasi-linear time algorithms exist for polylogarithmic weights on bounded-degree graphs
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of points with their pairwise distances, the traveling
salesman problem (TSP) asks for a shortest tour that visits each point exactly
once. A TSP instance is rectilinear when the points lie in the plane and the
distance considered between two points is the distance. In this paper, a
fixed-parameter algorithm for the Rectilinear TSP is presented and relies on
techniques for solving TSP on bounded-treewidth graphs. It proves that the
problem can be solved in where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time
It is well known that many local graph problems, like Vertex Cover and
Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E)
with a given tree decomposition of width tw. However, for nonlocal problems,
like the fundamental class of connectivity problems, for a long time we did not
know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al.
(FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity
problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for
Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether
randomization is necessary to achieve this runtime; furthermore, it is
desirable to also solve counting and weighted versions (the latter without
incurring a pseudo-polynomial cost in terms of the weights).
We present two new approaches rooted in linear algebra, based on matrix rank
and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms,
also for weighted and counting versions. For example, in this time we can solve
the traveling salesman problem or count the number of Hamiltonian cycles. The
rank-based ideas provide a rather general approach for speeding up even
straightforward dynamic programming formulations by identifying "small" sets of
representative partial solutions; we focus on the case of expressing
connectivity via sets of partitions, but the essential ideas should have
further applications. The determinant-based approach uses the matrix tree
theorem for deriving closed formulas for counting versions of connectivity
problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page
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