2,033 research outputs found

    Counting independent sets in graphs with bounded bipartite pathwidth

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    The Glauber dynamics can efficiently sample independent sets almost uniformly at random in polynomial time for graphs in a certain class. The class is determined by boundedness of a new graph parameter called bipartite pathwidth. This result, which we prove for the more general hardcore distribution with fugacity λ, can be viewed as a strong generalisation of Jerrum and Sinclair’s work on approximately counting matchings. The class of graphs with bounded bipartite path-width includes line graphs and claw-free graphs, which generalise line graphs. We consider two further generalisations of claw-free graphs and prove that these classes have bounded bipartite pathwidth

    Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

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    In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials C\mathcal C and show that if p∈Cp\in \cal C and there is a disk DD centered at zero in the complex plane such that p(G)p(G) does not vanish on DD for all bounded degree graphs GG, then for each zz in the interior of DD there exists a deterministic polynomial-time approximation algorithm for evaluating p(G)p(G) at zz. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by. Barvinok, which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In particular a tiny error in Proposition 4.4 has been fixed. The introduction and concluding remarks have also been rewritten to incorporate the most recent developments. Accepted for publication in SIAM Journal on Computatio

    Extremal Colorings and Independent Sets

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    We consider several extremal problems of maximizing the number of colorings and independent sets in some graph families with fixed chromatic number and order. First, we address the problem of maximizing the number of colorings in the family of connected graphs with chromatic number k and order n where k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. It was conjectured that extremal graphs are those which have clique number k and size (k2)+n−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(k2)+n−k(k2)+n−k. We affirm this conjecture for 4-chromatic claw-free graphs and for all k-chromatic line graphs with k≥4 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3ek≥4k≥4. We also reduce this extremal problem to a finite family of graphs when restricted to claw-free graphs. Secondly, we determine the maximum number of independent sets of each size in the family of n-vertex k-chromatic graphs (respectively connected n-vertex k-chromatic graphs and n-vertex k-chromatic graphs with c components). We show that the unique extremal graph is Kk∪En−k role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eKk∪En−kKk∪En−k, K1∨(Kk−1∪En−k) role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3eK1∨(Kk−1∪En−k)K1∨(Kk−1∪En−k) and (K1∨(Kk−1∪En−k−c+1))∪Ec−1 role= presentation style= box-sizing: inherit; display: inline; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative; \u3e(K1∨(Kk−1∪En−k−c+1))∪Ec−1(K1∨(Kk−1∪En−k−c+1))∪Ec−1 respectively

    The Complexity of Rerouting Shortest Paths

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    The Shortest Path Reconfiguration problem has as input a graph G (with unit edge lengths) with vertices s and t, and two shortest st-paths P and Q. The question is whether there exists a sequence of shortest st-paths that starts with P and ends with Q, such that subsequent paths differ in only one vertex. This is called a rerouting sequence. This problem is shown to be PSPACE-complete. For claw-free graphs and chordal graphs, it is shown that the problem can be solved in polynomial time, and that shortest rerouting sequences have linear length. For these classes, it is also shown that deciding whether a rerouting sequence exists between all pairs of shortest st-paths can be done in polynomial time. Finally, a polynomial time algorithm for counting the number of isolated paths is given.Comment: The results on claw-free graphs, chordal graphs and isolated paths have been added in version 2 (april 2012). Version 1 (September 2010) only contained the PSPACE-hardness result. (Version 2 has been submitted.

    Hard and Easy Instances of L-Tromino Tilings

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    We study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it either remains NP-complete or has a polynomial time algorithm. First, we characterize the possibility of when an Aztec rectangle and an Aztec diamond has an L-tromino tiling. Then, we study tilings of arbitrary regions where only 180∘180^\circ rotations of L-trominoes are available. For this particular case we show that deciding the existence of a tiling remains NP-complete; yet, if a region does not contains certain so-called "forbidden polyominoes" as sub-regions, then there exists a polynomial time algorithm for deciding a tiling.Comment: Full extended version of LNCS 11355:82-95 (WALCOM 2019
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