Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm

Given an undirected graph and two disjoint vertex pairs $s_1,t_1$ and $s_2,t_2$, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting $s_1$ with $t_1$, and $s_2$ with $t_2$, respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs with arbitrary placement of the terminals. In contrast, the randomized polynomial time algorithm by Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is serial in nature, and cannot count the solutions. Our results are built on an approach by Hirai and Namba, Algorithmica 2017, for a generalisation of S2DP, and fast algorithms for counting perfect matchings in planar graphs

Counting Matchings with k Unmatched Vertices in Planar Graphs

We consider the problem of counting matchings in planar graphs. While perfect matchings in planar graphs can be counted by a classical polynomial-time algorithm [Kasteleyn 1961], the problem of counting all matchings (possibly containing unmatched vertices, also known as defects) is known to be #P-complete on planar graphs [Jerrum 1987]. To interpolate between matchings and perfect matchings, we study the parameterized problem of counting matchings with k unmatched vertices in a planar graph G, on input G and k. This setting has a natural interpretation in statistical physics, and it is a special case of counting perfect matchings in k-apex graphs (graphs that become planar after removing k vertices). Starting from a recent #W[1]-hardness proof for counting perfect matchings on k-apex graphs [Curtican and Xia 2015], we obtain: - Counting matchings with k unmatched vertices in planar graphs is #W[1]-hard. - In contrast, given a plane graph G with s distinguished faces, there is an O(2^s n^3) time algorithm for counting those matchings with k unmatched vertices such that all unmatched vertices lie on the distinguished faces. This implies an f(k,s)n^O(1) time algorithm for counting perfect matchings in k-apex graphs whose apex neighborhood is covered by s faces

Distributed Dominating Set Approximations beyond Planar Graphs

The Minimum Dominating Set (MDS) problem is one of the most fundamental and challenging problems in distributed computing. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, there has been much progress on computing local approximations on sparse graphs, and in particular planar graphs. In this paper we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm and (2) a local $\mathcal{O}(\log^*{n})$-time approximation scheme. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299

Holographic Algorithm with Matchgates Is Universal for Planar #\#CSP Over Boolean Domain

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).Comment: 94 page

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in (Mulmuley et al. 1987) achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of non-uniform SPL by (Allender et al. 1999) and $NC^2$ by (Miller and Naor 1995, Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in non-bipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple

Kernelization for Counting Problems on Graphs: Preserving the Number of Minimum Solutions

A kernelization for a parameterized decision problem $\mathcal{Q}$ is a polynomial-time preprocessing algorithm that reduces any parameterized instance $(x,k)$ into an instance $(x',k')$ whose size is bounded by a function of $k$ alone and which has the same yes/no answer for $\mathcal{Q}$. Such preprocessing algorithms cannot exist in the context of counting problems, when the answer to be preserved is the number of solutions, since this number can be arbitrarily large compared to $k$. However, we show that for counting minimum feedback vertex sets of size at most $k$, and for counting minimum dominating sets of size at most $k$ in a planar graph, there is a polynomial-time algorithm that either outputs the answer or reduces to an instance $(G',k')$ of size polynomial in $k$ with the same number of minimum solutions. This shows that a meaningful theory of kernelization for counting problems is possible and opens the door for future developments. Our algorithms exploit that if the number of solutions exceeds $2^{\mathsf{poly}(k)}$, the size of the input is exponential in terms of $k$ so that the running time of a parameterized counting algorithm can be bounded by $\mathsf{poly}(n)$. Otherwise, we can use gadgets that slightly increase $k$ to represent choices among $2^{O(k)}$ options by only $\mathsf{poly}(k)$ vertices.Comment: Extended abstract appears in the proceedings of IPEC 202

New Polynomial Cases of the Weighted Efficient Domination Problem

Let G be a finite undirected graph. A vertex dominates itself and all its neighbors in G. A vertex set D is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete even for very restricted graph classes. In particular, the ED problem remains NP-complete for 2P3-free graphs and thus for P7-free graphs. We show that the weighted version of the problem (abbreviated WED) is solvable in polynomial time on various subclasses of 2P3-free and P7-free graphs, including (P2+P4)-free graphs, P5-free graphs and other classes. Furthermore, we show that a minimum weight e.d. consisting only of vertices of degree at most 2 (if one exists) can be found in polynomial time. This contrasts with our NP-completeness result for the ED problem on planar bipartite graphs with maximum degree 3