The Stackelberg minimum spanning tree game on planar and bounded-treewidth graphs

Consider property testing on bounded degree graphs and let $\\varepsilon>0$\ndenote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP\n2013) asserts that all properties of planar graphs (more generally hyperfinite)\nare testable with query complexity only depending on $\\varepsilon$. Recent\nadvances in testing minor-freeness have proven that all additive and monotone\nproperties of planar graphs can be tested in $poly(\\varepsilon^{-1})$ queries.\nSome properties falling outside this class, such as Hamiltonicity, also have a\nsimilar complexity for planar graphs. Motivated by these results, we ask: can\nall properties of planar graphs can be tested in $poly(\\varepsilon^{-1})$\nqueries? Is there a uniform query complexity upper bound for all planar\nproperties, and what is the "hardest" such property to test? We discover a\nsurprisingly clean and optimal answer. Any property of bounded degree planar\ngraphs can be tested in $\\exp(O(\\varepsilon^{-2}))$ queries. Moreover, there is\na matching lower bound, up to constant factors in the exponent. The natural\nproperty of testing isomorphism to a fixed graph needs\n$\\exp(\\Omega(\\varepsilon^{-2}))$ queries, thereby showing that (up to\npolynomial dependencies) isomorphism to an explicit fixed graph is the hardest\nproperty of planar graphs. The upper bound is a straightforward adapation of\nthe Newman-Sohler analysis that tracks dependencies on $\\varepsilon$ carefully.\nThe main technical contribution is the lower bound construction, which is\nachieved by a special family of planar graphs that are all mutually far from\neach other. We can also apply our techniques to get analogous results for\nbounded treewidth graphs. We prove that all properties of bounded treewidth\ngraphs can be tested in $\\exp(O(\\varepsilon^{-1}\\log \\varepsilon^{-1}))$\nqueries. Moreover, testing isomorphism to a fixed forest requires\n$\\exp(\\Omega(\\varepsilon^{-1}))$ queries.\n

Although Min-CDS in general graphs is hard to approximate, the restriction to certain special graph classes admits much better approximation results. Min-CDS in planar graphs remains NP-hard even for planar graphs that are regular of degree 4 [57]. The related problem, Min-DS in planar graphs, is also NP-hard even for planar graphs with maximum vertex degree 3 and planar graphs that are regular of degree 4 [57]. It is well known that Min-DS in planar graphs possesses a polynomial-time approximation scheme (PTAS) based on the shifting strategy [3]: For any constant e >0, there is a polynomial-time 1+e-approximation algorithm. Thus, it is immediate to conclude that Min-CDS in planar graphs can be approximated within a factor 3+e for any e>0 in polynomial time. However, the degree of the polynomial grows with 1∕e and hence, the approximation scheme is hardly practical.

On external-memory MST, SSSP and multi-way planar graph separation

Parameterized algorithms for load coloring problem

We introduce the Blue-Red Matching problem: given a graph with red and blue edges, and a bound w, find a maximum matching consisting of at most w edges of each color. We show that Blue-Red Matching is at least as hard as the problem Exact Matching (Papadimitriou and Yannakakis, 1982), for which it is still open whether it can be solved in polynomial time. We present an RNC algorithm for this problem as well as two fast approximation algorithms. We finally show the applicability of our results to the problem of routing and assigning wavelengths to a maximum number of requests in all-optical rings.

The mixed irredundant Ramsey number t(m, n) is the smallest natural number t such that if the edges of the complete graph Kt on t vertices are arbitrarily bi-coloured using the colours blue and red, then necessarily either the subgraph induced by the blue edges has an irredundant set of cardinality m or the subgraph induced by the red edges has an independent set of cardinality n (or both). Previously it was known that 18 ≤ t(3, 7) ≤ 22 and 18 ≤ t(3, 8) ≤ 28. In this paper we prove that t(3, 7) = 18 and t(3, 8) = 22.

We study the following all-or-nothing multicommodity flow problem in planar graphs. Input: A graph G with n vertices and k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Find: A largest subset W of {1, ...., k such that for every i in W, we can send one unit of flow between si and ti. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri et al. (STOC'04) give a poly-logarithmic factor approximation algorithm and show that a natural LP-relaxation has a poly-logarithmic integrality gap. This result is in contrast with the integrality gap Ω(√n) for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an O(1)-approximation algorithm and show that the integrality gap is O(1). In particular, in polynomial time, we can find an index set W with |W| = Ω(OPT) and eight si-ti paths for each i in W such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the recent result by S'eguin-Charbonneau and Shepherd (FOCS'11) who give an O(1)-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).

Valiant introduced matchgate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, matchgate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in the statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs and can be arbitrary real-valued symmetric functions. We prove that every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #P-hard on general graphs but tractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.

We study the computational complexity of the map redistricting problem (gerrymandering). Mathematically, the electoral district designer (gerrymanderer) attempts to partition a weighted graph into k connected components (districts) such that its candidate (party) wins as many districts as possible. Prior work has principally concerned the special cases where the graph is a path or a tree. Our focus concerns the realistic case where the graph is planar. We prove that the gerrymandering problem is solvable in polynomial time in λ-outerplanar graphs, when the number of candidates and λ are constants and the vertex weights (voting weights) are polynomially bounded. In contrast, the problem is NP-complete in general planar graphs even with just two candidates. This motivates the study of approximation algorithms for gerrymandering planar graphs. However, when the number of candidates is large, we prove it is hard to distinguish between instances where the gerrymanderer cannot win a single district and instances where the gerrymanderer can win at least one district. This immediately implies that the redistricting problem is inapproximable in polynomial time in planar graphs, unless P=NP. This conclusion appears terminal for the design of good approximation algorithms - but it is not. The inapproximability bound can be circumvented as it only applies when the maximum number of districts the gerrymanderer can win is extremely small, say one. Indeed, for a fixed number of candidates, our main result is that there is a constant factor approximation algorithm for redistricting unweighted planar graphs, provided the optimal value is a large enough constant.

In the exact matching problem we are given a graph G, some of whose edges are colored red, and a positive integer k. The goal is to determine if G has a perfect matching, exactly k edges of which are red. More generally if the matching number of G is m=m(G), the goal is to find a matching with m edges, exactly k edges of which are red, or determine that no such matching exists. This problem is one of the few remaining problems that have efficient randomized algorithms (in fact, this problem is in RNC), but for which no polynomial time deterministic algorithm is known. Our first result shows that, in a sense, this problem is as close to being in P as one can get. We give a polynomial time deterministic algorithm that either correctly decides that no maximum matching has exactly k red edges, or exhibits a matching with m(G)−1 edges having exactly k red edges. Hence, the additive error is one. We also present an efficient algorithm for the exact matching problem in families of graphs for which this problem is known to be tractable. We show how to count the number of exact perfect matchings in K 3,3-minor free graphs (these include all planar graphs as well as many others) in O(n 3.19) worst case time. Our algorithm can also count the number of perfect matchings in K 3,3-minor free graphs in O(n 2.19) time.

Given a graph with edges colored Red and Blue, we wish to sample and approximately count the number of perfect matchings with exactly k Red edges. We study a Markov chain on the space of all matchings of a graph that favors matchings with k Red edges. We show that it is rapidly mixing using non-traditional canonical paths that can backtrack, based on an algorithm for a simple combinatorial problem. We show that this chain can be used to sample dimer configurations on a 2-dimensional toroidal region with k Red edges.

Deciding whether a grid is a topological subgraph of a planar graph is NP-complete

The problem of determining the domination number of a graph is a well known NP-hard problem, even when restricted to planar graphs. By adding a further restriction on the diameter of the graph, we prove that planar graphs with diameter two and three have bounded domination numbers. This implies that the domination number of such a graph can be determined in polynomial time. We also give examples of planar graphs of diameter four, and nonplanar graphs of diameter two, having arbitrarily large domination numbers. © 1996 John Wiley & Sons, Inc.

Let G = (V,E) be a k-edge-connected multigraph with a designated vertex s ∈ V which has even degree. A splitting operation at s replaces two edges (s, u) and (s, v) incident to s with a single edge (u, v). A set of splitting operations at s is called complete if there is no edge incident to s in the resulting graph. It is known by Lovasz (1979) that there always exists a complete splitting at s such that the resulting graph G′ (neglecting the isolated vertex s) remains k-edge-connected. In this paper, we prove that, in the case where G is planar and k is an even integer or k = 3, there exists a complete splitting at s such that the resulting graph G′ remains k-edge-connected and planar, and present an O(|V|3 log|V|) time algorithm for finding such a splitting. However, for every odd k ≥ 5, there is a planar graph G with a vertex s which has no complete splitting at s which preserves both k-edge-connectivity and planarity. As an application of this result, we show that the problem of augmenting the edge-connectivity of a given outerplanar graph to an even integer k or to k = 3 can be solved in polynomial time.

A straight-line grid drawing of a plane graph G is a planar drawing of G, where each vertex is drawn at a grid point of an integer grid and each edge is drawn as a straight-line segment. The area of such a drawing is the area of the smallest axis-aligned rectangle on the grid which encloses the drawing. A minimum-area drawing of a plane graph G is a straight-line grid drawing of G where the area of the drawing is the minimum. Although it is NP-hard to find minimum-area drawings for general plane graphs, in this paper we obtain minimumarea drawings for plane 3-trees in polynomial time. Furthermore, we show a ⌊ 2n 3 − 1⌋ × 2⌈ n 3 ⌉ lower bound for the area of a straight-line grid drawing of a plane 3tree with n ≥ 6 vertices, which improves the previously known lower bound ⌊ 2(n−1) 3 ⌋×⌊ 2(n−1) 3 ⌋ for plane graphs.