Maximum Skew-Symmetric Flows and Matchings

The maximum integer skew-symmetric flow problem (MSFP) generalizes both the maximum flow and maximum matching problems. It was introduced by Tutte in terms of self-conjugate flows in antisymmetrical digraphs. He showed that for these objects there are natural analogs of classical theoretical results on usual network flows, such as the flow decomposition, augmenting path, and max-flow min-cut theorems. We give unified and shorter proofs for those theoretical results. We then extend to MSFP the shortest augmenting path method of Edmonds and Karp and the blocking flow method of Dinits, obtaining algorithms with similar time bounds in general case. Moreover, in the cases of unit arc capacities and unit ``node capacities'' the blocking skew-symmetric flow algorithm has time bounds similar to those established in Even and Tarjan (1975) and Karzanov (1973) for Dinits' algorithm. In particular, this implies an algorithm for finding a maximum matching in a nonbipartite graph in $O(\sqrt{n}m)$ time, which matches the time bound for the algorithm of Micali and Vazirani. Finally, extending a clique compression technique of Feder and Motwani to particular skew-symmetric graphs, we speed up the implied maximum matching algorithm to run in $O(\sqrt{n}m\log(n^2/m)/\log{n})$ time, improving the best known bound for dense nonbipartite graphs. Also other theoretical and algorithmic results on skew-symmetric flows and their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor stylistic corrections and shortenings to the original versio

Linear Time Parameterized Algorithms via Skew-Symmetric Multicuts

A skew-symmetric graph $(D=(V,A),\sigma)$ is a directed graph $D$ with an involution $\sigma$ on the set of vertices and arcs. In this paper, we introduce a separation problem, $d$-Skew-Symmetric Multicut, where we are given a skew-symmetric graph $D$, a family of $\cal T$ of $d$-sized subsets of vertices and an integer $k$. The objective is to decide if there is a set $X\subseteq A$ of $k$ arcs such that every set $J$ in the family has a vertex $v$ such that $v$ and $\sigma(v)$ are in different connected components of $D'=(V,A\setminus (X\cup \sigma(X))$. In this paper, we give an algorithm for this problem which runs in time $O((4d)^{k}(m+n+\ell))$, where $m$ is the number of arcs in the graph, $n$ the number of vertices and $\ell$ the length of the family given in the input. Using our algorithm, we show that Almost 2-SAT has an algorithm with running time $O(4^kk^4\ell)$ and we obtain algorithms for {\sc Odd Cycle Transversal} and {\sc Edge Bipartization} which run in time $O(4^kk^4(m+n))$ and $O(4^kk^5(m+n))$ respectively. This resolves an open problem posed by Reed, Smith and Vetta [Operations Research Letters, 2003] and improves upon the earlier almost linear time algorithm of Kawarabayashi and Reed [SODA, 2010]. We also show that Deletion q-Horn Backdoor Set Detection is a special case of 3-Skew-Symmetric Multicut, giving us an algorithm for Deletion q-Horn Backdoor Set Detection which runs in time $O(12^kk^5\ell)$. This gives the first fixed-parameter tractable algorithm for this problem answering a question posed in a paper by a superset of the authors [STACS, 2013]. Using this result, we get an algorithm for Satisfiability which runs in time $O(12^kk^5\ell)$ where $k$ is the size of the smallest q-Horn deletion backdoor set, with $\ell$ being the length of the input formula

Faster Algorithms for Half-Integral T-Path Packing

Let G = (V, E) be an undirected graph, a subset of vertices T be a set of terminals. Then a natural combinatorial problem consists in finding the maximum number of vertex-disjoint paths connecting distinct terminals. For this problem, a clever construction suggested by Gallai reduces it to computing a maximum non-bipartite matching and thus gives an O(mn^1/2 log(n^2/m)/log(n))-time algorithm (hereinafter n := |V|, m := |E|). Now let us consider the fractional relaxation, i.e. allow T-path packings with arbitrary nonnegative real weights. It is known that there always exists a half-integral solution, that is, one only needs to assign weights 0, 1/2, 1 to maximize the total weight of T-paths. It is also known that an optimum half-integral packing can be found in strongly-polynomial time but the actual time bounds are far from being satisfactory. In this paper we present a novel algorithm that solves the half-integral problem within O(mn^1/2 log(n^2/m)/log(n)) time, thus matching the complexities of integral and half-integral versions

On Adaptive Algorithms for Maximum Matching

In the fundamental Maximum Matching problem the task is to find a maximum cardinality set of pairwise disjoint edges in a given undirected graph. The fastest algorithm for this problem, due to Micali and Vazirani, runs in time O(sqrt{n}m) and stands unbeaten since 1980. It is complemented by faster, often linear-time, algorithms for various special graph classes. Moreover, there are fast parameterized algorithms, e.g., time O(km log n) relative to tree-width k, which outperform O(sqrt{n}m) when the parameter is sufficiently small. We show that the Micali-Vazirani algorithm, and in fact any algorithm following the phase framework of Hopcroft and Karp, is adaptive to beneficial input structure. We exhibit several graph classes for which such algorithms run in linear time O(n+m). More strongly, we show that they run in time O(sqrt{k}m) for graphs that are k vertex deletions away from any of several such classes, without explicitly computing an optimal or approximate deletion set; before, most such bounds were at least Omega(km). Thus, any phase-based matching algorithm with linear-time phases obliviously interpolates between linear time for k=O(1) and the worst case of O(sqrt{n}m) when k=Theta(n). We complement our findings by proving that the phase framework by itself still allows Omega(sqrt{n}) phases, and hence time Omega(sqrt{n}m), even on paths, cographs, and bipartite chain graphs