Min-Cost Flow in Unit-Capacity Planar Graphs

In this paper we give an O~((nm)^(2/3) log C) time algorithm for computing min-cost flow (or min-cost circulation) in unit capacity planar multigraphs where edge costs are integers bounded by C. For planar multigraphs, this improves upon the best known algorithms for general graphs: the O~(m^(10/7) log C) time algorithm of Cohen et al. [SODA 2017], the O(m^(3/2) log(nC)) time algorithm of Gabow and Tarjan [SIAM J. Comput. 1989] and the O~(sqrt(n) m log C) time algorithm of Lee and Sidford [FOCS 2014]. In particular, our result constitutes the first known fully combinatorial algorithm that breaks the Omega(m^(3/2)) time barrier for min-cost flow problem in planar graphs. To obtain our result we first give a very simple successive shortest paths based scaling algorithm for unit-capacity min-cost flow problem that does not explicitly operate on dual variables. This algorithm also runs in O~(m^(3/2) log C) time for general graphs, and, to the best of our knowledge, it has not been described before. We subsequently show how to implement this algorithm faster on planar graphs using well-established tools: r-divisions and efficient algorithms for computing (shortest) paths in so-called dense distance graphs

Shortest path and maximum flow problems in planar flow networks with additive gains and losses

In contrast to traditional flow networks, in additive flow networks, to every edge e is assigned a gain factor g(e) which represents the loss or gain of the flow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) is less than the designated capacity of e, then f(e) + g(e) = 0 units of flow reach the end point of e, provided e is used, i.e., provided f(e) != 0. In this report we study the maximum flow problem in additive flow networks, which we prove to be NP-hard even when the underlying graphs of additive flow networks are planar. We also investigate the shortest path problem, when to every edge e is assigned a cost value for every unit flow entering edge e, which we show to be NP-hard in the strong sense even when the additive flow networks are planar

Max-flow vitality in undirected unweighted planar graphs

We show a fast algorithm for determining the set of relevant edges in a planar undirected unweighted graph with respect to the maximum flow. This is a special case of the \emph{max flow vitality} problem, that has been efficiently solved for general undirected graphs and $st$-planar graphs. The \emph{vitality} of an edge of a graph with respect to the maximum flow between two fixed vertices $s$ and $t$ is defined as the reduction of the maximum flow caused by the removal of that edge. In this paper we show that the set of edges having vitality greater than zero in a planar undirected unweighted graph with $n$ vertices, can be found in $O(n \log n)$ worst-case time and $O(n)$ space.Comment: 9 pages, 4 figure