ACM Transactions on Algorithms最新文献

We present an undirected version of the recently introduced flow-augmentation technique: Given an undirected multigraph G with distinguished vertices s, t ∈ V(G) and an integer k, one can in randomized (k^{mathcal {O}(1)} cdot (|V(G)| + |E(G)|) ) time sample a set (A subseteq binom{V(G)}{2} ) such that the following holds: for every inclusion-wise minimal st-cut Z in G of cardinality at most k, Z becomes a minimum-cardinality cut between s and t in G + A (i.e., in the multigraph G with all edges of A added) with probability (2^{-mathcal {O}(k log k)} ).

Compared to the version for directed graphs [STOC 2022], the version presented here has improved success probability ((2^{-mathcal {O}(k log k)} ) instead of (2^{-mathcal {O}(k^4 log k)} )), linear dependency on the graph size in the running time bound, and an arguably simpler proof.

An immediate corollary is that the Bi-objective st-Cut problem can be solved in randomized FPT time (2^{mathcal {O}(k log k)} (|V(G)|+|E(G)|) ) on undirected graphs.

The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs G, H, and lists L(v)⊆V(H) for every v ∈ V(G), a list homomorphism is a function f: V(G) → V(H) that preserves the edges (i.e., uv ∈ E(G) implies f(u)f(v) ∈ E(H)) and respects the lists (i.e., f(v) ∈ L(v)). Standard techniques show that if G is given with a tree decomposition of width t, then the number of list homomorphisms can be counted in time (|V(H)|^tcdot n^{mathcal {O}(1)} ). Our main result is determining, for every fixed graph H, how much the base |V(H)| in the running time can be improved. For a connected graph H we define (operatorname{irr}(H) ) in the following way: if H has a loop or is nonbipartite, then (operatorname{irr}(H) ) is the maximum size of a set S⊆V(H) where any two vertices have different neighborhoods; if H is bipartite, then (operatorname{irr}(H) ) is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected H, we define (operatorname{irr}(H) ) as the maximum of (operatorname{irr}(C) ) over every connected component C of H. It follows from earlier results that if (operatorname{irr}(H)=1 ), then the problem of counting list homomorphisms to H is polynomial-time solvable, and otherwise it is #P-hard. We show that, for every fixed graph H, the number of list homomorphisms from (G, L) to H

Let G = (A∪B, E) be a bipartite graph where the set A consists of agents or main players and the set B consists of jobs or secondary players. Every vertex in A∪B has a strict ranking of its neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M to N is at least the number that prefer N to M. Popular matchings always exist in G since every stable matching is popular. A matching M is A-popular if for any matching N, the number of agents (i.e., vertices in A) that prefer M to N is at least the number of agents that prefer N to M. Unlike popular matchings, A-popular matchings need not exist in a given instance G and there is a simple linear time algorithm to decide if G admits an A-popular matching and compute one, if so.

We consider the problem of deciding if G admits a matching that is both popular and A-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when A is the more important side—so along with overall popularity, we would like to maintain “popularity within the set A”. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.

The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance.

Examples of results obtained for this restricted setting are:

The number one criticism of average-case analysis is that we do not actually know the probability distribution of real-world inputs. Thus, analyzing an algorithm on some random model has no implications for practical performance. At its core, this criticism doubts the existence of external validity, i.e., it assumes that algorithmic behavior on the somewhat simple and clean models does not translate beyond the models to practical performance real-world input.

With this paper, we provide a first step towards studying the question of external validity systematically. To this end, we evaluate the performance of six graph algorithms on a collection of 2740 sparse real-world networks depending on two properties; the heterogeneity (variance in the degree distribution) and locality (tendency of edges to connect vertices that are already close). We compare this with the performance on generated networks with varying locality and heterogeneity. We find that the performance in the idealized setting of network models translates surprisingly well to real-world networks. Moreover, heterogeneity and locality appear to be the core properties impacting the performance of many graph algorithms.