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Locating source of diffusion in networks is crucial for controlling and preventing epidemic risks. It has been studied under various probabilistic models. In this paper, we study source location from a deterministic point of view by modeling it as the minimum weighted doubly resolving set (DRS) problem, which is a strengthening of the well-known metric dimension problem.Let G be a vertex weighted undirected graph on n vertices. A vertex subset S of G is DRS of G if for every pair of vertices u, v in G, there exist x, y ∈ S such that the difference of distances (in terms of number of edges) between u and x, y is not equal to the difference of distances between v and x, y. The minimum weighted DRS problem consists of finding a DRS in G with minimum total weight. We establish Θ(ln n) approximability of the minimum DRS problem on general graphs for both weighted and unweighted versions. This is the first work providing explicit approximation lower and upper bounds for minimum (weighted) DRS problem, which are nearly tight. Moreover, we design first known strongly polynomial time algorithms for the minimum weighted DRS problem on general wheels and trees with additional constant k ≥ 0 edges.

We study a network congestion game of discrete-time dynamic traffic of atomic agents with a single origin-destination pair. Any agent freely makes a dynamic decision at each vertex (e.g., road crossing) and traffic is regulated with given priorities on edges (e.g., road segments). We first constructively prove that there always exists a subgame perfect equilibrium (SPE) in this game. We then study the relationship between this model and a simplified model, in which agents select and fix an origin-destination path simultaneously. We show that the set of Nash equilibrium (NE) flows of the simplified model is a proper subset of the set of SPE flows of our main model. We prove that each NE is also a strong NE and hence weakly Pareto optimal. We establish several other nice properties of NE flows, including global First-In-First-Out. Then for two classes of networks, including series-parallel ones, we show that the queue lengths at equilibrium are bounded at any given instance, which means the price of anarchy of any given game instance is bounded, provided that the inflow size never exceeds the network capacity.

We design a Copula-based generic randomized truthful mechanism for scheduling on two unrelated machines with approximation ratio within [1.5852, 1.58606], offering an improved upper bound for the two-machine case. Moreover, we provide an upper bound 1.5067711 for the two-machine two-task case, which is almost tight in view of the lower bound of 1.506 for the scale-free truthful mechanisms [4]. Of independent interest is the explicit incorporation of the concept of Copula in the design and analysis of the proposed approximation algorithm. We hope that techniques like this one will also prove useful in solving other problems in the future.

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a b s t r a c tLet G be a graph and let A be its cutset-edge incidence matrix. We prove that the linear system 1 2 Ax ≥ 1, x ≥ 0 is box totally dual integral (box-TDI) if and only if G is a seriesparallel graph; a by-product of this characterization is a structural description of a box-TDI system on matroids. Our results strengthen two previous theorems obtained respectively by Cornuéjols, Fonlupt, and Naddef and by Mahjoub which assert that both polyhedra

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