Deciding whether or not a feasible solution to the Traveling Salesman Problem with Pickups and Deliveries (TSPPD) exists is polynomially solvable. We prove that counting the number of feasible solutions of the TSPPD is hard by showing the problem is #P-complete.

We prove that the problem of counting the number of colourings of the vertices of a graph with at most two colours, such that the colour classes induce connected subgraphs is #P-complete. We also show that the closely related problem of counting the number of cocircuits of a graph is #P-complete.

Answering a problem posed by Nakhleh, we prove that counting the number of phylogenetic trees inferred by a (binary) phylogenetic network is #P-complete. An immediate consequence of this result is that counting the number of phylogenetic trees commonly inferred by two (binary) phylogenetic networks is also #P-complete.

Many commonly-used auction mechanisms are “maximal-in-range”. We show that any maximalin-range mechanism for n bidders and m items cannot both approximate the social welfare with a ratio better than min(n,m) for any constant η < 1/2 and run in polynomial time, unless NP ⊆ P/poly. This significantly improves upon a previous bound on the achievable social welfare of polynomial time maximal-in-ran...

We show that a problem of deciding whether a formula for a multivariate polynomial of n variables over a finite field of characteristic 2 has degree n when reduced modulo a certain Boolean ideal belongs to ⊕ P. When the formula is allowed to have succinct representations as sums of monomials, the problem becomes ⊕ P-complete. key words: degree of multivariate polynomials, finite field of charac...

Consider a directed network in which each arc can fail with some specified probability. An entity arrives on this network at a designated origin node and traverses the network in a random-walk fashion until it either terminates at a destination node, or until an arc fails while being traversed. We study the problem of assessing the probability that the random walk reaches the destination node, ...

We solve the problem of computing the residual reliability (the RES problem) for all classes of P-threshold graphs for which efficient structural characterizations based on decomposition to indecomposable components have been established. In particular, we give a constructive proof of existence of linear algorithms for computing residual reliability of pseudodomishold, domishold, matrogenic and...

We consider the computational complexity of the market equilibrium problem by exploring the structural properties of the Leontief exchange economy. We prove that, for economies guaranteed to have a market equilibrium, finding one with maximum social welfare or maximum individual welfare is NP-hard. In addition, we prove that counting the number of equilibrium prices is #P-hard.

We consider the complexity of combining bodies of evidence according to the rules of the Dempster{Shafer theory of evidence. We prove that, given as input a set of tables representing basic probability assignments m1; : : : ; mn over a frame of discernment , and a set A , the problem of computing the combined basic probability value (m1: : :mn)(A) is #P-complete. As a corollary, we obtain that ...

Given a function f over n binary variables, and an ordering of the n variables, Added abstract we consider the Expected Decision Depth problem. Namely, what is the expected number of bits that need to be observed until the value of the function is determined, when bits of the input are observed according to the given order. Our main finding is that this problem is #P-hard. In particular, hardne...